direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C6×C3⋊C8, C6⋊C24, C12.3C12, C12.66D6, C62.4C4, C12.11Dic3, (C3×C6)⋊3C8, C3⋊2(C2×C24), C32⋊8(C2×C8), C4.14(S3×C6), (C2×C12).7C6, (C3×C12).8C4, (C2×C6).5C12, C6.5(C2×C12), C12.15(C2×C6), (C6×C12).10C2, (C2×C12).21S3, C4.3(C3×Dic3), C2.1(C6×Dic3), (C2×C6).9Dic3, C6.17(C2×Dic3), (C3×C12).44C22, C22.2(C3×Dic3), (C2×C4).5(C3×S3), (C3×C6).27(C2×C4), SmallGroup(144,74)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C6×C3⋊C8 |
Generators and relations for C6×C3⋊C8
G = < a,b,c | a6=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C6×C3⋊C8
►On 48 points
(1 33 13 30 47 18)(2 34 14 31 48 19)(3 35 15 32 41 20)(4 36 16 25 42 21)(5 37 9 26 43 22)(6 38 10 27 44 23)(7 39 11 28 45 24)(8 40 12 29 46 17)
(1 13 47)(2 48 14)(3 15 41)(4 42 16)(5 9 43)(6 44 10)(7 11 45)(8 46 12)(17 29 40)(18 33 30)(19 31 34)(20 35 32)(21 25 36)(22 37 26)(23 27 38)(24 39 28)
(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)| (1,33,13,30,47,18)(2,34,14,31,48,19)(3,35,15,32,41,20)(4,36,16,25,42,21)(5,37,9,26,43,22)(6,38,10,27,44,23)(7,39,11,28,45,24)(8,40,12,29,46,17), (1,13,47)(2,48,14)(3,15,41)(4,42,16)(5,9,43)(6,44,10)(7,11,45)(8,46,12)(17,29,40)(18,33,30)(19,31,34)(20,35,32)(21,25,36)(22,37,26)(23,27,38)(24,39,28), (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( (1,33,13,30,47,18)(2,34,14,31,48,19)(3,35,15,32,41,20)(4,36,16,25,42,21)(5,37,9,26,43,22)(6,38,10,27,44,23)(7,39,11,28,45,24)(8,40,12,29,46,17), (1,13,47)(2,48,14)(3,15,41)(4,42,16)(5,9,43)(6,44,10)(7,11,45)(8,46,12)(17,29,40)(18,33,30)(19,31,34)(20,35,32)(21,25,36)(22,37,26)(23,27,38)(24,39,28), (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([[(1,33,13,30,47,18),(2,34,14,31,48,19),(3,35,15,32,41,20),(4,36,16,25,42,21),(5,37,9,26,43,22),(6,38,10,27,44,23),(7,39,11,28,45,24),(8,40,12,29,46,17)], [(1,13,47),(2,48,14),(3,15,41),(4,42,16),(5,9,43),(6,44,10),(7,11,45),(8,46,12),(17,29,40),(18,33,30),(19,31,34),(20,35,32),(21,25,36),(22,37,26),(23,27,38),(24,39,28)], [(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)]])
C6×C3⋊C8 is a maximal subgroup of
C6.(S3×C8) C3⋊C8⋊Dic3 C2.Dic32 C12.77D12 C12.78D12 C6.16D24 C6.17D24 C6.Dic12 C12.73D12 C12.81D12 C12.15Dic6 C12.Dic6 C6.18D24 C12.82D12 Dic3×C24 D12.2Dic3 C3⋊C8.22D6 D12.27D6 S3×C2×C24
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6O | 8A | ··· | 8H | 12A | ··· | 12H | 12I | ··· | 12T | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 | S3 | Dic3 | D6 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | S3×C6 | C3×Dic3 | C3×C3⋊C8 |
| kernel | C6×C3⋊C8 | C3×C3⋊C8 | C6×C12 | C2×C3⋊C8 | C3×C12 | C62 | C3⋊C8 | C2×C12 | C3×C6 | C12 | C2×C6 | C6 | C2×C12 | C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of C6×C3⋊C8 ►in GL4(𝔽73) generated by
| 65 | 0 | 0 | 0 |
| 0 | 65 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 7 |
| 0 | 0 | 0 | 64 |
| 63 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 63 | 4 |
| 0 | 0 | 66 | 10 |
G:=sub<GL(4,GF(73))| [65,0,0,0,0,65,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,8,0,0,0,7,64],[63,0,0,0,0,72,0,0,0,0,63,66,0,0,4,10] >;
C6×C3⋊C8 in GAP, Magma, Sage, TeX
C_6\times C_3\rtimes C_8
% in TeX
G:=Group("C6xC3:C8"); // GroupNames label
G:=SmallGroup(144,74);
// by ID
G=gap.SmallGroup(144,74);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,72,69,3461]);
// Polycyclic
G:=Group<a,b,c|a^6=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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