metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.4D8, C24.81D4, C4.12D24, Dic12.2C4, M5(2).4S3, C8.5(C4×S3), C24.2(C2×C4), (C2×C8).47D6, (C2×C4).10D12, (C2×C6).9SD16, C4.19(D6⋊C4), (C2×C12).100D4, C8.38(C3⋊D4), C3⋊2(C8.17D4), C24.C4.6C2, (C2×C24).51C22, (C2×Dic12).6C2, (C3×M5(2)).5C2, C6.18(D4⋊C4), C12.43(C22⋊C4), C22.7(C24⋊C2), C2.10(C2.D24), SmallGroup(192,76)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.4D8
G = < a,b,c | a12=1, b8=a6, c2=a9, bab-1=a7, cac-1=a5, cbc-1=a9b7 >
Smallest permutation representation of C12.4D8
►On 96 points
Generators in S96
(1 43 78 5 47 66 9 35 70 13 39 74)(2 36 79 14 48 75 10 44 71 6 40 67)(3 45 80 7 33 68 11 37 72 15 41 76)(4 38 65 16 34 77 12 46 73 8 42 69)(17 58 83 29 54 95 25 50 91 21 62 87)(18 51 84 22 55 88 26 59 92 30 63 96)(19 60 85 31 56 81 27 52 93 23 64 89)(20 53 86 24 57 90 28 61 94 32 49 82)
(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 60 13 64 9 52 5 56)(2 63 6 59 10 55 14 51)(3 58 15 62 11 50 7 54)(4 61 8 57 12 53 16 49)(17 33 21 45 25 41 29 37)(18 44 30 48 26 36 22 40)(19 47 23 43 27 39 31 35)(20 42 32 46 28 34 24 38)(65 90 69 86 73 82 77 94)(66 85 78 89 74 93 70 81)(67 88 71 84 75 96 79 92)(68 83 80 87 76 91 72 95)
G:=sub<Sym(96)| (1,43,78,5,47,66,9,35,70,13,39,74)(2,36,79,14,48,75,10,44,71,6,40,67)(3,45,80,7,33,68,11,37,72,15,41,76)(4,38,65,16,34,77,12,46,73,8,42,69)(17,58,83,29,54,95,25,50,91,21,62,87)(18,51,84,22,55,88,26,59,92,30,63,96)(19,60,85,31,56,81,27,52,93,23,64,89)(20,53,86,24,57,90,28,61,94,32,49,82), (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), (1,60,13,64,9,52,5,56)(2,63,6,59,10,55,14,51)(3,58,15,62,11,50,7,54)(4,61,8,57,12,53,16,49)(17,33,21,45,25,41,29,37)(18,44,30,48,26,36,22,40)(19,47,23,43,27,39,31,35)(20,42,32,46,28,34,24,38)(65,90,69,86,73,82,77,94)(66,85,78,89,74,93,70,81)(67,88,71,84,75,96,79,92)(68,83,80,87,76,91,72,95)>;
G:=Group( (1,43,78,5,47,66,9,35,70,13,39,74)(2,36,79,14,48,75,10,44,71,6,40,67)(3,45,80,7,33,68,11,37,72,15,41,76)(4,38,65,16,34,77,12,46,73,8,42,69)(17,58,83,29,54,95,25,50,91,21,62,87)(18,51,84,22,55,88,26,59,92,30,63,96)(19,60,85,31,56,81,27,52,93,23,64,89)(20,53,86,24,57,90,28,61,94,32,49,82), (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), (1,60,13,64,9,52,5,56)(2,63,6,59,10,55,14,51)(3,58,15,62,11,50,7,54)(4,61,8,57,12,53,16,49)(17,33,21,45,25,41,29,37)(18,44,30,48,26,36,22,40)(19,47,23,43,27,39,31,35)(20,42,32,46,28,34,24,38)(65,90,69,86,73,82,77,94)(66,85,78,89,74,93,70,81)(67,88,71,84,75,96,79,92)(68,83,80,87,76,91,72,95) );
G=PermutationGroup([[(1,43,78,5,47,66,9,35,70,13,39,74),(2,36,79,14,48,75,10,44,71,6,40,67),(3,45,80,7,33,68,11,37,72,15,41,76),(4,38,65,16,34,77,12,46,73,8,42,69),(17,58,83,29,54,95,25,50,91,21,62,87),(18,51,84,22,55,88,26,59,92,30,63,96),(19,60,85,31,56,81,27,52,93,23,64,89),(20,53,86,24,57,90,28,61,94,32,49,82)], [(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)], [(1,60,13,64,9,52,5,56),(2,63,6,59,10,55,14,51),(3,58,15,62,11,50,7,54),(4,61,8,57,12,53,16,49),(17,33,21,45,25,41,29,37),(18,44,30,48,26,36,22,40),(19,47,23,43,27,39,31,35),(20,42,32,46,28,34,24,38),(65,90,69,86,73,82,77,94),(66,85,78,89,74,93,70,81),(67,88,71,84,75,96,79,92),(68,83,80,87,76,91,72,95)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 24E | 24F | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 24 | 24 | 2 | 4 | 2 | 2 | 4 | 24 | 24 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D8 | SD16 | C4×S3 | C3⋊D4 | D12 | D24 | C24⋊C2 | C8.17D4 | C12.4D8 |
| kernel | C12.4D8 | C24.C4 | C3×M5(2) | C2×Dic12 | Dic12 | M5(2) | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C12.4D8 ►in GL4(𝔽97) generated by
| 39 | 68 | 0 | 0 |
| 29 | 68 | 0 | 0 |
| 0 | 0 | 58 | 29 |
| 0 | 0 | 68 | 29 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 95 | 79 | 0 | 0 |
| 18 | 16 | 0 | 0 |
| 0 | 0 | 52 | 54 |
| 0 | 0 | 2 | 45 |
| 72 | 5 | 0 | 0 |
| 30 | 25 | 0 | 0 |
G:=sub<GL(4,GF(97))| [39,29,0,0,68,68,0,0,0,0,58,68,0,0,29,29],[0,0,95,18,0,0,79,16,1,0,0,0,0,1,0,0],[0,0,72,30,0,0,5,25,52,2,0,0,54,45,0,0] >;
C12.4D8 in GAP, Magma, Sage, TeX
C_{12}._4D_8 % in TeX
G:=Group("C12.4D8"); // GroupNames label
G:=SmallGroup(192,76);
// by ID
G=gap.SmallGroup(192,76);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,85,92,422,387,268,570,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^8=a^6,c^2=a^9,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^9*b^7>;
// generators/relations
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