direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×C42⋊C4, C42⋊4C12, (C4×C12)⋊5C4, (C6×D4)⋊4C4, (C2×D4)⋊2C12, C23⋊C4⋊2C6, C4⋊1D4.2C6, C23.3(C3×D4), (C22×C6).3D4, C6.34(C23⋊C4), (C6×D4).176C22, (C3×C23⋊C4)⋊8C2, (C2×C4).1(C2×C12), (C2×D4).3(C2×C6), C2.8(C3×C23⋊C4), (C2×C12).12(C2×C4), (C3×C4⋊1D4).9C2, (C2×C6).75(C22⋊C4), C22.12(C3×C22⋊C4), SmallGroup(192,159)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42⋊C4
G = < a,b,c,d | a3=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b2c >
Subgroups: 242 in 86 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, C23, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×D4, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C23⋊C4, C4⋊1D4, C4×C12, C3×C22⋊C4, C6×D4, C6×D4, C42⋊C4, C3×C23⋊C4, C3×C4⋊1D4, C3×C42⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, C2×C12, C3×D4, C23⋊C4, C3×C22⋊C4, C42⋊C4, C3×C23⋊C4, C3×C42⋊C4
►On 24 points - transitive group 24T353
(1 5 11)(2 6 9)(3 7 10)(4 8 12)(13 17 23)(14 18 24)(15 19 21)(16 20 22)
(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 2 3 4)(5 6 7 8)(9 10 12 11)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 16 2 13)(3 14 4 15)(5 20 6 17)(7 18 8 19)(9 23 11 22)(10 24 12 21)
G:=sub<Sym(24)| (1,5,11)(2,6,9)(3,7,10)(4,8,12)(13,17,23)(14,18,24)(15,19,21)(16,20,22), (13,14,15,16)(17,18,19,20)(21,22,23,24), (1,2,3,4)(5,6,7,8)(9,10,12,11)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,16,2,13)(3,14,4,15)(5,20,6,17)(7,18,8,19)(9,23,11,22)(10,24,12,21)>;
G:=Group( (1,5,11)(2,6,9)(3,7,10)(4,8,12)(13,17,23)(14,18,24)(15,19,21)(16,20,22), (13,14,15,16)(17,18,19,20)(21,22,23,24), (1,2,3,4)(5,6,7,8)(9,10,12,11)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,16,2,13)(3,14,4,15)(5,20,6,17)(7,18,8,19)(9,23,11,22)(10,24,12,21) );
G=PermutationGroup([[(1,5,11),(2,6,9),(3,7,10),(4,8,12),(13,17,23),(14,18,24),(15,19,21),(16,20,22)], [(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,2,3,4),(5,6,7,8),(9,10,12,11),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,16,2,13),(3,14,4,15),(5,20,6,17),(7,18,8,19),(9,23,11,22),(10,24,12,21)]])
G:=TransitiveGroup(24,353);
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 12A | ··· | 12F | 12G | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | D4 | C3×D4 | C23⋊C4 | C42⋊C4 | C3×C23⋊C4 | C3×C42⋊C4 |
| kernel | C3×C42⋊C4 | C3×C23⋊C4 | C3×C4⋊1D4 | C42⋊C4 | C4×C12 | C6×D4 | C23⋊C4 | C4⋊1D4 | C42 | C2×D4 | C22×C6 | C23 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 2 | 4 | 1 | 2 | 2 | 4 |
Matrix representation of C3×C42⋊C4 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 3 | 0 | 1 |
| 1 | 0 | 1 | 5 |
| 4 | 4 | 4 | 6 |
| 0 | 0 | 0 | 6 |
| 4 | 5 | 4 | 6 |
| 6 | 0 | 6 | 0 |
| 3 | 3 | 3 | 1 |
| 3 | 4 | 2 | 0 |
| 1 | 6 | 1 | 4 |
| 6 | 6 | 6 | 1 |
| 2 | 5 | 6 | 3 |
| 3 | 3 | 2 | 1 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,1,4,0,3,0,4,0,0,1,4,0,1,5,6,6],[4,6,3,3,5,0,3,4,4,6,3,2,6,0,1,0],[1,6,2,3,6,6,5,3,1,6,6,2,4,1,3,1] >;
C3×C42⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes C_4
% in TeX
G:=Group("C3xC4^2:C4"); // GroupNames label
G:=SmallGroup(192,159);
// by ID
G=gap.SmallGroup(192,159);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,1522,248,2951,375,6053]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^2*c>;
// generators/relations