direct product, metabelian, soluble, monomial, A-group
Aliases: C2×C42⋊C3, C42⋊3C6, C23.4A4, (C2×C42)⋊C3, C22.1(C2×A4), SmallGroup(96,68)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C2×C42⋊C3 |
Generators and relations for C2×C42⋊C3
G = < a,b,c,d | a2=b4=c4=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >
Character table of C2×C42⋊C3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | |
| size | 1 | 1 | 3 | 3 | 16 | 16 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 16 | 16 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ζ6 | ζ65 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ζ65 | ζ6 | linear of order 6 |
| ρ7 | 3 | -3 | -3 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ8 | 3 | 3 | 3 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
| ρ9 | 3 | 3 | -1 | -1 | 0 | 0 | -1-2i | 1 | -1+2i | 1 | -1-2i | 1 | 1 | -1+2i | 0 | 0 | complex lifted from C42⋊C3 |
| ρ10 | 3 | 3 | -1 | -1 | 0 | 0 | 1 | -1-2i | 1 | -1+2i | 1 | -1+2i | -1-2i | 1 | 0 | 0 | complex lifted from C42⋊C3 |
| ρ11 | 3 | 3 | -1 | -1 | 0 | 0 | 1 | -1+2i | 1 | -1-2i | 1 | -1-2i | -1+2i | 1 | 0 | 0 | complex lifted from C42⋊C3 |
| ρ12 | 3 | -3 | 1 | -1 | 0 | 0 | -1+2i | 1 | -1-2i | 1 | 1-2i | -1 | -1 | 1+2i | 0 | 0 | complex faithful |
| ρ13 | 3 | -3 | 1 | -1 | 0 | 0 | 1 | -1+2i | 1 | -1-2i | -1 | 1+2i | 1-2i | -1 | 0 | 0 | complex faithful |
| ρ14 | 3 | -3 | 1 | -1 | 0 | 0 | -1-2i | 1 | -1+2i | 1 | 1+2i | -1 | -1 | 1-2i | 0 | 0 | complex faithful |
| ρ15 | 3 | -3 | 1 | -1 | 0 | 0 | 1 | -1-2i | 1 | -1+2i | -1 | 1-2i | 1+2i | -1 | 0 | 0 | complex faithful |
| ρ16 | 3 | 3 | -1 | -1 | 0 | 0 | -1+2i | 1 | -1-2i | 1 | -1+2i | 1 | 1 | -1-2i | 0 | 0 | complex lifted from C42⋊C3 |
►On 12 points - transitive group 12T55
Generators in S12
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)
(5 6 7 8)(9 10 11 12)
(1 2 3 4)(5 7)(6 8)(9 10 11 12)
(1 7 12)(2 6 9)(3 5 10)(4 8 11)
G:=sub<Sym(12)| (1,3)(2,4)(5,7)(6,8)(9,11)(10,12), (5,6,7,8)(9,10,11,12), (1,2,3,4)(5,7)(6,8)(9,10,11,12), (1,7,12)(2,6,9)(3,5,10)(4,8,11)>;
G:=Group( (1,3)(2,4)(5,7)(6,8)(9,11)(10,12), (5,6,7,8)(9,10,11,12), (1,2,3,4)(5,7)(6,8)(9,10,11,12), (1,7,12)(2,6,9)(3,5,10)(4,8,11) );
G=PermutationGroup([[(1,3),(2,4),(5,7),(6,8),(9,11),(10,12)], [(5,6,7,8),(9,10,11,12)], [(1,2,3,4),(5,7),(6,8),(9,10,11,12)], [(1,7,12),(2,6,9),(3,5,10),(4,8,11)]])
G:=TransitiveGroup(12,55);
►On 24 points - transitive group 24T173
Generators in S24
(1 7)(2 8)(3 6)(4 5)(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 6 7 3)(2 5 8 4)(9 14 11 16)(10 15 12 13)(17 21)(18 22)(19 23)(20 24)
(1 21 10)(2 19 14)(3 20 9)(4 22 13)(5 24 15)(6 18 11)(7 23 12)(8 17 16)
G:=sub<Sym(24)| (1,7)(2,8)(3,6)(4,5)(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), (1,6,7,3)(2,5,8,4)(9,14,11,16)(10,15,12,13)(17,21)(18,22)(19,23)(20,24), (1,21,10)(2,19,14)(3,20,9)(4,22,13)(5,24,15)(6,18,11)(7,23,12)(8,17,16)>;
G:=Group( (1,7)(2,8)(3,6)(4,5)(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), (1,6,7,3)(2,5,8,4)(9,14,11,16)(10,15,12,13)(17,21)(18,22)(19,23)(20,24), (1,21,10)(2,19,14)(3,20,9)(4,22,13)(5,24,15)(6,18,11)(7,23,12)(8,17,16) );
G=PermutationGroup([[(1,7),(2,8),(3,6),(4,5),(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)], [(1,6,7,3),(2,5,8,4),(9,14,11,16),(10,15,12,13),(17,21),(18,22),(19,23),(20,24)], [(1,21,10),(2,19,14),(3,20,9),(4,22,13),(5,24,15),(6,18,11),(7,23,12),(8,17,16)]])
G:=TransitiveGroup(24,173);
►On 24 points - transitive group 24T174
Generators in S24
(1 5)(2 6)(3 7)(4 8)(9 14)(10 15)(11 16)(12 13)(17 24)(18 21)(19 22)(20 23)
(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 12 11 10)(13 16 15 14)
(1 19 15)(2 17 13)(3 18 14)(4 20 16)(5 22 10)(6 24 12)(7 21 9)(8 23 11)
G:=sub<Sym(24)| (1,5)(2,6)(3,7)(4,8)(9,14)(10,15)(11,16)(12,13)(17,24)(18,21)(19,22)(20,23), (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)(9,12,11,10)(13,16,15,14), (1,19,15)(2,17,13)(3,18,14)(4,20,16)(5,22,10)(6,24,12)(7,21,9)(8,23,11)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,14)(10,15)(11,16)(12,13)(17,24)(18,21)(19,22)(20,23), (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)(9,12,11,10)(13,16,15,14), (1,19,15)(2,17,13)(3,18,14)(4,20,16)(5,22,10)(6,24,12)(7,21,9)(8,23,11) );
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,14),(10,15),(11,16),(12,13),(17,24),(18,21),(19,22),(20,23)], [(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),(9,12,11,10),(13,16,15,14)], [(1,19,15),(2,17,13),(3,18,14),(4,20,16),(5,22,10),(6,24,12),(7,21,9),(8,23,11)]])
G:=TransitiveGroup(24,174);
►On 24 points - transitive group 24T175
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 24)(10 21)(11 22)(12 23)(13 17)(14 18)(15 19)(16 20)
(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 3 6)(2 7 4 5)(9 21 11 23)(10 22 12 24)(13 19)(14 20)(15 17)(16 18)
(1 17 11)(2 13 22)(3 19 9)(4 15 24)(5 18 21)(6 14 10)(7 20 23)(8 16 12)
G:=sub<Sym(24)| (1,2)(3,4)(5,6)(7,8)(9,24)(10,21)(11,22)(12,23)(13,17)(14,18)(15,19)(16,20), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8,3,6)(2,7,4,5)(9,21,11,23)(10,22,12,24)(13,19)(14,20)(15,17)(16,18), (1,17,11)(2,13,22)(3,19,9)(4,15,24)(5,18,21)(6,14,10)(7,20,23)(8,16,12)>;
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,24)(10,21)(11,22)(12,23)(13,17)(14,18)(15,19)(16,20), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8,3,6)(2,7,4,5)(9,21,11,23)(10,22,12,24)(13,19)(14,20)(15,17)(16,18), (1,17,11)(2,13,22)(3,19,9)(4,15,24)(5,18,21)(6,14,10)(7,20,23)(8,16,12) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,24),(10,21),(11,22),(12,23),(13,17),(14,18),(15,19),(16,20)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,8,3,6),(2,7,4,5),(9,21,11,23),(10,22,12,24),(13,19),(14,20),(15,17),(16,18)], [(1,17,11),(2,13,22),(3,19,9),(4,15,24),(5,18,21),(6,14,10),(7,20,23),(8,16,12)]])
G:=TransitiveGroup(24,175);
C2×C42⋊C3 is a maximal subgroup of
C23.9S4 C42⋊C12 C42⋊2C12
C2×C42⋊C3 is a maximal quotient of C42⋊4C4⋊C3
| action | f(x) | Disc(f) |
|---|---|---|
| 12T55 | x12-38x10+538x8-3458x6+9659x4-8788x2+169 | 224·312·712·1314·2234 |
Matrix representation of C2×C42⋊C3 ►in GL3(𝔽5) generated by
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 2 | 1 | 3 |
| 3 | 0 | 1 |
| 0 | 0 | 3 |
| 0 | 4 | 0 |
| 4 | 1 | 4 |
| 1 | 2 | 0 |
| 1 | 4 | 1 |
| 0 | 3 | 4 |
| 0 | 3 | 1 |
G:=sub<GL(3,GF(5))| [4,0,0,0,4,0,0,0,4],[2,3,0,1,0,0,3,1,3],[0,4,1,4,1,2,0,4,0],[1,0,0,4,3,3,1,4,1] >;
C2×C42⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes C_3
% in TeX
G:=Group("C2xC4^2:C3"); // GroupNames label
G:=SmallGroup(96,68);
// by ID
G=gap.SmallGroup(96,68);
# by ID
G:=PCGroup([6,-2,-3,-2,2,-2,2,116,230,801,69,730,1307]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,d*c*d^-1=b^-1*c^2>;
// generators/relations
Export
Subgroup lattice of C2×C42⋊C3 in TeX
Character table of C2×C42⋊C3 in TeX