metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.8D12, C24.58D4, C42.263D6, (C4×C8)⋊9S3, (C4×C24)⋊14C2, (C2×D24).2C2, C6.4(C4○D8), (C2×C4).62D12, (C2×C8).318D6, C4.32(C2×D12), C42⋊7S3⋊1C2, (C2×Dic12)⋊1C2, C2.7(C4○D24), (C2×C12).352D4, C12.275(C2×D4), C6.5(C4⋊1D4), C3⋊1(C8.12D4), C2.7(C4⋊D12), (C2×D12).4C22, C22.92(C2×D12), (C4×C12).309C22, (C2×C24).390C22, (C2×C12).725C23, (C2×Dic6).3C22, (C2×C24⋊C2)⋊8C2, (C2×C6).108(C2×D4), (C2×C4).668(C22×S3), SmallGroup(192,255)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.8D12
G = < a,b,c | a8=b12=1, c2=a4, ab=ba, cac-1=a-1, cbc-1=a4b-1 >
Subgroups: 472 in 130 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, C24, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C4×C8, C4.4D4, C2×D8, C2×SD16, C2×Q16, C24⋊C2, D24, Dic12, D6⋊C4, C4×C12, C2×C24, C2×Dic6, C2×D12, C8.12D4, C4×C24, C42⋊7S3, C2×C24⋊C2, C2×D24, C2×Dic12, C8.8D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C4⋊1D4, C4○D8, C2×D12, C8.12D4, C4⋊D12, C4○D24, C8.8D12
►On 96 points
(1 66 75 16 43 57 85 35)(2 67 76 17 44 58 86 36)(3 68 77 18 45 59 87 25)(4 69 78 19 46 60 88 26)(5 70 79 20 47 49 89 27)(6 71 80 21 48 50 90 28)(7 72 81 22 37 51 91 29)(8 61 82 23 38 52 92 30)(9 62 83 24 39 53 93 31)(10 63 84 13 40 54 94 32)(11 64 73 14 41 55 95 33)(12 65 74 15 42 56 96 34)
(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 42 43 12)(2 11 44 41)(3 40 45 10)(4 9 46 39)(5 38 47 8)(6 7 48 37)(13 59 32 68)(14 67 33 58)(15 57 34 66)(16 65 35 56)(17 55 36 64)(18 63 25 54)(19 53 26 62)(20 61 27 52)(21 51 28 72)(22 71 29 50)(23 49 30 70)(24 69 31 60)(73 76 95 86)(74 85 96 75)(77 84 87 94)(78 93 88 83)(79 82 89 92)(80 91 90 81)
G:=sub<Sym(96)| (1,66,75,16,43,57,85,35)(2,67,76,17,44,58,86,36)(3,68,77,18,45,59,87,25)(4,69,78,19,46,60,88,26)(5,70,79,20,47,49,89,27)(6,71,80,21,48,50,90,28)(7,72,81,22,37,51,91,29)(8,61,82,23,38,52,92,30)(9,62,83,24,39,53,93,31)(10,63,84,13,40,54,94,32)(11,64,73,14,41,55,95,33)(12,65,74,15,42,56,96,34), (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), (1,42,43,12)(2,11,44,41)(3,40,45,10)(4,9,46,39)(5,38,47,8)(6,7,48,37)(13,59,32,68)(14,67,33,58)(15,57,34,66)(16,65,35,56)(17,55,36,64)(18,63,25,54)(19,53,26,62)(20,61,27,52)(21,51,28,72)(22,71,29,50)(23,49,30,70)(24,69,31,60)(73,76,95,86)(74,85,96,75)(77,84,87,94)(78,93,88,83)(79,82,89,92)(80,91,90,81)>;
G:=Group( (1,66,75,16,43,57,85,35)(2,67,76,17,44,58,86,36)(3,68,77,18,45,59,87,25)(4,69,78,19,46,60,88,26)(5,70,79,20,47,49,89,27)(6,71,80,21,48,50,90,28)(7,72,81,22,37,51,91,29)(8,61,82,23,38,52,92,30)(9,62,83,24,39,53,93,31)(10,63,84,13,40,54,94,32)(11,64,73,14,41,55,95,33)(12,65,74,15,42,56,96,34), (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), (1,42,43,12)(2,11,44,41)(3,40,45,10)(4,9,46,39)(5,38,47,8)(6,7,48,37)(13,59,32,68)(14,67,33,58)(15,57,34,66)(16,65,35,56)(17,55,36,64)(18,63,25,54)(19,53,26,62)(20,61,27,52)(21,51,28,72)(22,71,29,50)(23,49,30,70)(24,69,31,60)(73,76,95,86)(74,85,96,75)(77,84,87,94)(78,93,88,83)(79,82,89,92)(80,91,90,81) );
G=PermutationGroup([[(1,66,75,16,43,57,85,35),(2,67,76,17,44,58,86,36),(3,68,77,18,45,59,87,25),(4,69,78,19,46,60,88,26),(5,70,79,20,47,49,89,27),(6,71,80,21,48,50,90,28),(7,72,81,22,37,51,91,29),(8,61,82,23,38,52,92,30),(9,62,83,24,39,53,93,31),(10,63,84,13,40,54,94,32),(11,64,73,14,41,55,95,33),(12,65,74,15,42,56,96,34)], [(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)], [(1,42,43,12),(2,11,44,41),(3,40,45,10),(4,9,46,39),(5,38,47,8),(6,7,48,37),(13,59,32,68),(14,67,33,58),(15,57,34,66),(16,65,35,56),(17,55,36,64),(18,63,25,54),(19,53,26,62),(20,61,27,52),(21,51,28,72),(22,71,29,50),(23,49,30,70),(24,69,31,60),(73,76,95,86),(74,85,96,75),(77,84,87,94),(78,93,88,83),(79,82,89,92),(80,91,90,81)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 24 | 24 | 2 | 2 | ··· | 2 | 24 | 24 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D12 | D12 | C4○D8 | C4○D24 |
| kernel | C8.8D12 | C4×C24 | C42⋊7S3 | C2×C24⋊C2 | C2×D24 | C2×Dic12 | C4×C8 | C24 | C2×C12 | C42 | C2×C8 | C8 | C2×C4 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 8 | 16 |
Matrix representation of C8.8D12 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 48 |
| 0 | 0 | 38 | 32 |
| 7 | 66 | 0 | 0 |
| 7 | 14 | 0 | 0 |
| 0 | 0 | 46 | 8 |
| 0 | 0 | 55 | 27 |
| 7 | 66 | 0 | 0 |
| 59 | 66 | 0 | 0 |
| 0 | 0 | 46 | 8 |
| 0 | 0 | 0 | 27 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,38,0,0,48,32],[7,7,0,0,66,14,0,0,0,0,46,55,0,0,8,27],[7,59,0,0,66,66,0,0,0,0,46,0,0,0,8,27] >;
C8.8D12 in GAP, Magma, Sage, TeX
C_8._8D_{12} % in TeX
G:=Group("C8.8D12"); // GroupNames label
G:=SmallGroup(192,255);
// by ID
G=gap.SmallGroup(192,255);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,58,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^8=b^12=1,c^2=a^4,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;
// generators/relations