metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.1D12, C24.1D4, C42.21D6, C8⋊C4⋊4S3, (C2×C8).56D6, C12⋊2Q8⋊4C2, C4.35(C2×D12), (C2×C12).39D4, (C2×C4).28D12, C3⋊1(C8.2D4), C12.278(C2×D4), C6.8(C4⋊1D4), (C4×C12).6C22, (C2×Dic12)⋊10C2, C42⋊7S3.3C2, (C2×C24).57C22, C2.9(C8.D6), (C2×D12).9C22, C6.5(C8.C22), C2.10(C4⋊D12), (C2×C12).736C23, C22.100(C2×D12), (C2×Dic6).9C22, (C3×C8⋊C4)⋊5C2, (C2×C24⋊C2).2C2, (C2×C6).119(C2×D4), (C2×C4).680(C22×S3), SmallGroup(192,274)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.D12
G = < a,b,c | a8=b12=1, c2=a4, bab-1=a5, cac-1=a-1, cbc-1=a4b-1 >
Subgroups: 408 in 124 conjugacy classes, 47 normal (17 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, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C24, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C24⋊C2, Dic12, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, C2×Dic6, C2×Dic6, C2×D12, C8.2D4, C3×C8⋊C4, C12⋊2Q8, C42⋊7S3, C2×C24⋊C2, C2×Dic12, C8.D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C4⋊1D4, C8.C22, C2×D12, C8.2D4, C4⋊D12, C8.D6, C8.D12
►On 96 points
(1 53 91 20 82 28 61 42)(2 29 92 43 83 54 62 21)(3 55 93 22 84 30 63 44)(4 31 94 45 73 56 64 23)(5 57 95 24 74 32 65 46)(6 33 96 47 75 58 66 13)(7 59 85 14 76 34 67 48)(8 35 86 37 77 60 68 15)(9 49 87 16 78 36 69 38)(10 25 88 39 79 50 70 17)(11 51 89 18 80 26 71 40)(12 27 90 41 81 52 72 19)
(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 81 82 12)(2 11 83 80)(3 79 84 10)(4 9 73 78)(5 77 74 8)(6 7 75 76)(13 59 47 34)(14 33 48 58)(15 57 37 32)(16 31 38 56)(17 55 39 30)(18 29 40 54)(19 53 41 28)(20 27 42 52)(21 51 43 26)(22 25 44 50)(23 49 45 36)(24 35 46 60)(61 72 91 90)(62 89 92 71)(63 70 93 88)(64 87 94 69)(65 68 95 86)(66 85 96 67)
G:=sub<Sym(96)| (1,53,91,20,82,28,61,42)(2,29,92,43,83,54,62,21)(3,55,93,22,84,30,63,44)(4,31,94,45,73,56,64,23)(5,57,95,24,74,32,65,46)(6,33,96,47,75,58,66,13)(7,59,85,14,76,34,67,48)(8,35,86,37,77,60,68,15)(9,49,87,16,78,36,69,38)(10,25,88,39,79,50,70,17)(11,51,89,18,80,26,71,40)(12,27,90,41,81,52,72,19), (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), (1,81,82,12)(2,11,83,80)(3,79,84,10)(4,9,73,78)(5,77,74,8)(6,7,75,76)(13,59,47,34)(14,33,48,58)(15,57,37,32)(16,31,38,56)(17,55,39,30)(18,29,40,54)(19,53,41,28)(20,27,42,52)(21,51,43,26)(22,25,44,50)(23,49,45,36)(24,35,46,60)(61,72,91,90)(62,89,92,71)(63,70,93,88)(64,87,94,69)(65,68,95,86)(66,85,96,67)>;
G:=Group( (1,53,91,20,82,28,61,42)(2,29,92,43,83,54,62,21)(3,55,93,22,84,30,63,44)(4,31,94,45,73,56,64,23)(5,57,95,24,74,32,65,46)(6,33,96,47,75,58,66,13)(7,59,85,14,76,34,67,48)(8,35,86,37,77,60,68,15)(9,49,87,16,78,36,69,38)(10,25,88,39,79,50,70,17)(11,51,89,18,80,26,71,40)(12,27,90,41,81,52,72,19), (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), (1,81,82,12)(2,11,83,80)(3,79,84,10)(4,9,73,78)(5,77,74,8)(6,7,75,76)(13,59,47,34)(14,33,48,58)(15,57,37,32)(16,31,38,56)(17,55,39,30)(18,29,40,54)(19,53,41,28)(20,27,42,52)(21,51,43,26)(22,25,44,50)(23,49,45,36)(24,35,46,60)(61,72,91,90)(62,89,92,71)(63,70,93,88)(64,87,94,69)(65,68,95,86)(66,85,96,67) );
G=PermutationGroup([[(1,53,91,20,82,28,61,42),(2,29,92,43,83,54,62,21),(3,55,93,22,84,30,63,44),(4,31,94,45,73,56,64,23),(5,57,95,24,74,32,65,46),(6,33,96,47,75,58,66,13),(7,59,85,14,76,34,67,48),(8,35,86,37,77,60,68,15),(9,49,87,16,78,36,69,38),(10,25,88,39,79,50,70,17),(11,51,89,18,80,26,71,40),(12,27,90,41,81,52,72,19)], [(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)], [(1,81,82,12),(2,11,83,80),(3,79,84,10),(4,9,73,78),(5,77,74,8),(6,7,75,76),(13,59,47,34),(14,33,48,58),(15,57,37,32),(16,31,38,56),(17,55,39,30),(18,29,40,54),(19,53,41,28),(20,27,42,52),(21,51,43,26),(22,25,44,50),(23,49,45,36),(24,35,46,60),(61,72,91,90),(62,89,92,71),(63,70,93,88),(64,87,94,69),(65,68,95,86),(66,85,96,67)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 24 | 2 | 2 | 2 | 4 | 4 | 24 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D12 | D12 | C8.C22 | C8.D6 |
| kernel | C8.D12 | C3×C8⋊C4 | C12⋊2Q8 | C42⋊7S3 | C2×C24⋊C2 | C2×Dic12 | C8⋊C4 | C24 | C2×C12 | C42 | C2×C8 | C8 | C2×C4 | C6 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 2 | 4 |
Matrix representation of C8.D12 ►in GL6(𝔽73)
| 48 | 67 | 0 | 0 | 0 | 0 |
| 7 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 43 | 13 | 0 | 0 |
| 0 | 0 | 60 | 30 | 0 | 0 |
| 25 | 6 | 0 | 0 | 0 | 0 |
| 66 | 48 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 27 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 46 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 25 | 6 | 0 | 0 | 0 | 0 |
| 42 | 48 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 27 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 43 |
| 0 | 0 | 0 | 0 | 13 | 43 |
G:=sub<GL(6,GF(73))| [48,7,0,0,0,0,67,25,0,0,0,0,0,0,0,0,43,60,0,0,0,0,13,30,0,0,27,0,0,0,0,0,0,27,0,0],[25,66,0,0,0,0,6,48,0,0,0,0,0,0,27,46,0,0,0,0,27,0,0,0,0,0,0,0,46,27,0,0,0,0,46,0],[25,42,0,0,0,0,6,48,0,0,0,0,0,0,27,0,0,0,0,0,27,46,0,0,0,0,0,0,30,13,0,0,0,0,43,43] >;
C8.D12 in GAP, Magma, Sage, TeX
C_8.D_{12} % in TeX
G:=Group("C8.D12"); // GroupNames label
G:=SmallGroup(192,274);
// by ID
G=gap.SmallGroup(192,274);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,387,58,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^8=b^12=1,c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;
// generators/relations