direct product, non-abelian, soluble, monomial
Aliases: C2×C42⋊S3, C42⋊2D6, C23.10S4, (C2×C42)⋊S3, C42⋊C3⋊4C22, C22.1(C2×S4), (C2×C42⋊C3)⋊3C2, SmallGroup(192,944)
Series: Derived ►Chief ►Lower central ►Upper central
| C42⋊C3 — C2×C42⋊S3 |
Generators and relations for C2×C42⋊S3
G = < a,b,c,d,e | a2=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=c, dcd-1=b-1c-1, ece=b, ede=d-1 >
Subgroups: 408 in 84 conjugacy classes, 11 normal (9 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, D4, Q8, C23, C23, A4, D6, C42, C42, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, S4, C2×A4, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C42⋊C3, C2×S4, C2×C4≀C2, C42⋊S3, C2×C42⋊C3, C2×C42⋊S3
Quotients: C1, C2, C22, S3, D6, S4, C2×S4, C42⋊S3, C2×C42⋊S3
Character table of C2×C42⋊S3
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6 | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 3 | 3 | 12 | 12 | 32 | 3 | 3 | 3 | 3 | 6 | 6 | 12 | 12 | 32 | 12 | 12 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ7 | 3 | 3 | 3 | 3 | 1 | 1 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ8 | 3 | -3 | 3 | -3 | 1 | -1 | 0 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from C2×S4 |
| ρ9 | 3 | -3 | 3 | -3 | -1 | 1 | 0 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from C2×S4 |
| ρ10 | 3 | 3 | 3 | 3 | -1 | -1 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S4 |
| ρ11 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | -1+2i | -1+2i | -1-2i | -1-2i | 1 | 1 | 1 | 1 | 0 | -i | -i | i | i | complex lifted from C42⋊S3 |
| ρ12 | 3 | -3 | -1 | 1 | 1 | -1 | 0 | 1+2i | -1-2i | -1+2i | 1-2i | -1 | 1 | 1 | -1 | 0 | -i | i | i | -i | complex faithful |
| ρ13 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | -1-2i | -1-2i | -1+2i | -1+2i | 1 | 1 | 1 | 1 | 0 | i | i | -i | -i | complex lifted from C42⋊S3 |
| ρ14 | 3 | -3 | -1 | 1 | -1 | 1 | 0 | 1-2i | -1+2i | -1-2i | 1+2i | -1 | 1 | -1 | 1 | 0 | -i | i | i | -i | complex faithful |
| ρ15 | 3 | 3 | -1 | -1 | 1 | 1 | 0 | -1-2i | -1-2i | -1+2i | -1+2i | 1 | 1 | -1 | -1 | 0 | -i | -i | i | i | complex lifted from C42⋊S3 |
| ρ16 | 3 | 3 | -1 | -1 | 1 | 1 | 0 | -1+2i | -1+2i | -1-2i | -1-2i | 1 | 1 | -1 | -1 | 0 | i | i | -i | -i | complex lifted from C42⋊S3 |
| ρ17 | 3 | -3 | -1 | 1 | -1 | 1 | 0 | 1+2i | -1-2i | -1+2i | 1-2i | -1 | 1 | -1 | 1 | 0 | i | -i | -i | i | complex faithful |
| ρ18 | 3 | -3 | -1 | 1 | 1 | -1 | 0 | 1-2i | -1+2i | -1-2i | 1+2i | -1 | 1 | 1 | -1 | 0 | i | -i | -i | i | complex faithful |
| ρ19 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C42⋊S3 |
| ρ20 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 12 points - transitive group 12T95
Generators in S12
(1 2)(3 4)(5 7)(6 8)(9 11)(10 12)
(1 2)(3 4)(5 6 7 8)(9 10 11 12)
(1 4 2 3)(5 6 7 8)(9 11)(10 12)
(1 5 12)(2 7 10)(3 8 11)(4 6 9)
(1 12)(2 10)(3 11)(4 9)
G:=sub<Sym(12)| (1,2)(3,4)(5,7)(6,8)(9,11)(10,12), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,4,2,3)(5,6,7,8)(9,11)(10,12), (1,5,12)(2,7,10)(3,8,11)(4,6,9), (1,12)(2,10)(3,11)(4,9)>;
G:=Group( (1,2)(3,4)(5,7)(6,8)(9,11)(10,12), (1,2)(3,4)(5,6,7,8)(9,10,11,12), (1,4,2,3)(5,6,7,8)(9,11)(10,12), (1,5,12)(2,7,10)(3,8,11)(4,6,9), (1,12)(2,10)(3,11)(4,9) );
G=PermutationGroup([[(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)], [(1,2),(3,4),(5,6,7,8),(9,10,11,12)], [(1,4,2,3),(5,6,7,8),(9,11),(10,12)], [(1,5,12),(2,7,10),(3,8,11),(4,6,9)], [(1,12),(2,10),(3,11),(4,9)]])
G:=TransitiveGroup(12,95);
►On 12 points - transitive group 12T96
Generators in S12
(1 4)(2 3)(5 7)(6 8)(9 11)(10 12)
(5 6 7 8)(9 10 11 12)
(1 2 4 3)(9 12 11 10)
(1 9 8)(2 10 7)(3 12 5)(4 11 6)
(1 6)(2 7)(3 5)(4 8)(9 11)
G:=sub<Sym(12)| (1,4)(2,3)(5,7)(6,8)(9,11)(10,12), (5,6,7,8)(9,10,11,12), (1,2,4,3)(9,12,11,10), (1,9,8)(2,10,7)(3,12,5)(4,11,6), (1,6)(2,7)(3,5)(4,8)(9,11)>;
G:=Group( (1,4)(2,3)(5,7)(6,8)(9,11)(10,12), (5,6,7,8)(9,10,11,12), (1,2,4,3)(9,12,11,10), (1,9,8)(2,10,7)(3,12,5)(4,11,6), (1,6)(2,7)(3,5)(4,8)(9,11) );
G=PermutationGroup([[(1,4),(2,3),(5,7),(6,8),(9,11),(10,12)], [(5,6,7,8),(9,10,11,12)], [(1,2,4,3),(9,12,11,10)], [(1,9,8),(2,10,7),(3,12,5),(4,11,6)], [(1,6),(2,7),(3,5),(4,8),(9,11)]])
G:=TransitiveGroup(12,96);
►On 12 points - transitive group 12T97
Generators in S12
(1 4)(2 3)(5 7)(6 8)(9 11)(10 12)
(5 6 7 8)(9 10 11 12)
(1 2 4 3)(9 12 11 10)
(1 10 5)(2 11 8)(3 9 6)(4 12 7)
(1 8)(2 5)(3 7)(4 6)(9 12)(10 11)
G:=sub<Sym(12)| (1,4)(2,3)(5,7)(6,8)(9,11)(10,12), (5,6,7,8)(9,10,11,12), (1,2,4,3)(9,12,11,10), (1,10,5)(2,11,8)(3,9,6)(4,12,7), (1,8)(2,5)(3,7)(4,6)(9,12)(10,11)>;
G:=Group( (1,4)(2,3)(5,7)(6,8)(9,11)(10,12), (5,6,7,8)(9,10,11,12), (1,2,4,3)(9,12,11,10), (1,10,5)(2,11,8)(3,9,6)(4,12,7), (1,8)(2,5)(3,7)(4,6)(9,12)(10,11) );
G=PermutationGroup([[(1,4),(2,3),(5,7),(6,8),(9,11),(10,12)], [(5,6,7,8),(9,10,11,12)], [(1,2,4,3),(9,12,11,10)], [(1,10,5),(2,11,8),(3,9,6),(4,12,7)], [(1,8),(2,5),(3,7),(4,6),(9,12),(10,11)]])
G:=TransitiveGroup(12,97);
►On 24 points - transitive group 24T471
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4 2 3)(5 7 6 8)(9 10 11 12)(13 15)(14 16)(17 18 19 20)(21 23)(22 24)
(1 18 24)(2 20 22)(3 17 23)(4 19 21)(5 10 14)(6 12 16)(7 11 15)(8 9 13)
(1 13)(2 15)(3 16)(4 14)(5 21)(6 23)(7 22)(8 24)(9 18)(10 19)(11 20)(12 17)
G:=sub<Sym(24)| (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,7,6,8)(9,10,11,12)(13,15)(14,16)(17,18,19,20)(21,23)(22,24), (1,18,24)(2,20,22)(3,17,23)(4,19,21)(5,10,14)(6,12,16)(7,11,15)(8,9,13), (1,13)(2,15)(3,16)(4,14)(5,21)(6,23)(7,22)(8,24)(9,18)(10,19)(11,20)(12,17)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,7,6,8)(9,10,11,12)(13,15)(14,16)(17,18,19,20)(21,23)(22,24), (1,18,24)(2,20,22)(3,17,23)(4,19,21)(5,10,14)(6,12,16)(7,11,15)(8,9,13), (1,13)(2,15)(3,16)(4,14)(5,21)(6,23)(7,22)(8,24)(9,18)(10,19)(11,20)(12,17) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4,2,3),(5,7,6,8),(9,10,11,12),(13,15),(14,16),(17,18,19,20),(21,23),(22,24)], [(1,18,24),(2,20,22),(3,17,23),(4,19,21),(5,10,14),(6,12,16),(7,11,15),(8,9,13)], [(1,13),(2,15),(3,16),(4,14),(5,21),(6,23),(7,22),(8,24),(9,18),(10,19),(11,20),(12,17)]])
G:=TransitiveGroup(24,471);
►On 24 points - transitive group 24T472
Generators in S24
(1 4)(2 3)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 8 4 6)(2 7 3 5)(9 21 11 23)(10 22 12 24)(13 20)(14 17)(15 18)(16 19)
(1 9 20)(2 22 13)(3 24 15)(4 11 18)(5 21 17)(6 12 14)(7 23 19)(8 10 16)
(1 20)(2 13)(3 15)(4 18)(5 16)(6 19)(7 14)(8 17)(10 21)(12 23)
G:=sub<Sym(24)| (1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8,4,6)(2,7,3,5)(9,21,11,23)(10,22,12,24)(13,20)(14,17)(15,18)(16,19), (1,9,20)(2,22,13)(3,24,15)(4,11,18)(5,21,17)(6,12,14)(7,23,19)(8,10,16), (1,20)(2,13)(3,15)(4,18)(5,16)(6,19)(7,14)(8,17)(10,21)(12,23)>;
G:=Group( (1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8,4,6)(2,7,3,5)(9,21,11,23)(10,22,12,24)(13,20)(14,17)(15,18)(16,19), (1,9,20)(2,22,13)(3,24,15)(4,11,18)(5,21,17)(6,12,14)(7,23,19)(8,10,16), (1,20)(2,13)(3,15)(4,18)(5,16)(6,19)(7,14)(8,17)(10,21)(12,23) );
G=PermutationGroup([[(1,4),(2,3),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,8,4,6),(2,7,3,5),(9,21,11,23),(10,22,12,24),(13,20),(14,17),(15,18),(16,19)], [(1,9,20),(2,22,13),(3,24,15),(4,11,18),(5,21,17),(6,12,14),(7,23,19),(8,10,16)], [(1,20),(2,13),(3,15),(4,18),(5,16),(6,19),(7,14),(8,17),(10,21),(12,23)]])
G:=TransitiveGroup(24,472);
►On 24 points - transitive group 24T473
Generators in S24
(1 6)(2 5)(3 8)(4 7)(9 21)(10 22)(11 23)(12 24)(13 19)(14 20)(15 17)(16 18)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4 2 3)(5 8 6 7)(9 11)(10 12)(13 14 15 16)(17 18 19 20)(21 23)(22 24)
(1 17 21)(2 19 23)(3 20 24)(4 18 22)(5 13 11)(6 15 9)(7 16 10)(8 14 12)
(1 21)(2 23)(3 24)(4 22)(5 11)(6 9)(7 10)(8 12)
G:=sub<Sym(24)| (1,6)(2,5)(3,8)(4,7)(9,21)(10,22)(11,23)(12,24)(13,19)(14,20)(15,17)(16,18), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,8,6,7)(9,11)(10,12)(13,14,15,16)(17,18,19,20)(21,23)(22,24), (1,17,21)(2,19,23)(3,20,24)(4,18,22)(5,13,11)(6,15,9)(7,16,10)(8,14,12), (1,21)(2,23)(3,24)(4,22)(5,11)(6,9)(7,10)(8,12)>;
G:=Group( (1,6)(2,5)(3,8)(4,7)(9,21)(10,22)(11,23)(12,24)(13,19)(14,20)(15,17)(16,18), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,8,6,7)(9,11)(10,12)(13,14,15,16)(17,18,19,20)(21,23)(22,24), (1,17,21)(2,19,23)(3,20,24)(4,18,22)(5,13,11)(6,15,9)(7,16,10)(8,14,12), (1,21)(2,23)(3,24)(4,22)(5,11)(6,9)(7,10)(8,12) );
G=PermutationGroup([[(1,6),(2,5),(3,8),(4,7),(9,21),(10,22),(11,23),(12,24),(13,19),(14,20),(15,17),(16,18)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4,2,3),(5,8,6,7),(9,11),(10,12),(13,14,15,16),(17,18,19,20),(21,23),(22,24)], [(1,17,21),(2,19,23),(3,20,24),(4,18,22),(5,13,11),(6,15,9),(7,16,10),(8,14,12)], [(1,21),(2,23),(3,24),(4,22),(5,11),(6,9),(7,10),(8,12)]])
G:=TransitiveGroup(24,473);
►On 24 points - transitive group 24T474
Generators in S24
(1 8)(2 7)(3 5)(4 6)(9 24)(10 21)(11 22)(12 23)(13 20)(14 17)(15 18)(16 19)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 4 2 3)(5 8 6 7)(9 11)(10 12)(13 14 15 16)(17 18 19 20)(21 23)(22 24)
(1 20 21)(2 18 23)(3 19 24)(4 17 22)(5 16 9)(6 14 11)(7 15 12)(8 13 10)
(1 23)(2 21)(3 22)(4 24)(5 11)(6 9)(7 10)(8 12)(13 15)(14 16)(17 19)(18 20)
G:=sub<Sym(24)| (1,8)(2,7)(3,5)(4,6)(9,24)(10,21)(11,22)(12,23)(13,20)(14,17)(15,18)(16,19), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,8,6,7)(9,11)(10,12)(13,14,15,16)(17,18,19,20)(21,23)(22,24), (1,20,21)(2,18,23)(3,19,24)(4,17,22)(5,16,9)(6,14,11)(7,15,12)(8,13,10), (1,23)(2,21)(3,22)(4,24)(5,11)(6,9)(7,10)(8,12)(13,15)(14,16)(17,19)(18,20)>;
G:=Group( (1,8)(2,7)(3,5)(4,6)(9,24)(10,21)(11,22)(12,23)(13,20)(14,17)(15,18)(16,19), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,3)(5,8,6,7)(9,11)(10,12)(13,14,15,16)(17,18,19,20)(21,23)(22,24), (1,20,21)(2,18,23)(3,19,24)(4,17,22)(5,16,9)(6,14,11)(7,15,12)(8,13,10), (1,23)(2,21)(3,22)(4,24)(5,11)(6,9)(7,10)(8,12)(13,15)(14,16)(17,19)(18,20) );
G=PermutationGroup([[(1,8),(2,7),(3,5),(4,6),(9,24),(10,21),(11,22),(12,23),(13,20),(14,17),(15,18),(16,19)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4,2,3),(5,8,6,7),(9,11),(10,12),(13,14,15,16),(17,18,19,20),(21,23),(22,24)], [(1,20,21),(2,18,23),(3,19,24),(4,17,22),(5,16,9),(6,14,11),(7,15,12),(8,13,10)], [(1,23),(2,21),(3,22),(4,24),(5,11),(6,9),(7,10),(8,12),(13,15),(14,16),(17,19),(18,20)]])
G:=TransitiveGroup(24,474);
►On 24 points - transitive group 24T475
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 21)(10 22)(11 23)(12 24)(13 19)(14 20)(15 17)(16 18)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 8 5 3)(2 7 6 4)(9 21)(10 22)(11 23)(12 24)(13 18 15 20)(14 19 16 17)
(1 15 11)(2 17 23)(3 14 24)(4 20 12)(5 13 9)(6 19 21)(7 18 10)(8 16 22)
(1 21)(2 9)(3 24)(4 12)(5 23)(6 11)(7 10)(8 22)(13 17)(15 19)
G:=sub<Sym(24)| (1,2)(3,4)(5,6)(7,8)(9,21)(10,22)(11,23)(12,24)(13,19)(14,20)(15,17)(16,18), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8,5,3)(2,7,6,4)(9,21)(10,22)(11,23)(12,24)(13,18,15,20)(14,19,16,17), (1,15,11)(2,17,23)(3,14,24)(4,20,12)(5,13,9)(6,19,21)(7,18,10)(8,16,22), (1,21)(2,9)(3,24)(4,12)(5,23)(6,11)(7,10)(8,22)(13,17)(15,19)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,21)(10,22)(11,23)(12,24)(13,19)(14,20)(15,17)(16,18), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8,5,3)(2,7,6,4)(9,21)(10,22)(11,23)(12,24)(13,18,15,20)(14,19,16,17), (1,15,11)(2,17,23)(3,14,24)(4,20,12)(5,13,9)(6,19,21)(7,18,10)(8,16,22), (1,21)(2,9)(3,24)(4,12)(5,23)(6,11)(7,10)(8,22)(13,17)(15,19) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,21),(10,22),(11,23),(12,24),(13,19),(14,20),(15,17),(16,18)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,8,5,3),(2,7,6,4),(9,21),(10,22),(11,23),(12,24),(13,18,15,20),(14,19,16,17)], [(1,15,11),(2,17,23),(3,14,24),(4,20,12),(5,13,9),(6,19,21),(7,18,10),(8,16,22)], [(1,21),(2,9),(3,24),(4,12),(5,23),(6,11),(7,10),(8,22),(13,17),(15,19)]])
G:=TransitiveGroup(24,475);
►On 24 points - transitive group 24T476
Generators in S24
(1 4)(2 3)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 3 4 2)(5 6 7 8)(13 16 15 14)(21 24 23 22)
(1 23 9)(2 22 10)(3 24 12)(4 21 11)(5 16 20)(6 13 19)(7 14 18)(8 15 17)
(1 20)(2 19)(3 17)(4 18)(5 9)(6 10)(7 11)(8 12)(13 22)(14 21)(15 24)(16 23)
G:=sub<Sym(24)| (1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3,4,2)(5,6,7,8)(13,16,15,14)(21,24,23,22), (1,23,9)(2,22,10)(3,24,12)(4,21,11)(5,16,20)(6,13,19)(7,14,18)(8,15,17), (1,20)(2,19)(3,17)(4,18)(5,9)(6,10)(7,11)(8,12)(13,22)(14,21)(15,24)(16,23)>;
G:=Group( (1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3,4,2)(5,6,7,8)(13,16,15,14)(21,24,23,22), (1,23,9)(2,22,10)(3,24,12)(4,21,11)(5,16,20)(6,13,19)(7,14,18)(8,15,17), (1,20)(2,19)(3,17)(4,18)(5,9)(6,10)(7,11)(8,12)(13,22)(14,21)(15,24)(16,23) );
G=PermutationGroup([[(1,4),(2,3),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,3,4,2),(5,6,7,8),(13,16,15,14),(21,24,23,22)], [(1,23,9),(2,22,10),(3,24,12),(4,21,11),(5,16,20),(6,13,19),(7,14,18),(8,15,17)], [(1,20),(2,19),(3,17),(4,18),(5,9),(6,10),(7,11),(8,12),(13,22),(14,21),(15,24),(16,23)]])
G:=TransitiveGroup(24,476);
►On 24 points - transitive group 24T477
Generators in S24
(1 3)(2 4)(5 7)(6 8)(9 13)(10 14)(11 15)(12 16)(17 22)(18 23)(19 24)(20 21)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 6 4 7)(2 5 3 8)(9 14 11 16)(10 15 12 13)(17 24)(18 21)(19 22)(20 23)
(1 12 22)(2 14 19)(3 16 17)(4 10 24)(5 15 21)(6 9 18)(7 11 20)(8 13 23)
(1 22)(2 19)(3 17)(4 24)(5 20)(6 23)(7 21)(8 18)(9 13)(11 15)
G:=sub<Sym(24)| (1,3)(2,4)(5,7)(6,8)(9,13)(10,14)(11,15)(12,16)(17,22)(18,23)(19,24)(20,21), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6,4,7)(2,5,3,8)(9,14,11,16)(10,15,12,13)(17,24)(18,21)(19,22)(20,23), (1,12,22)(2,14,19)(3,16,17)(4,10,24)(5,15,21)(6,9,18)(7,11,20)(8,13,23), (1,22)(2,19)(3,17)(4,24)(5,20)(6,23)(7,21)(8,18)(9,13)(11,15)>;
G:=Group( (1,3)(2,4)(5,7)(6,8)(9,13)(10,14)(11,15)(12,16)(17,22)(18,23)(19,24)(20,21), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6,4,7)(2,5,3,8)(9,14,11,16)(10,15,12,13)(17,24)(18,21)(19,22)(20,23), (1,12,22)(2,14,19)(3,16,17)(4,10,24)(5,15,21)(6,9,18)(7,11,20)(8,13,23), (1,22)(2,19)(3,17)(4,24)(5,20)(6,23)(7,21)(8,18)(9,13)(11,15) );
G=PermutationGroup([[(1,3),(2,4),(5,7),(6,8),(9,13),(10,14),(11,15),(12,16),(17,22),(18,23),(19,24),(20,21)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,6,4,7),(2,5,3,8),(9,14,11,16),(10,15,12,13),(17,24),(18,21),(19,22),(20,23)], [(1,12,22),(2,14,19),(3,16,17),(4,10,24),(5,15,21),(6,9,18),(7,11,20),(8,13,23)], [(1,22),(2,19),(3,17),(4,24),(5,20),(6,23),(7,21),(8,18),(9,13),(11,15)]])
G:=TransitiveGroup(24,477);
►On 24 points - transitive group 24T478
Generators in S24
(1 3)(2 4)(5 8)(6 7)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 7 4 5)(2 8 3 6)(9 15 11 13)(10 16 12 14)(17 24)(18 21)(19 22)(20 23)
(1 16 18)(2 9 21)(3 11 23)(4 14 20)(5 15 22)(6 12 19)(7 13 24)(8 10 17)
(1 20)(2 23)(3 21)(4 18)(5 19)(6 22)(7 17)(8 24)(9 11)(10 13)(12 15)(14 16)
G:=sub<Sym(24)| (1,3)(2,4)(5,8)(6,7)(9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,7,4,5)(2,8,3,6)(9,15,11,13)(10,16,12,14)(17,24)(18,21)(19,22)(20,23), (1,16,18)(2,9,21)(3,11,23)(4,14,20)(5,15,22)(6,12,19)(7,13,24)(8,10,17), (1,20)(2,23)(3,21)(4,18)(5,19)(6,22)(7,17)(8,24)(9,11)(10,13)(12,15)(14,16)>;
G:=Group( (1,3)(2,4)(5,8)(6,7)(9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,7,4,5)(2,8,3,6)(9,15,11,13)(10,16,12,14)(17,24)(18,21)(19,22)(20,23), (1,16,18)(2,9,21)(3,11,23)(4,14,20)(5,15,22)(6,12,19)(7,13,24)(8,10,17), (1,20)(2,23)(3,21)(4,18)(5,19)(6,22)(7,17)(8,24)(9,11)(10,13)(12,15)(14,16) );
G=PermutationGroup([[(1,3),(2,4),(5,8),(6,7),(9,14),(10,15),(11,16),(12,13),(17,22),(18,23),(19,24),(20,21)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,7,4,5),(2,8,3,6),(9,15,11,13),(10,16,12,14),(17,24),(18,21),(19,22),(20,23)], [(1,16,18),(2,9,21),(3,11,23),(4,14,20),(5,15,22),(6,12,19),(7,13,24),(8,10,17)], [(1,20),(2,23),(3,21),(4,18),(5,19),(6,22),(7,17),(8,24),(9,11),(10,13),(12,15),(14,16)]])
G:=TransitiveGroup(24,478);
►On 24 points - transitive group 24T479
Generators in S24
(1 6)(2 7)(3 8)(4 5)(9 24)(10 21)(11 22)(12 23)(13 17)(14 18)(15 19)(16 20)
(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 2 4 3)(5 8 6 7)(13 16 15 14)(17 20 19 18)
(1 20 22)(2 17 21)(3 19 23)(4 18 24)(5 14 9)(6 16 11)(7 13 10)(8 15 12)
(1 11)(2 12)(3 10)(4 9)(5 24)(6 22)(7 23)(8 21)(13 19)(14 18)(15 17)(16 20)
G:=sub<Sym(24)| (1,6)(2,7)(3,8)(4,5)(9,24)(10,21)(11,22)(12,23)(13,17)(14,18)(15,19)(16,20), (9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,2,4,3)(5,8,6,7)(13,16,15,14)(17,20,19,18), (1,20,22)(2,17,21)(3,19,23)(4,18,24)(5,14,9)(6,16,11)(7,13,10)(8,15,12), (1,11)(2,12)(3,10)(4,9)(5,24)(6,22)(7,23)(8,21)(13,19)(14,18)(15,17)(16,20)>;
G:=Group( (1,6)(2,7)(3,8)(4,5)(9,24)(10,21)(11,22)(12,23)(13,17)(14,18)(15,19)(16,20), (9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,2,4,3)(5,8,6,7)(13,16,15,14)(17,20,19,18), (1,20,22)(2,17,21)(3,19,23)(4,18,24)(5,14,9)(6,16,11)(7,13,10)(8,15,12), (1,11)(2,12)(3,10)(4,9)(5,24)(6,22)(7,23)(8,21)(13,19)(14,18)(15,17)(16,20) );
G=PermutationGroup([[(1,6),(2,7),(3,8),(4,5),(9,24),(10,21),(11,22),(12,23),(13,17),(14,18),(15,19),(16,20)], [(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,2,4,3),(5,8,6,7),(13,16,15,14),(17,20,19,18)], [(1,20,22),(2,17,21),(3,19,23),(4,18,24),(5,14,9),(6,16,11),(7,13,10),(8,15,12)], [(1,11),(2,12),(3,10),(4,9),(5,24),(6,22),(7,23),(8,21),(13,19),(14,18),(15,17),(16,20)]])
G:=TransitiveGroup(24,479);
►On 24 points - transitive group 24T480
Generators in S24
(1 8)(2 6)(3 5)(4 7)(9 21)(10 22)(11 23)(12 24)(13 20)(14 17)(15 18)(16 19)
(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 2 4 3)(5 8 6 7)(13 16 15 14)(17 20 19 18)
(1 20 24)(2 17 23)(3 19 21)(4 18 22)(5 16 9)(6 14 11)(7 15 10)(8 13 12)
(1 21)(2 22)(3 24)(4 23)(5 12)(6 10)(7 11)(8 9)(13 16)(14 15)(17 18)(19 20)
G:=sub<Sym(24)| (1,8)(2,6)(3,5)(4,7)(9,21)(10,22)(11,23)(12,24)(13,20)(14,17)(15,18)(16,19), (9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,2,4,3)(5,8,6,7)(13,16,15,14)(17,20,19,18), (1,20,24)(2,17,23)(3,19,21)(4,18,22)(5,16,9)(6,14,11)(7,15,10)(8,13,12), (1,21)(2,22)(3,24)(4,23)(5,12)(6,10)(7,11)(8,9)(13,16)(14,15)(17,18)(19,20)>;
G:=Group( (1,8)(2,6)(3,5)(4,7)(9,21)(10,22)(11,23)(12,24)(13,20)(14,17)(15,18)(16,19), (9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,2,4,3)(5,8,6,7)(13,16,15,14)(17,20,19,18), (1,20,24)(2,17,23)(3,19,21)(4,18,22)(5,16,9)(6,14,11)(7,15,10)(8,13,12), (1,21)(2,22)(3,24)(4,23)(5,12)(6,10)(7,11)(8,9)(13,16)(14,15)(17,18)(19,20) );
G=PermutationGroup([[(1,8),(2,6),(3,5),(4,7),(9,21),(10,22),(11,23),(12,24),(13,20),(14,17),(15,18),(16,19)], [(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,2,4,3),(5,8,6,7),(13,16,15,14),(17,20,19,18)], [(1,20,24),(2,17,23),(3,19,21),(4,18,22),(5,16,9),(6,14,11),(7,15,10),(8,13,12)], [(1,21),(2,22),(3,24),(4,23),(5,12),(6,10),(7,11),(8,9),(13,16),(14,15),(17,18),(19,20)]])
G:=TransitiveGroup(24,480);
Polynomial with Galois group C2×C42⋊S3 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 12T95 | x12-20x10+107x8-196x6+112x4-22x2+1 | 224·312·134·174·414 |
| 12T96 | x12-3x4-4 | -238·316 |
| 12T97 | x12-32x10+391x8-2248x6+5951x4-5622x2+81 | 224·38·58·536·15594 |
Matrix representation of C2×C42⋊S3 ►in GL3(𝔽5) generated by
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 3 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 3 |
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 4 |
| 0 | 4 | 0 |
| 0 | 0 | 1 |
| 4 | 0 | 0 |
| 4 | 0 | 0 |
| 0 | 0 | 4 |
| 0 | 4 | 0 |
G:=sub<GL(3,GF(5))| [4,0,0,0,4,0,0,0,4],[3,0,0,0,4,0,0,0,3],[3,0,0,0,3,0,0,0,4],[0,0,4,4,0,0,0,1,0],[4,0,0,0,0,4,0,4,0] >;
C2×C42⋊S3 in GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes S_3
% in TeX
G:=Group("C2xC4^2:S3"); // GroupNames label
G:=SmallGroup(192,944);
// by ID
G=gap.SmallGroup(192,944);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,2,-2,2,170,675,185,360,424,1173,102,6053,1027,1784]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e=c,d*c*d^-1=b^-1*c^-1,e*c*e=b,e*d*e=d^-1>;
// generators/relations
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