metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.7Q8, C12.55D8, C8.7Dic6, C3⋊C16.1C4, C8.26(C4×S3), C12.4(C4⋊C4), (C2×C6).2Q16, C24.11(C2×C4), (C2×C8).248D6, (C2×C12).93D4, C6.4(C2.D8), C3⋊1(C8.4Q8), C4.28(D4⋊S3), C8.C4.1S3, C24.C4.3C2, C4.4(Dic3⋊C4), (C2×C24).36C22, C2.5(C6.Q16), C22.1(C3⋊Q16), (C2×C3⋊C16).1C2, (C3×C8.C4).1C2, (C2×C4).117(C3⋊D4), SmallGroup(192,52)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.7Q8
G = < a,b,c | a24=1, b4=a12, c2=a21b2, bab-1=a19, cac-1=a17, cbc-1=a21b3 >
Smallest permutation representation of C24.7Q8
►On 96 points
Generators in S96
(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 31 7 25 13 43 19 37)(2 26 8 44 14 38 20 32)(3 45 9 39 15 33 21 27)(4 40 10 34 16 28 22 46)(5 35 11 29 17 47 23 41)(6 30 12 48 18 42 24 36)(49 90 55 84 61 78 67 96)(50 85 56 79 62 73 68 91)(51 80 57 74 63 92 69 86)(52 75 58 93 64 87 70 81)(53 94 59 88 65 82 71 76)(54 89 60 83 66 77 72 95)
(1 68 4 71 7 50 10 53 13 56 16 59 19 62 22 65)(2 61 5 64 8 67 11 70 14 49 17 52 20 55 23 58)(3 54 6 57 9 60 12 63 15 66 18 69 21 72 24 51)(25 76 40 91 31 82 46 73 37 88 28 79 43 94 34 85)(26 93 41 84 32 75 47 90 38 81 29 96 44 87 35 78)(27 86 42 77 33 92 48 83 39 74 30 89 45 80 36 95)
G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,31,7,25,13,43,19,37)(2,26,8,44,14,38,20,32)(3,45,9,39,15,33,21,27)(4,40,10,34,16,28,22,46)(5,35,11,29,17,47,23,41)(6,30,12,48,18,42,24,36)(49,90,55,84,61,78,67,96)(50,85,56,79,62,73,68,91)(51,80,57,74,63,92,69,86)(52,75,58,93,64,87,70,81)(53,94,59,88,65,82,71,76)(54,89,60,83,66,77,72,95), (1,68,4,71,7,50,10,53,13,56,16,59,19,62,22,65)(2,61,5,64,8,67,11,70,14,49,17,52,20,55,23,58)(3,54,6,57,9,60,12,63,15,66,18,69,21,72,24,51)(25,76,40,91,31,82,46,73,37,88,28,79,43,94,34,85)(26,93,41,84,32,75,47,90,38,81,29,96,44,87,35,78)(27,86,42,77,33,92,48,83,39,74,30,89,45,80,36,95)>;
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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,31,7,25,13,43,19,37)(2,26,8,44,14,38,20,32)(3,45,9,39,15,33,21,27)(4,40,10,34,16,28,22,46)(5,35,11,29,17,47,23,41)(6,30,12,48,18,42,24,36)(49,90,55,84,61,78,67,96)(50,85,56,79,62,73,68,91)(51,80,57,74,63,92,69,86)(52,75,58,93,64,87,70,81)(53,94,59,88,65,82,71,76)(54,89,60,83,66,77,72,95), (1,68,4,71,7,50,10,53,13,56,16,59,19,62,22,65)(2,61,5,64,8,67,11,70,14,49,17,52,20,55,23,58)(3,54,6,57,9,60,12,63,15,66,18,69,21,72,24,51)(25,76,40,91,31,82,46,73,37,88,28,79,43,94,34,85)(26,93,41,84,32,75,47,90,38,81,29,96,44,87,35,78)(27,86,42,77,33,92,48,83,39,74,30,89,45,80,36,95) );
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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,31,7,25,13,43,19,37),(2,26,8,44,14,38,20,32),(3,45,9,39,15,33,21,27),(4,40,10,34,16,28,22,46),(5,35,11,29,17,47,23,41),(6,30,12,48,18,42,24,36),(49,90,55,84,61,78,67,96),(50,85,56,79,62,73,68,91),(51,80,57,74,63,92,69,86),(52,75,58,93,64,87,70,81),(53,94,59,88,65,82,71,76),(54,89,60,83,66,77,72,95)], [(1,68,4,71,7,50,10,53,13,56,16,59,19,62,22,65),(2,61,5,64,8,67,11,70,14,49,17,52,20,55,23,58),(3,54,6,57,9,60,12,63,15,66,18,69,21,72,24,51),(25,76,40,91,31,82,46,73,37,88,28,79,43,94,34,85),(26,93,41,84,32,75,47,90,38,81,29,96,44,87,35,78),(27,86,42,77,33,92,48,83,39,74,30,89,45,80,36,95)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 16A | ··· | 16H | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 24 | 24 | 2 | 2 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | - | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | Q8 | D4 | D6 | D8 | Q16 | Dic6 | C4×S3 | C3⋊D4 | C8.4Q8 | D4⋊S3 | C3⋊Q16 | C24.7Q8 |
| kernel | C24.7Q8 | C2×C3⋊C16 | C24.C4 | C3×C8.C4 | C3⋊C16 | C8.C4 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C8 | C2×C4 | C3 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 4 |
Matrix representation of C24.7Q8 ►in GL4(𝔽97) generated by
| 1 | 1 | 0 | 0 |
| 96 | 0 | 0 | 0 |
| 0 | 0 | 33 | 0 |
| 0 | 0 | 8 | 47 |
| 56 | 15 | 0 | 0 |
| 82 | 41 | 0 | 0 |
| 0 | 0 | 71 | 3 |
| 0 | 0 | 26 | 26 |
| 18 | 94 | 0 | 0 |
| 76 | 79 | 0 | 0 |
| 0 | 0 | 85 | 0 |
| 0 | 0 | 53 | 8 |
G:=sub<GL(4,GF(97))| [1,96,0,0,1,0,0,0,0,0,33,8,0,0,0,47],[56,82,0,0,15,41,0,0,0,0,71,26,0,0,3,26],[18,76,0,0,94,79,0,0,0,0,85,53,0,0,0,8] >;
C24.7Q8 in GAP, Magma, Sage, TeX
C_{24}._7Q_8 % in TeX
G:=Group("C24.7Q8"); // GroupNames label
G:=SmallGroup(192,52);
// by ID
G=gap.SmallGroup(192,52);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,141,36,184,346,192,851,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=1,b^4=a^12,c^2=a^21*b^2,b*a*b^-1=a^19,c*a*c^-1=a^17,c*b*c^-1=a^21*b^3>;
// generators/relations
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