metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊1(D6⋊C4), C6.59(C4×D4), (C2×D12)⋊10C4, C12⋊2(C22⋊C4), (C2×Dic3)⋊11D4, (C2×C4).144D12, (C2×C12).142D4, C2.5(C12⋊D4), C2.3(C12⋊3D4), C6.15(C4⋊1D4), C6.54(C4⋊D4), C22.111(S3×D4), C22.48(C2×D12), (C22×C4).351D6, C6.51(C4.4D4), (C22×D12).11C2, C2.18(Dic3⋊5D4), C2.2(C12.23D4), (S3×C23).18C22, C23.306(C22×S3), (C22×C6).352C23, C3⋊3(C24.3C22), (C22×C12).144C22, C22.28(Q8⋊3S3), (C22×Dic3).192C22, (C6×C4⋊C4)⋊6C2, (C2×C4⋊C4)⋊6S3, (C2×C4×Dic3)⋊1C2, (C2×D6⋊C4)⋊10C2, (C2×C4).79(C4×S3), C2.16(C2×D6⋊C4), (C2×C12).86(C2×C4), (C2×C6).452(C2×D4), C6.43(C2×C22⋊C4), C22.137(S3×C2×C4), C22.67(C2×C3⋊D4), (C2×C6).189(C4○D4), (C2×C4).129(C3⋊D4), (C22×S3).21(C2×C4), (C2×C6).120(C22×C4), SmallGroup(192,547)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×D12)⋊10C4
G = < a,b,c,d | a2=b12=c2=d4=1, ab=ba, dcd-1=ac=ca, ad=da, cbc=b-1, dbd-1=b7 >
Subgroups: 824 in 258 conjugacy classes, 83 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C4×Dic3, D6⋊C4, C3×C4⋊C4, C2×D12, C2×D12, C22×Dic3, C22×C12, C22×C12, S3×C23, C24.3C22, C2×C4×Dic3, C2×D6⋊C4, C6×C4⋊C4, C22×D12, (C2×D12)⋊10C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4○D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, D6⋊C4, S3×C2×C4, C2×D12, S3×D4, Q8⋊3S3, C2×C3⋊D4, C24.3C22, Dic3⋊5D4, C12⋊D4, C2×D6⋊C4, C12⋊3D4, C12.23D4, (C2×D12)⋊10C4
►On 96 points
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 61)(11 62)(12 63)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 92)(26 93)(27 94)(28 95)(29 96)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(49 80)(50 81)(51 82)(52 83)(53 84)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 48)(25 86)(26 85)(27 96)(28 95)(29 94)(30 93)(31 92)(32 91)(33 90)(34 89)(35 88)(36 87)(49 59)(50 58)(51 57)(52 56)(53 55)(61 69)(62 68)(63 67)(64 66)(70 72)(74 84)(75 83)(76 82)(77 81)(78 80)
(1 88 84 41)(2 95 73 48)(3 90 74 43)(4 85 75 38)(5 92 76 45)(6 87 77 40)(7 94 78 47)(8 89 79 42)(9 96 80 37)(10 91 81 44)(11 86 82 39)(12 93 83 46)(13 72 29 49)(14 67 30 56)(15 62 31 51)(16 69 32 58)(17 64 33 53)(18 71 34 60)(19 66 35 55)(20 61 36 50)(21 68 25 57)(22 63 26 52)(23 70 27 59)(24 65 28 54)
G:=sub<Sym(96)| (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,61)(11,62)(12,63)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,92)(26,93)(27,94)(28,95)(29,96)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(49,80)(50,81)(51,82)(52,83)(53,84)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,48)(25,86)(26,85)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(49,59)(50,58)(51,57)(52,56)(53,55)(61,69)(62,68)(63,67)(64,66)(70,72)(74,84)(75,83)(76,82)(77,81)(78,80), (1,88,84,41)(2,95,73,48)(3,90,74,43)(4,85,75,38)(5,92,76,45)(6,87,77,40)(7,94,78,47)(8,89,79,42)(9,96,80,37)(10,91,81,44)(11,86,82,39)(12,93,83,46)(13,72,29,49)(14,67,30,56)(15,62,31,51)(16,69,32,58)(17,64,33,53)(18,71,34,60)(19,66,35,55)(20,61,36,50)(21,68,25,57)(22,63,26,52)(23,70,27,59)(24,65,28,54)>;
G:=Group( (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,61)(11,62)(12,63)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,92)(26,93)(27,94)(28,95)(29,96)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(49,80)(50,81)(51,82)(52,83)(53,84)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,48)(25,86)(26,85)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(49,59)(50,58)(51,57)(52,56)(53,55)(61,69)(62,68)(63,67)(64,66)(70,72)(74,84)(75,83)(76,82)(77,81)(78,80), (1,88,84,41)(2,95,73,48)(3,90,74,43)(4,85,75,38)(5,92,76,45)(6,87,77,40)(7,94,78,47)(8,89,79,42)(9,96,80,37)(10,91,81,44)(11,86,82,39)(12,93,83,46)(13,72,29,49)(14,67,30,56)(15,62,31,51)(16,69,32,58)(17,64,33,53)(18,71,34,60)(19,66,35,55)(20,61,36,50)(21,68,25,57)(22,63,26,52)(23,70,27,59)(24,65,28,54) );
G=PermutationGroup([[(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,61),(11,62),(12,63),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,92),(26,93),(27,94),(28,95),(29,96),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(49,80),(50,81),(51,82),(52,83),(53,84),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,47),(14,46),(15,45),(16,44),(17,43),(18,42),(19,41),(20,40),(21,39),(22,38),(23,37),(24,48),(25,86),(26,85),(27,96),(28,95),(29,94),(30,93),(31,92),(32,91),(33,90),(34,89),(35,88),(36,87),(49,59),(50,58),(51,57),(52,56),(53,55),(61,69),(62,68),(63,67),(64,66),(70,72),(74,84),(75,83),(76,82),(77,81),(78,80)], [(1,88,84,41),(2,95,73,48),(3,90,74,43),(4,85,75,38),(5,92,76,45),(6,87,77,40),(7,94,78,47),(8,89,79,42),(9,96,80,37),(10,91,81,44),(11,86,82,39),(12,93,83,46),(13,72,29,49),(14,67,30,56),(15,62,31,51),(16,69,32,58),(17,64,33,53),(18,71,34,60),(19,66,35,55),(20,61,36,50),(21,68,25,57),(22,63,26,52),(23,70,27,59),(24,65,28,54)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | C4○D4 | C4×S3 | D12 | C3⋊D4 | S3×D4 | Q8⋊3S3 |
| kernel | (C2×D12)⋊10C4 | C2×C4×Dic3 | C2×D6⋊C4 | C6×C4⋊C4 | C22×D12 | C2×D12 | C2×C4⋊C4 | C2×Dic3 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 4 | 1 | 1 | 8 | 1 | 4 | 4 | 3 | 4 | 4 | 4 | 4 | 2 | 2 |
Matrix representation of (C2×D12)⋊10C4 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,3,7,0,0,0,0,6,10,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
(C2×D12)⋊10C4 in GAP, Magma, Sage, TeX
(C_2\times D_{12})\rtimes_{10}C_4 % in TeX
G:=Group("(C2xD12):10C4"); // GroupNames label
G:=SmallGroup(192,547);
// by ID
G=gap.SmallGroup(192,547);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,422,387,58,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^2=d^4=1,a*b=b*a,d*c*d^-1=a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^7>;
// generators/relations