Aliases: (C6×C12)⋊5C4, C62.18(C2×C4), C62⋊C4.2C2, C32⋊1(C42⋊C2), (C4×C3⋊S3)⋊10C4, (C4×C32⋊C4)⋊8C2, C4⋊(C32⋊C4)⋊6C2, (C2×C4)⋊4(C32⋊C4), C4.14(C2×C32⋊C4), (C3×C12).21(C2×C4), (C2×C3⋊Dic3)⋊14C4, C22.7(C2×C32⋊C4), C3⋊S3.10(C4○D4), C2.7(C22×C32⋊C4), (C2×C3⋊S3).36C23, C3⋊Dic3.53(C2×C4), (C3×C6).29(C22×C4), (C4×C3⋊S3).100C22, (C2×C32⋊C4).23C22, (C22×C3⋊S3).97C22, (C2×C4×C3⋊S3).30C2, (C2×C3⋊S3).49(C2×C4), SmallGroup(288,934)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C6×C12)⋊5C4
G = < a,b,c | a6=b12=c4=1, ab=ba, cac-1=a-1b2, cbc-1=a2b >
Subgroups: 656 in 130 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C42⋊C2, C3⋊Dic3, C3×C12, C32⋊C4, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C2×C32⋊C4, C22×C3⋊S3, C4×C32⋊C4, C4⋊(C32⋊C4), C62⋊C4, C2×C4×C3⋊S3, (C6×C12)⋊5C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C42⋊C2, C32⋊C4, C2×C32⋊C4, C22×C32⋊C4, (C6×C12)⋊5C4
►On 24 points - transitive group 24T621
(13 15 17 19 21 23)(14 16 18 20 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 19)(2 20 6 24)(3 21 11 17)(4 22)(5 23 9 15)(7 13)(8 14 12 18)(10 16)
G:=sub<Sym(24)| (13,15,17,19,21,23)(14,16,18,20,22,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,19)(2,20,6,24)(3,21,11,17)(4,22)(5,23,9,15)(7,13)(8,14,12,18)(10,16)>;
G:=Group( (13,15,17,19,21,23)(14,16,18,20,22,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,19)(2,20,6,24)(3,21,11,17)(4,22)(5,23,9,15)(7,13)(8,14,12,18)(10,16) );
G=PermutationGroup([[(13,15,17,19,21,23),(14,16,18,20,22,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,19),(2,20,6,24),(3,21,11,17),(4,22),(5,23,9,15),(7,13),(8,14,12,18),(10,16)]])
G:=TransitiveGroup(24,621);
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4N | 6A | ··· | 6F | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 9 | 9 | 18 | 4 | 4 | 1 | 1 | 2 | 9 | 9 | 18 | ··· | 18 | 4 | ··· | 4 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4○D4 | C32⋊C4 | C2×C32⋊C4 | C2×C32⋊C4 | (C6×C12)⋊5C4 |
| kernel | (C6×C12)⋊5C4 | C4×C32⋊C4 | C4⋊(C32⋊C4) | C62⋊C4 | C2×C4×C3⋊S3 | C4×C3⋊S3 | C2×C3⋊Dic3 | C6×C12 | C3⋊S3 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 4 | 2 | 4 | 2 | 8 |
Matrix representation of (C6×C12)⋊5C4 ►in GL4(𝔽5) generated by
| 0 | 1 | 1 | 2 |
| 1 | 3 | 2 | 2 |
| 0 | 2 | 0 | 4 |
| 0 | 2 | 4 | 2 |
| 3 | 3 | 3 | 3 |
| 4 | 1 | 2 | 0 |
| 2 | 2 | 3 | 0 |
| 1 | 3 | 3 | 0 |
| 0 | 1 | 1 | 2 |
| 0 | 2 | 4 | 3 |
| 3 | 1 | 1 | 2 |
| 1 | 2 | 4 | 2 |
G:=sub<GL(4,GF(5))| [0,1,0,0,1,3,2,2,1,2,0,4,2,2,4,2],[3,4,2,1,3,1,2,3,3,2,3,3,3,0,0,0],[0,0,3,1,1,2,1,2,1,4,1,4,2,3,2,2] >;
(C6×C12)⋊5C4 in GAP, Magma, Sage, TeX
(C_6\times C_{12})\rtimes_5C_4 % in TeX
G:=Group("(C6xC12):5C4"); // GroupNames label
G:=SmallGroup(288,934);
// by ID
G=gap.SmallGroup(288,934);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,120,422,9413,362,12550,1203]);
// Polycyclic
G:=Group<a,b,c|a^6=b^12=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b^2,c*b*c^-1=a^2*b>;
// generators/relations