metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.91D6, C6.492- 1+4, C6.942+ 1+4, (C4×D12)⋊6C2, C4○D12⋊11C4, D12⋊27(C2×C4), C4⋊C4.310D6, (C4×Dic6)⋊8C2, C42⋊C2⋊7S3, Dic6⋊25(C2×C4), C2.2(D4○D12), C6.19(C23×C4), (C2×C6).67C24, C2.2(Q8○D12), D6.6(C22×C4), C12.90(C22×C4), (C4×C12).23C22, C22⋊C4.127D6, (C22×C4).205D6, Dic3⋊4D4⋊42C2, (C2×C12).489C23, D6⋊C4.119C22, C22.29(S3×C23), Dic3.8(C22×C4), (C2×D12).286C22, C4⋊Dic3.397C22, C23.165(C22×S3), (C22×C6).137C23, Dic3⋊C4.132C22, (C22×S3).164C23, (C22×C12).227C22, C3⋊2(C23.33C23), (C4×Dic3).195C22, (C2×Dic3).196C23, (C2×Dic6).315C22, (C22×Dic3).86C22, (C2×C4)⋊7(C4×S3), C4.94(S3×C2×C4), (S3×C4⋊C4)⋊11C2, (C4×S3)⋊2(C2×C4), C3⋊D4⋊7(C2×C4), (C2×C12)⋊13(C2×C4), C22.7(S3×C2×C4), C4⋊C4⋊7S3⋊11C2, C2.21(S3×C22×C4), (C2×C4⋊Dic3)⋊39C2, (S3×C2×C4).58C22, (C3×C42⋊C2)⋊9C2, (C2×C4○D12).18C2, (C2×C6).23(C22×C4), (C3×C4⋊C4).306C22, (C2×C4).273(C22×S3), (C2×C3⋊D4).98C22, (C3×C22⋊C4).137C22, SmallGroup(192,1082)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.91D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=a2b-1, bd=db, dcd-1=c5 >
Subgroups: 648 in 294 conjugacy classes, 151 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C22×Dic3, C2×C3⋊D4, C22×C12, C23.33C23, C4×Dic6, C4×D12, Dic3⋊4D4, S3×C4⋊C4, C4⋊C4⋊7S3, C2×C4⋊Dic3, C3×C42⋊C2, C2×C4○D12, C42.91D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C23×C4, 2+ 1+4, 2- 1+4, S3×C2×C4, S3×C23, C23.33C23, S3×C22×C4, D4○D12, Q8○D12, C42.91D6
►On 96 points
(1 93 26 64)(2 88 27 71)(3 95 28 66)(4 90 29 61)(5 85 30 68)(6 92 31 63)(7 87 32 70)(8 94 33 65)(9 89 34 72)(10 96 35 67)(11 91 36 62)(12 86 25 69)(13 82 59 41)(14 77 60 48)(15 84 49 43)(16 79 50 38)(17 74 51 45)(18 81 52 40)(19 76 53 47)(20 83 54 42)(21 78 55 37)(22 73 56 44)(23 80 57 39)(24 75 58 46)
(1 13 32 53)(2 20 33 60)(3 15 34 55)(4 22 35 50)(5 17 36 57)(6 24 25 52)(7 19 26 59)(8 14 27 54)(9 21 28 49)(10 16 29 56)(11 23 30 51)(12 18 31 58)(37 95 84 72)(38 90 73 67)(39 85 74 62)(40 92 75 69)(41 87 76 64)(42 94 77 71)(43 89 78 66)(44 96 79 61)(45 91 80 68)(46 86 81 63)(47 93 82 70)(48 88 83 65)
(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 75 7 81)(2 80 8 74)(3 73 9 79)(4 78 10 84)(5 83 11 77)(6 76 12 82)(13 69 19 63)(14 62 20 68)(15 67 21 61)(16 72 22 66)(17 65 23 71)(18 70 24 64)(25 41 31 47)(26 46 32 40)(27 39 33 45)(28 44 34 38)(29 37 35 43)(30 42 36 48)(49 96 55 90)(50 89 56 95)(51 94 57 88)(52 87 58 93)(53 92 59 86)(54 85 60 91)
G:=sub<Sym(96)| (1,93,26,64)(2,88,27,71)(3,95,28,66)(4,90,29,61)(5,85,30,68)(6,92,31,63)(7,87,32,70)(8,94,33,65)(9,89,34,72)(10,96,35,67)(11,91,36,62)(12,86,25,69)(13,82,59,41)(14,77,60,48)(15,84,49,43)(16,79,50,38)(17,74,51,45)(18,81,52,40)(19,76,53,47)(20,83,54,42)(21,78,55,37)(22,73,56,44)(23,80,57,39)(24,75,58,46), (1,13,32,53)(2,20,33,60)(3,15,34,55)(4,22,35,50)(5,17,36,57)(6,24,25,52)(7,19,26,59)(8,14,27,54)(9,21,28,49)(10,16,29,56)(11,23,30,51)(12,18,31,58)(37,95,84,72)(38,90,73,67)(39,85,74,62)(40,92,75,69)(41,87,76,64)(42,94,77,71)(43,89,78,66)(44,96,79,61)(45,91,80,68)(46,86,81,63)(47,93,82,70)(48,88,83,65), (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), (1,75,7,81)(2,80,8,74)(3,73,9,79)(4,78,10,84)(5,83,11,77)(6,76,12,82)(13,69,19,63)(14,62,20,68)(15,67,21,61)(16,72,22,66)(17,65,23,71)(18,70,24,64)(25,41,31,47)(26,46,32,40)(27,39,33,45)(28,44,34,38)(29,37,35,43)(30,42,36,48)(49,96,55,90)(50,89,56,95)(51,94,57,88)(52,87,58,93)(53,92,59,86)(54,85,60,91)>;
G:=Group( (1,93,26,64)(2,88,27,71)(3,95,28,66)(4,90,29,61)(5,85,30,68)(6,92,31,63)(7,87,32,70)(8,94,33,65)(9,89,34,72)(10,96,35,67)(11,91,36,62)(12,86,25,69)(13,82,59,41)(14,77,60,48)(15,84,49,43)(16,79,50,38)(17,74,51,45)(18,81,52,40)(19,76,53,47)(20,83,54,42)(21,78,55,37)(22,73,56,44)(23,80,57,39)(24,75,58,46), (1,13,32,53)(2,20,33,60)(3,15,34,55)(4,22,35,50)(5,17,36,57)(6,24,25,52)(7,19,26,59)(8,14,27,54)(9,21,28,49)(10,16,29,56)(11,23,30,51)(12,18,31,58)(37,95,84,72)(38,90,73,67)(39,85,74,62)(40,92,75,69)(41,87,76,64)(42,94,77,71)(43,89,78,66)(44,96,79,61)(45,91,80,68)(46,86,81,63)(47,93,82,70)(48,88,83,65), (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), (1,75,7,81)(2,80,8,74)(3,73,9,79)(4,78,10,84)(5,83,11,77)(6,76,12,82)(13,69,19,63)(14,62,20,68)(15,67,21,61)(16,72,22,66)(17,65,23,71)(18,70,24,64)(25,41,31,47)(26,46,32,40)(27,39,33,45)(28,44,34,38)(29,37,35,43)(30,42,36,48)(49,96,55,90)(50,89,56,95)(51,94,57,88)(52,87,58,93)(53,92,59,86)(54,85,60,91) );
G=PermutationGroup([[(1,93,26,64),(2,88,27,71),(3,95,28,66),(4,90,29,61),(5,85,30,68),(6,92,31,63),(7,87,32,70),(8,94,33,65),(9,89,34,72),(10,96,35,67),(11,91,36,62),(12,86,25,69),(13,82,59,41),(14,77,60,48),(15,84,49,43),(16,79,50,38),(17,74,51,45),(18,81,52,40),(19,76,53,47),(20,83,54,42),(21,78,55,37),(22,73,56,44),(23,80,57,39),(24,75,58,46)], [(1,13,32,53),(2,20,33,60),(3,15,34,55),(4,22,35,50),(5,17,36,57),(6,24,25,52),(7,19,26,59),(8,14,27,54),(9,21,28,49),(10,16,29,56),(11,23,30,51),(12,18,31,58),(37,95,84,72),(38,90,73,67),(39,85,74,62),(40,92,75,69),(41,87,76,64),(42,94,77,71),(43,89,78,66),(44,96,79,61),(45,91,80,68),(46,86,81,63),(47,93,82,70),(48,88,83,65)], [(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)], [(1,75,7,81),(2,80,8,74),(3,73,9,79),(4,78,10,84),(5,83,11,77),(6,76,12,82),(13,69,19,63),(14,62,20,68),(15,67,21,61),(16,72,22,66),(17,65,23,71),(18,70,24,64),(25,41,31,47),(26,46,32,40),(27,39,33,45),(28,44,34,38),(29,37,35,43),(30,42,36,48),(49,96,55,90),(50,89,56,95),(51,94,57,88),(52,87,58,93),(53,92,59,86),(54,85,60,91)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | ··· | 4L | 4M | ··· | 4X | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | D6 | D6 | C4×S3 | 2+ 1+4 | 2- 1+4 | D4○D12 | Q8○D12 |
| kernel | C42.91D6 | C4×Dic6 | C4×D12 | Dic3⋊4D4 | S3×C4⋊C4 | C4⋊C4⋊7S3 | C2×C4⋊Dic3 | C3×C42⋊C2 | C2×C4○D12 | C4○D12 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 16 | 1 | 2 | 2 | 2 | 1 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C42.91D6 ►in GL6(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 10 | 0 |
| 0 | 0 | 8 | 1 | 11 | 11 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 3 | 0 | 0 |
| 0 | 0 | 8 | 1 | 0 | 0 |
| 0 | 0 | 5 | 11 | 1 | 2 |
| 0 | 0 | 8 | 1 | 12 | 12 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 10 | 0 |
| 0 | 0 | 0 | 12 | 0 | 2 |
| 0 | 0 | 5 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 10 | 0 | 0 |
| 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 12 | 12 |
G:=sub<GL(6,GF(13))| [5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,8,0,0,0,0,0,1,0,0,0,0,10,11,1,12,0,0,0,11,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,8,5,8,0,0,3,1,11,1,0,0,0,0,1,12,0,0,0,0,2,12],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,5,0,0,0,0,12,0,12,0,0,10,0,1,0,0,0,0,2,0,1],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,5,0,0,0,0,10,12,0,0,0,0,0,0,1,12,0,0,0,0,2,12] >;
C42.91D6 in GAP, Magma, Sage, TeX
C_4^2._{91}D_6 % in TeX
G:=Group("C4^2.91D6"); // GroupNames label
G:=SmallGroup(192,1082);
// by ID
G=gap.SmallGroup(192,1082);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,570,297,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations