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), (C4×C12).23C22, C12.90(C22×C4), 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, (C22×C6).137C23, C23.165(C22×S3), Dic3⋊C4.132C22, (C22×S3).164C23, (C22×C12).227C22, C3⋊2(C23.33C23), (C2×Dic3).196C23, (C4×Dic3).195C22, (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
Subgroups: 648 in 294 conjugacy classes, 151 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×4], C4 [×12], C22, C22 [×2], C22 [×10], S3 [×4], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×20], D4 [×12], Q8 [×4], C23, C23 [×2], Dic3 [×4], Dic3 [×4], C12 [×4], C12 [×4], D6 [×4], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×8], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic6 [×4], C4×S3 [×8], C4×S3 [×4], D12 [×4], C2×Dic3 [×6], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×8], C22×S3 [×2], C22×C6, C2×C4⋊C4 [×3], C42⋊C2, C42⋊C2 [×2], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C4×Dic3 [×4], Dic3⋊C4 [×4], C4⋊Dic3 [×4], D6⋊C4 [×4], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, S3×C2×C4 [×6], C2×D12, C4○D12 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C22×C12, C23.33C23, C4×Dic6 [×2], C4×D12 [×2], Dic3⋊4D4 [×4], S3×C4⋊C4 [×2], C4⋊C4⋊7S3 [×2], C2×C4⋊Dic3, C3×C42⋊C2, C2×C4○D12, C42.91D6
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C23×C4, 2+ (1+4), 2- (1+4), S3×C2×C4 [×6], S3×C23, C23.33C23, S3×C22×C4, D4○D12, Q8○D12, C42.91D6
Generators and relations
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 >
►On 96 points
(1 16 36 64)(2 23 25 71)(3 18 26 66)(4 13 27 61)(5 20 28 68)(6 15 29 63)(7 22 30 70)(8 17 31 65)(9 24 32 72)(10 19 33 67)(11 14 34 62)(12 21 35 69)(37 81 89 53)(38 76 90 60)(39 83 91 55)(40 78 92 50)(41 73 93 57)(42 80 94 52)(43 75 95 59)(44 82 96 54)(45 77 85 49)(46 84 86 56)(47 79 87 51)(48 74 88 58)
(1 52 30 74)(2 59 31 81)(3 54 32 76)(4 49 33 83)(5 56 34 78)(6 51 35 73)(7 58 36 80)(8 53 25 75)(9 60 26 82)(10 55 27 77)(11 50 28 84)(12 57 29 79)(13 45 67 91)(14 40 68 86)(15 47 69 93)(16 42 70 88)(17 37 71 95)(18 44 72 90)(19 39 61 85)(20 46 62 92)(21 41 63 87)(22 48 64 94)(23 43 65 89)(24 38 66 96)
(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 47 7 41)(2 40 8 46)(3 45 9 39)(4 38 10 44)(5 43 11 37)(6 48 12 42)(13 82 19 76)(14 75 20 81)(15 80 21 74)(16 73 22 79)(17 78 23 84)(18 83 24 77)(25 92 31 86)(26 85 32 91)(27 90 33 96)(28 95 34 89)(29 88 35 94)(30 93 36 87)(49 66 55 72)(50 71 56 65)(51 64 57 70)(52 69 58 63)(53 62 59 68)(54 67 60 61)
G:=sub<Sym(96)| (1,16,36,64)(2,23,25,71)(3,18,26,66)(4,13,27,61)(5,20,28,68)(6,15,29,63)(7,22,30,70)(8,17,31,65)(9,24,32,72)(10,19,33,67)(11,14,34,62)(12,21,35,69)(37,81,89,53)(38,76,90,60)(39,83,91,55)(40,78,92,50)(41,73,93,57)(42,80,94,52)(43,75,95,59)(44,82,96,54)(45,77,85,49)(46,84,86,56)(47,79,87,51)(48,74,88,58), (1,52,30,74)(2,59,31,81)(3,54,32,76)(4,49,33,83)(5,56,34,78)(6,51,35,73)(7,58,36,80)(8,53,25,75)(9,60,26,82)(10,55,27,77)(11,50,28,84)(12,57,29,79)(13,45,67,91)(14,40,68,86)(15,47,69,93)(16,42,70,88)(17,37,71,95)(18,44,72,90)(19,39,61,85)(20,46,62,92)(21,41,63,87)(22,48,64,94)(23,43,65,89)(24,38,66,96), (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), (1,47,7,41)(2,40,8,46)(3,45,9,39)(4,38,10,44)(5,43,11,37)(6,48,12,42)(13,82,19,76)(14,75,20,81)(15,80,21,74)(16,73,22,79)(17,78,23,84)(18,83,24,77)(25,92,31,86)(26,85,32,91)(27,90,33,96)(28,95,34,89)(29,88,35,94)(30,93,36,87)(49,66,55,72)(50,71,56,65)(51,64,57,70)(52,69,58,63)(53,62,59,68)(54,67,60,61)>;
G:=Group( (1,16,36,64)(2,23,25,71)(3,18,26,66)(4,13,27,61)(5,20,28,68)(6,15,29,63)(7,22,30,70)(8,17,31,65)(9,24,32,72)(10,19,33,67)(11,14,34,62)(12,21,35,69)(37,81,89,53)(38,76,90,60)(39,83,91,55)(40,78,92,50)(41,73,93,57)(42,80,94,52)(43,75,95,59)(44,82,96,54)(45,77,85,49)(46,84,86,56)(47,79,87,51)(48,74,88,58), (1,52,30,74)(2,59,31,81)(3,54,32,76)(4,49,33,83)(5,56,34,78)(6,51,35,73)(7,58,36,80)(8,53,25,75)(9,60,26,82)(10,55,27,77)(11,50,28,84)(12,57,29,79)(13,45,67,91)(14,40,68,86)(15,47,69,93)(16,42,70,88)(17,37,71,95)(18,44,72,90)(19,39,61,85)(20,46,62,92)(21,41,63,87)(22,48,64,94)(23,43,65,89)(24,38,66,96), (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), (1,47,7,41)(2,40,8,46)(3,45,9,39)(4,38,10,44)(5,43,11,37)(6,48,12,42)(13,82,19,76)(14,75,20,81)(15,80,21,74)(16,73,22,79)(17,78,23,84)(18,83,24,77)(25,92,31,86)(26,85,32,91)(27,90,33,96)(28,95,34,89)(29,88,35,94)(30,93,36,87)(49,66,55,72)(50,71,56,65)(51,64,57,70)(52,69,58,63)(53,62,59,68)(54,67,60,61) );
G=PermutationGroup([(1,16,36,64),(2,23,25,71),(3,18,26,66),(4,13,27,61),(5,20,28,68),(6,15,29,63),(7,22,30,70),(8,17,31,65),(9,24,32,72),(10,19,33,67),(11,14,34,62),(12,21,35,69),(37,81,89,53),(38,76,90,60),(39,83,91,55),(40,78,92,50),(41,73,93,57),(42,80,94,52),(43,75,95,59),(44,82,96,54),(45,77,85,49),(46,84,86,56),(47,79,87,51),(48,74,88,58)], [(1,52,30,74),(2,59,31,81),(3,54,32,76),(4,49,33,83),(5,56,34,78),(6,51,35,73),(7,58,36,80),(8,53,25,75),(9,60,26,82),(10,55,27,77),(11,50,28,84),(12,57,29,79),(13,45,67,91),(14,40,68,86),(15,47,69,93),(16,42,70,88),(17,37,71,95),(18,44,72,90),(19,39,61,85),(20,46,62,92),(21,41,63,87),(22,48,64,94),(23,43,65,89),(24,38,66,96)], [(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)], [(1,47,7,41),(2,40,8,46),(3,45,9,39),(4,38,10,44),(5,43,11,37),(6,48,12,42),(13,82,19,76),(14,75,20,81),(15,80,21,74),(16,73,22,79),(17,78,23,84),(18,83,24,77),(25,92,31,86),(26,85,32,91),(27,90,33,96),(28,95,34,89),(29,88,35,94),(30,93,36,87),(49,66,55,72),(50,71,56,65),(51,64,57,70),(52,69,58,63),(53,62,59,68),(54,67,60,61)])
Matrix representation ►G ⊆ 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] >;
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 |
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