direct product, metabelian, soluble, monomial, A-group
Aliases: C4×C42⋊C3, C4≀C3, C43⋊C3, C42⋊6C12, C23.8(C2×A4), (C22×C4).7A4, (C2×C42).3C6, C22.1(C4×A4), C2.1(C2×C42⋊C3), (C2×C42⋊C3).4C2, SmallGroup(192,188)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C4×C42⋊C3 |
Generators and relations for C4×C42⋊C3
G = < a,b,c,d | a4=b4=c4=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >
Subgroups: 192 in 58 conjugacy classes, 12 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C2×C4, C23, C12, A4, C42, C42, C22×C4, C22×C4, C2×A4, C2×C42, C2×C42, C42⋊C3, C4×A4, C43, C2×C42⋊C3, C4×C42⋊C3
Quotients: C1, C2, C3, C4, C6, C12, A4, C2×A4, C42⋊C3, C4×A4, C2×C42⋊C3, C4×C42⋊C3
►On 12 points - transitive group 12T94
Generators in S12
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(1 4 3 2)(5 8 7 6)(9 11)(10 12)
(1 2 3 4)(9 12 11 10)
(1 11 7)(2 12 8)(3 9 5)(4 10 6)
G:=sub<Sym(12)| (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4,3,2)(5,8,7,6)(9,11)(10,12), (1,2,3,4)(9,12,11,10), (1,11,7)(2,12,8)(3,9,5)(4,10,6)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4,3,2)(5,8,7,6)(9,11)(10,12), (1,2,3,4)(9,12,11,10), (1,11,7)(2,12,8)(3,9,5)(4,10,6) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(1,4,3,2),(5,8,7,6),(9,11),(10,12)], [(1,2,3,4),(9,12,11,10)], [(1,11,7),(2,12,8),(3,9,5),(4,10,6)]])
G:=TransitiveGroup(12,94);
►On 24 points - transitive group 24T469
Generators in S24
(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 9)(2 10)(3 11)(4 12)(5 6 7 8)(13 17 15 19)(14 18 16 20)(21 22 23 24)
(1 12 3 10)(2 9 4 11)(5 24)(6 21)(7 22)(8 23)(13 16 15 14)(17 20 19 18)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 19 11)(6 20 12)(7 17 9)(8 18 10)
G:=sub<Sym(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), (1,9)(2,10)(3,11)(4,12)(5,6,7,8)(13,17,15,19)(14,18,16,20)(21,22,23,24), (1,12,3,10)(2,9,4,11)(5,24)(6,21)(7,22)(8,23)(13,16,15,14)(17,20,19,18), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,19,11)(6,20,12)(7,17,9)(8,18,10)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,9)(2,10)(3,11)(4,12)(5,6,7,8)(13,17,15,19)(14,18,16,20)(21,22,23,24), (1,12,3,10)(2,9,4,11)(5,24)(6,21)(7,22)(8,23)(13,16,15,14)(17,20,19,18), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,19,11)(6,20,12)(7,17,9)(8,18,10) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,9),(2,10),(3,11),(4,12),(5,6,7,8),(13,17,15,19),(14,18,16,20),(21,22,23,24)], [(1,12,3,10),(2,9,4,11),(5,24),(6,21),(7,22),(8,23),(13,16,15,14),(17,20,19,18)], [(1,24,16),(2,21,13),(3,22,14),(4,23,15),(5,19,11),(6,20,12),(7,17,9),(8,18,10)]])
G:=TransitiveGroup(24,469);
►On 24 points - transitive group 24T470
Generators in S24
(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 11 3 9)(2 12 4 10)(5 24 7 22)(6 21 8 23)
(1 11 3 9)(2 12 4 10)(5 7)(6 8)(13 17 15 19)(14 18 16 20)(21 23)(22 24)
(1 19 6)(2 20 7)(3 17 8)(4 18 5)(9 15 21)(10 16 22)(11 13 23)(12 14 24)
G:=sub<Sym(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), (1,11,3,9)(2,12,4,10)(5,24,7,22)(6,21,8,23), (1,11,3,9)(2,12,4,10)(5,7)(6,8)(13,17,15,19)(14,18,16,20)(21,23)(22,24), (1,19,6)(2,20,7)(3,17,8)(4,18,5)(9,15,21)(10,16,22)(11,13,23)(12,14,24)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,11,3,9)(2,12,4,10)(5,24,7,22)(6,21,8,23), (1,11,3,9)(2,12,4,10)(5,7)(6,8)(13,17,15,19)(14,18,16,20)(21,23)(22,24), (1,19,6)(2,20,7)(3,17,8)(4,18,5)(9,15,21)(10,16,22)(11,13,23)(12,14,24) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,11,3,9),(2,12,4,10),(5,24,7,22),(6,21,8,23)], [(1,11,3,9),(2,12,4,10),(5,7),(6,8),(13,17,15,19),(14,18,16,20),(21,23),(22,24)], [(1,19,6),(2,20,7),(3,17,8),(4,18,5),(9,15,21),(10,16,22),(11,13,23),(12,14,24)]])
G:=TransitiveGroup(24,470);
Polynomial with Galois group C4×C42⋊C3 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 12T94 | x12-18x10+119x8-383x6+644x4-543x2+181 | 212·78·1813 |
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | ··· | 4T | 6A | 6B | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 3 | 3 | 16 | 16 | 1 | 1 | 3 | ··· | 3 | 16 | 16 | 16 | 16 | 16 | 16 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | A4 | C2×A4 | C42⋊C3 | C4×A4 | C2×C42⋊C3 | C4×C42⋊C3 |
| kernel | C4×C42⋊C3 | C2×C42⋊C3 | C43 | C42⋊C3 | C2×C42 | C42 | C22×C4 | C23 | C4 | C22 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 4 | 2 | 4 | 8 |
Matrix representation of C4×C42⋊C3 ►in GL3(𝔽5) generated by
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 4 | 0 | 1 |
| 0 | 0 | 3 |
| 1 | 2 | 2 |
| 4 | 3 | 3 |
| 2 | 4 | 1 |
| 3 | 4 | 2 |
| 1 | 1 | 1 |
| 0 | 4 | 2 |
| 0 | 2 | 0 |
G:=sub<GL(3,GF(5))| [3,0,0,0,3,0,0,0,3],[4,0,1,0,0,2,1,3,2],[4,2,3,3,4,4,3,1,2],[1,0,0,1,4,2,1,2,0] >;
C4×C42⋊C3 in GAP, Magma, Sage, TeX
C_4\times C_4^2\rtimes C_3
% in TeX
G:=Group("C4xC4^2:C3"); // GroupNames label
G:=SmallGroup(192,188);
// by ID
G=gap.SmallGroup(192,188);
# by ID
G:=PCGroup([7,-2,-3,-2,-2,2,-2,2,42,346,360,2321,102,2028,3541]);
// Polycyclic
G:=Group<a,b,c,d|a^4=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