Aliases: (C6×C12).7C4, C62.6(C2×C4), C32⋊3(C22⋊C8), C3⋊Dic3.51D4, (C3×C6).2M4(2), C2.1(C62⋊C4), C2.3(C32⋊M4(2)), (C2×C3⋊S3)⋊5C8, (C3×C6).10(C2×C8), (C2×C32⋊2C8)⋊2C2, (C22×C3⋊S3).8C4, (C2×C4).3(C32⋊C4), C2.4(C3⋊S3⋊3C8), C22.11(C2×C32⋊C4), (C3×C6).13(C22⋊C4), (C2×C3⋊Dic3).110C22, (C2×C4×C3⋊S3).13C2, SmallGroup(288,426)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C2×C32⋊2C8 — C62.6(C2×C4) |
Generators and relations for C62.6(C2×C4)
G = < a,b,c,d | a6=b6=1, c2=d4=b3, ab=ba, ac=ca, dad-1=a-1b4, bc=cb, dbd-1=a4b, dcd-1=a3c >
Subgroups: 536 in 104 conjugacy classes, 22 normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C2×C8, C22×C4, C3⋊S3, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22⋊C8, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, C32⋊2C8, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C2×C32⋊2C8, C2×C4×C3⋊S3, C62.6(C2×C4)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), C22⋊C8, C32⋊C4, C2×C32⋊C4, C3⋊S3⋊3C8, C32⋊M4(2), C62⋊C4, C62.6(C2×C4)
►On 48 points
(1 15 43 23 29 39)(2 24)(3 33 31 17 45 9)(4 18)(5 11 47 19 25 35)(6 20)(7 37 27 21 41 13)(8 22)(10 32)(12 26)(14 28)(16 30)(34 46)(36 48)(38 42)(40 44)
(1 25 43 5 29 47)(2 48 30 6 44 26)(3 41 31 7 45 27)(4 28 46 8 32 42)(9 21 33 13 17 37)(10 38 18 14 34 22)(11 39 19 15 35 23)(12 24 36 16 20 40)
(1 7 5 3)(2 22 6 18)(4 24 8 20)(9 39 13 35)(10 44 14 48)(11 33 15 37)(12 46 16 42)(17 23 21 19)(25 45 29 41)(26 34 30 38)(27 47 31 43)(28 36 32 40)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)
G:=sub<Sym(48)| (1,15,43,23,29,39)(2,24)(3,33,31,17,45,9)(4,18)(5,11,47,19,25,35)(6,20)(7,37,27,21,41,13)(8,22)(10,32)(12,26)(14,28)(16,30)(34,46)(36,48)(38,42)(40,44), (1,25,43,5,29,47)(2,48,30,6,44,26)(3,41,31,7,45,27)(4,28,46,8,32,42)(9,21,33,13,17,37)(10,38,18,14,34,22)(11,39,19,15,35,23)(12,24,36,16,20,40), (1,7,5,3)(2,22,6,18)(4,24,8,20)(9,39,13,35)(10,44,14,48)(11,33,15,37)(12,46,16,42)(17,23,21,19)(25,45,29,41)(26,34,30,38)(27,47,31,43)(28,36,32,40), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)>;
G:=Group( (1,15,43,23,29,39)(2,24)(3,33,31,17,45,9)(4,18)(5,11,47,19,25,35)(6,20)(7,37,27,21,41,13)(8,22)(10,32)(12,26)(14,28)(16,30)(34,46)(36,48)(38,42)(40,44), (1,25,43,5,29,47)(2,48,30,6,44,26)(3,41,31,7,45,27)(4,28,46,8,32,42)(9,21,33,13,17,37)(10,38,18,14,34,22)(11,39,19,15,35,23)(12,24,36,16,20,40), (1,7,5,3)(2,22,6,18)(4,24,8,20)(9,39,13,35)(10,44,14,48)(11,33,15,37)(12,46,16,42)(17,23,21,19)(25,45,29,41)(26,34,30,38)(27,47,31,43)(28,36,32,40), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48) );
G=PermutationGroup([[(1,15,43,23,29,39),(2,24),(3,33,31,17,45,9),(4,18),(5,11,47,19,25,35),(6,20),(7,37,27,21,41,13),(8,22),(10,32),(12,26),(14,28),(16,30),(34,46),(36,48),(38,42),(40,44)], [(1,25,43,5,29,47),(2,48,30,6,44,26),(3,41,31,7,45,27),(4,28,46,8,32,42),(9,21,33,13,17,37),(10,38,18,14,34,22),(11,39,19,15,35,23),(12,24,36,16,20,40)], [(1,7,5,3),(2,22,6,18),(4,24,8,20),(9,39,13,35),(10,44,14,48),(11,33,15,37),(12,46,16,42),(17,23,21,19),(25,45,29,41),(26,34,30,38),(27,47,31,43),(28,36,32,40)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 8A | ··· | 8H | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 4 | 4 | 2 | 2 | 9 | 9 | 9 | 9 | 4 | ··· | 4 | 18 | ··· | 18 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | M4(2) | C32⋊C4 | C2×C32⋊C4 | C3⋊S3⋊3C8 | C32⋊M4(2) | C62⋊C4 |
| kernel | C62.6(C2×C4) | C2×C32⋊2C8 | C2×C4×C3⋊S3 | C6×C12 | C22×C3⋊S3 | C2×C3⋊S3 | C3⋊Dic3 | C3×C6 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C62.6(C2×C4) ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 7 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 63 | 48 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[46,7,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[63,0,0,0,0,0,48,10,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0] >;
C62.6(C2×C4) in GAP, Magma, Sage, TeX
C_6^2._6(C_2\times C_4)
% in TeX
G:=Group("C6^2.6(C2xC4)"); // GroupNames label
G:=SmallGroup(288,426);
// by ID
G=gap.SmallGroup(288,426);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,120,100,9413,691,12550,2372]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=1,c^2=d^4=b^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^4,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=a^3*c>;
// generators/relations