Aliases: C62.4C8, C32⋊6M5(2), (C3×C12).6C8, (C6×C12).3C4, C4.(C32⋊2C8), C32⋊2C16⋊4C2, C32⋊4C8.10C4, C22.(C32⋊2C8), C32⋊4C8.35C22, (C3×C6).23(C2×C8), C4.20(C2×C32⋊C4), (C3×C12).17(C2×C4), (C2×C4).5(C32⋊C4), C2.3(C2×C32⋊2C8), (C2×C32⋊4C8).20C2, SmallGroup(288,421)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C62.4C8
G = < a,b,c | a6=b6=1, c8=b3, ab=ba, cac-1=a-1b, cbc-1=a4b >
Smallest permutation representation of C62.4C8
►On 48 points
Generators in S48
(1 27 47)(2 40 28 10 48 20)(3 33 29)(4 22 34 12 30 42)(5 31 35)(6 44 32 14 36 24)(7 37 17)(8 26 38 16 18 46)(9 19 39)(11 41 21)(13 23 43)(15 45 25)
(1 9)(2 20 48 10 28 40)(3 11)(4 42 30 12 34 22)(5 13)(6 24 36 14 32 44)(7 15)(8 46 18 16 38 26)(17 25)(19 27)(21 29)(23 31)(33 41)(35 43)(37 45)(39 47)
(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)| (1,27,47)(2,40,28,10,48,20)(3,33,29)(4,22,34,12,30,42)(5,31,35)(6,44,32,14,36,24)(7,37,17)(8,26,38,16,18,46)(9,19,39)(11,41,21)(13,23,43)(15,45,25), (1,9)(2,20,48,10,28,40)(3,11)(4,42,30,12,34,22)(5,13)(6,24,36,14,32,44)(7,15)(8,46,18,16,38,26)(17,25)(19,27)(21,29)(23,31)(33,41)(35,43)(37,45)(39,47), (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( (1,27,47)(2,40,28,10,48,20)(3,33,29)(4,22,34,12,30,42)(5,31,35)(6,44,32,14,36,24)(7,37,17)(8,26,38,16,18,46)(9,19,39)(11,41,21)(13,23,43)(15,45,25), (1,9)(2,20,48,10,28,40)(3,11)(4,42,30,12,34,22)(5,13)(6,24,36,14,32,44)(7,15)(8,46,18,16,38,26)(17,25)(19,27)(21,29)(23,31)(33,41)(35,43)(37,45)(39,47), (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([[(1,27,47),(2,40,28,10,48,20),(3,33,29),(4,22,34,12,30,42),(5,31,35),(6,44,32,14,36,24),(7,37,17),(8,26,38,16,18,46),(9,19,39),(11,41,21),(13,23,43),(15,45,25)], [(1,9),(2,20,48,10,28,40),(3,11),(4,42,30,12,34,22),(5,13),(6,24,36,14,32,44),(7,15),(8,46,18,16,38,26),(17,25),(19,27),(21,29),(23,31),(33,41),(35,43),(37,45),(39,47)], [(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 | 3A | 3B | 4A | 4B | 4C | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 8E | 8F | 12A | ··· | 12H | 16A | ··· | 16H |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 16 | ··· | 16 |
| size | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | ··· | 4 | 9 | 9 | 9 | 9 | 18 | 18 | 4 | ··· | 4 | 18 | ··· | 18 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | M5(2) | C32⋊C4 | C32⋊2C8 | C2×C32⋊C4 | C32⋊2C8 | C62.4C8 |
| kernel | C62.4C8 | C32⋊2C16 | C2×C32⋊4C8 | C32⋊4C8 | C6×C12 | C3×C12 | C62 | C32 | C2×C4 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C62.4C8 ►in GL4(𝔽5) generated by
| 1 | 0 | 0 | 3 |
| 0 | 2 | 4 | 0 |
| 0 | 3 | 4 | 0 |
| 4 | 0 | 0 | 3 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 1 | 0 |
| 0 | 2 | 2 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(5))| [1,0,0,4,0,2,3,0,0,4,4,0,3,0,0,3],[4,0,0,0,0,4,2,0,0,1,2,0,0,0,0,4],[0,1,0,0,0,0,0,1,4,0,0,0,0,0,2,0] >;
C62.4C8 in GAP, Magma, Sage, TeX
C_6^2._4C_8
% in TeX
G:=Group("C6^2.4C8"); // GroupNames label
G:=SmallGroup(288,421);
// by ID
G=gap.SmallGroup(288,421);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,58,80,9413,691,12550,2372]);
// Polycyclic
G:=Group<a,b,c|a^6=b^6=1,c^8=b^3,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a^4*b>;
// generators/relations
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