metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.97D4, C24.10Q8, C8.9Dic6, M5(2).2S3, C3⋊C8.1C8, C6.9(C4⋊C8), C4.13(S3×C8), C12.5(C2×C8), C3⋊2(C8.C8), (C2×C8).266D6, C12.41(C4⋊C4), C8.50(C3⋊D4), (C4×Dic3).4C4, (C2×C6).3M4(2), C12.C8.7C2, C2.5(Dic3⋊C8), (C8×Dic3).15C2, (C3×M5(2)).4C2, C4.28(Dic3⋊C4), (C2×C24).261C22, C22.5(C8⋊S3), (C2×C3⋊C8).6C4, (C2×C4).135(C4×S3), (C2×C12).50(C2×C4), SmallGroup(192,70)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.97D4
G = < a,b,c | a24=1, b4=a12, c2=a9, bab-1=cac-1=a17, cbc-1=a12b3 >
Smallest permutation representation of C24.97D4
►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 4 7 10 13 16 19 22)(2 21 8 3 14 9 20 15)(5 24 11 6 17 12 23 18)(25 46 43 40 37 34 31 28)(26 39 44 33 38 27 32 45)(29 42 47 36 41 30 35 48)
(1 46 10 31 19 40 4 25 13 34 22 43 7 28 16 37)(2 39 11 48 20 33 5 42 14 27 23 36 8 45 17 30)(3 32 12 41 21 26 6 35 15 44 24 29 9 38 18 47)
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), (1,4,7,10,13,16,19,22)(2,21,8,3,14,9,20,15)(5,24,11,6,17,12,23,18)(25,46,43,40,37,34,31,28)(26,39,44,33,38,27,32,45)(29,42,47,36,41,30,35,48), (1,46,10,31,19,40,4,25,13,34,22,43,7,28,16,37)(2,39,11,48,20,33,5,42,14,27,23,36,8,45,17,30)(3,32,12,41,21,26,6,35,15,44,24,29,9,38,18,47)>;
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), (1,4,7,10,13,16,19,22)(2,21,8,3,14,9,20,15)(5,24,11,6,17,12,23,18)(25,46,43,40,37,34,31,28)(26,39,44,33,38,27,32,45)(29,42,47,36,41,30,35,48), (1,46,10,31,19,40,4,25,13,34,22,43,7,28,16,37)(2,39,11,48,20,33,5,42,14,27,23,36,8,45,17,30)(3,32,12,41,21,26,6,35,15,44,24,29,9,38,18,47) );
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)], [(1,4,7,10,13,16,19,22),(2,21,8,3,14,9,20,15),(5,24,11,6,17,12,23,18),(25,46,43,40,37,34,31,28),(26,39,44,33,38,27,32,45),(29,42,47,36,41,30,35,48)], [(1,46,10,31,19,40,4,25,13,34,22,43,7,28,16,37),(2,39,11,48,20,33,5,42,14,27,23,36,8,45,17,30),(3,32,12,41,21,26,6,35,15,44,24,29,9,38,18,47)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 12A | 12B | 12C | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 24A | 24B | 24C | 24D | 24E | 24F | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 6 | 6 | 6 | 6 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | Q8 | D6 | M4(2) | Dic6 | C3⋊D4 | C4×S3 | S3×C8 | C8⋊S3 | C8.C8 | C24.97D4 |
| kernel | C24.97D4 | C12.C8 | C8×Dic3 | C3×M5(2) | C2×C3⋊C8 | C4×Dic3 | C3⋊C8 | M5(2) | C24 | C24 | C2×C8 | C2×C6 | C8 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 4 |
Matrix representation of C24.97D4 ►in GL4(𝔽97) generated by
| 0 | 96 | 0 | 0 |
| 1 | 96 | 0 | 0 |
| 0 | 0 | 33 | 0 |
| 0 | 0 | 0 | 33 |
| 0 | 22 | 0 | 0 |
| 22 | 0 | 0 | 0 |
| 0 | 0 | 47 | 0 |
| 0 | 0 | 40 | 64 |
| 39 | 29 | 0 | 0 |
| 68 | 58 | 0 | 0 |
| 0 | 0 | 64 | 66 |
| 0 | 0 | 81 | 33 |
G:=sub<GL(4,GF(97))| [0,1,0,0,96,96,0,0,0,0,33,0,0,0,0,33],[0,22,0,0,22,0,0,0,0,0,47,40,0,0,0,64],[39,68,0,0,29,58,0,0,0,0,64,81,0,0,66,33] >;
C24.97D4 in GAP, Magma, Sage, TeX
C_{24}._{97}D_4 % in TeX
G:=Group("C24.97D4"); // GroupNames label
G:=SmallGroup(192,70);
// by ID
G=gap.SmallGroup(192,70);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,141,36,100,570,136,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=1,b^4=a^12,c^2=a^9,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^12*b^3>;
// generators/relations
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