metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C12.1C8, C24.4C4, C8.22D6, C3⋊2M5(2), C8.2Dic3, C24.26C22, C4.(C3⋊C8), C3⋊C16⋊5C2, (C2×C8).7S3, C6.9(C2×C8), (C2×C6).3C8, C22.(C3⋊C8), (C2×C12).8C4, (C2×C24).13C2, C12.39(C2×C4), (C2×C4).5Dic3, C4.11(C2×Dic3), C2.4(C2×C3⋊C8), SmallGroup(96,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.C8
G = < a,b | a24=1, b4=a18, bab-1=a5 >
Smallest permutation representation of C12.C8
►On 48 points
(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)
(1 41 10 38 19 35 4 32 13 29 22 26 7 47 16 44)(2 46 11 43 20 40 5 37 14 34 23 31 8 28 17 25)(3 27 12 48 21 45 6 42 15 39 24 36 9 33 18 30)
G:=sub<Sym(48)| (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), (1,41,10,38,19,35,4,32,13,29,22,26,7,47,16,44)(2,46,11,43,20,40,5,37,14,34,23,31,8,28,17,25)(3,27,12,48,21,45,6,42,15,39,24,36,9,33,18,30)>;
G:=Group( (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), (1,41,10,38,19,35,4,32,13,29,22,26,7,47,16,44)(2,46,11,43,20,40,5,37,14,34,23,31,8,28,17,25)(3,27,12,48,21,45,6,42,15,39,24,36,9,33,18,30) );
G=PermutationGroup([[(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)], [(1,41,10,38,19,35,4,32,13,29,22,26,7,47,16,44),(2,46,11,43,20,40,5,37,14,34,23,31,8,28,17,25),(3,27,12,48,21,45,6,42,15,39,24,36,9,33,18,30)]])
C12.C8 is a maximal subgroup of
C24.1C8 C12.15C42 C8.Dic6 D24⋊8C4 C24.6Q8 D24.C4 C24.8D4 D12.C8 C24.97D4 C48⋊C4 C8.25D12 C24.D4 C24.99D4 D8.Dic3 Q16.Dic3 D8⋊2Dic3 D12.4C8 S3×M5(2) C24.78C23 D8.D6 C24.27C23 Q16⋊D6 D8.9D6 C36.C8 C24.61D6 C24.94D6 C40.51D6 C60.7C8 C120.C4 C60.C8
C12.C8 is a maximal quotient of
C24.C8 C12⋊C16 C24.98D4 C36.C8 C24.61D6 C24.94D6 C40.51D6 C60.7C8 C120.C4 C60.C8
36 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 12D | 16A | ··· | 16H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | Dic3 | D6 | Dic3 | C3⋊C8 | C3⋊C8 | M5(2) | C12.C8 |
| kernel | C12.C8 | C3⋊C16 | C2×C24 | C24 | C2×C12 | C12 | C2×C6 | C2×C8 | C8 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 |
Matrix representation of C12.C8 ►in GL2(𝔽97) generated by
| 73 | 0 |
| 0 | 9 |
| 0 | 1 |
| 33 | 0 |
G:=sub<GL(2,GF(97))| [73,0,0,9],[0,33,1,0] >;
C12.C8 in GAP, Magma, Sage, TeX
C_{12}.C_8 % in TeX
G:=Group("C12.C8"); // GroupNames label
G:=SmallGroup(96,19);
// by ID
G=gap.SmallGroup(96,19);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,217,50,69,2309]);
// Polycyclic
G:=Group<a,b|a^24=1,b^4=a^18,b*a*b^-1=a^5>;
// generators/relations
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