metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C24.1C8, C24.72D4, C8.29D12, C24.18Q8, C8.14Dic6, C42.9Dic3, C8.1(C3⋊C8), C6.4(C4⋊C8), (C4×C8).11S3, C3⋊1(C8.C8), (C4×C12).14C4, (C2×C24).18C4, (C4×C24).13C2, C12.37(C2×C8), (C2×C8).321D6, C12.39(C4⋊C4), (C2×C8).9Dic3, C2.5(C12⋊C8), C12.C8.5C2, C4.19(C4⋊Dic3), (C2×C6).19M4(2), (C2×C24).416C22, C22.2(C4.Dic3), C4.8(C2×C3⋊C8), (C2×C12).301(C2×C4), (C2×C4).69(C2×Dic3), SmallGroup(192,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.1C8
G = < a,b | a24=1, b8=a12, bab-1=a-1 >
Smallest permutation representation of C24.1C8
►On 48 points
Generators in S48
(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 33 22 36 19 39 16 42 13 45 10 48 7 27 4 30)(2 32 23 35 20 38 17 41 14 44 11 47 8 26 5 29)(3 31 24 34 21 37 18 40 15 43 12 46 9 25 6 28)
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), (1,33,22,36,19,39,16,42,13,45,10,48,7,27,4,30)(2,32,23,35,20,38,17,41,14,44,11,47,8,26,5,29)(3,31,24,34,21,37,18,40,15,43,12,46,9,25,6,28)>;
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), (1,33,22,36,19,39,16,42,13,45,10,48,7,27,4,30)(2,32,23,35,20,38,17,41,14,44,11,47,8,26,5,29)(3,31,24,34,21,37,18,40,15,43,12,46,9,25,6,28) );
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)], [(1,33,22,36,19,39,16,42,13,45,10,48,7,27,4,30),(2,32,23,35,20,38,17,41,14,44,11,47,8,26,5,29),(3,31,24,34,21,37,18,40,15,43,12,46,9,25,6,28)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | ··· | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 12A | ··· | 12L | 16A | ··· | 16H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | - | + | - | + | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | Q8 | Dic3 | Dic3 | D6 | M4(2) | C3⋊C8 | Dic6 | D12 | C4.Dic3 | C8.C8 | C24.1C8 |
| kernel | C24.1C8 | C12.C8 | C4×C24 | C4×C12 | C2×C24 | C24 | C4×C8 | C24 | C24 | C42 | C2×C8 | C2×C8 | C2×C6 | C8 | C8 | C8 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 8 | 16 |
Matrix representation of C24.1C8 ►in GL2(𝔽97) generated by
| 4 | 0 |
| 0 | 73 |
| 0 | 1 |
| 47 | 0 |
G:=sub<GL(2,GF(97))| [4,0,0,73],[0,47,1,0] >;
C24.1C8 in GAP, Magma, Sage, TeX
C_{24}._1C_8 % in TeX
G:=Group("C24.1C8"); // GroupNames label
G:=SmallGroup(192,22);
// by ID
G=gap.SmallGroup(192,22);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,64,100,1123,136,102,6278]);
// Polycyclic
G:=Group<a,b|a^24=1,b^8=a^12,b*a*b^-1=a^-1>;
// generators/relations
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