metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.125D6, C6.92- (1+4), (S3×Q8)⋊5C4, (C4×Q8)⋊10S3, (Q8×C12)⋊7C2, C4⋊C4.323D6, (Q8×Dic3)⋊8C2, Q8.17(C4×S3), (C4×Dic6)⋊38C2, C6.25(C23×C4), (C2×Q8).224D6, C2.4(Q8○D12), C12.35(C22×C4), (C2×C6).116C24, C42⋊2S3.3C2, Dic6.20(C2×C4), D6.19(C22×C4), Dic6⋊C4⋊18C2, (C4×C12).168C22, (C2×C12).495C23, D6⋊C4.124C22, C22.35(S3×C23), (C6×Q8).216C22, C4⋊Dic3.366C22, C2.2(Q8.15D6), (C4×Dic3).84C22, Dic3.11(C22×C4), Dic3⋊C4.137C22, (C22×S3).175C23, C3⋊2(C23.32C23), (C2×Dic3).212C23, (C2×Dic6).290C22, C4.35(S3×C2×C4), (C2×S3×Q8).6C2, (C4×S3).9(C2×C4), C2.27(S3×C22×C4), (S3×C2×C4).69C22, (C3×Q8).16(C2×C4), C4⋊C4⋊7S3.10C2, (C3×C4⋊C4).344C22, (C2×C4).288(C22×S3), SmallGroup(192,1131)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 504 in 266 conjugacy classes, 151 normal (22 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×6], C4 [×14], C22, C22 [×4], S3 [×2], C6 [×3], C2×C4, C2×C4 [×6], C2×C4 [×19], Q8 [×4], Q8 [×12], C23, Dic3 [×6], Dic3 [×4], C12 [×6], C12 [×4], D6 [×2], D6 [×2], C2×C6, C42 [×3], C42 [×9], C22⋊C4 [×4], C4⋊C4 [×3], C4⋊C4 [×9], C22×C4 [×3], C2×Q8, C2×Q8 [×11], Dic6 [×12], C4×S3 [×12], C2×Dic3, C2×Dic3 [×6], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3, C42⋊C2 [×6], C4×Q8, C4×Q8 [×7], C22×Q8, C4×Dic3 [×9], Dic3⋊C4 [×6], C4⋊Dic3 [×3], D6⋊C4, D6⋊C4 [×3], C4×C12 [×3], C3×C4⋊C4 [×3], C2×Dic6 [×3], S3×C2×C4 [×3], S3×Q8 [×8], C6×Q8, C23.32C23, C4×Dic6 [×3], C42⋊2S3 [×3], Dic6⋊C4 [×3], C4⋊C4⋊7S3 [×3], Q8×Dic3, Q8×C12, C2×S3×Q8, C42.125D6
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], S3×C2×C4 [×6], S3×C23, C23.32C23, S3×C22×C4, Q8.15D6, Q8○D12, C42.125D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c5 >
►On 96 points
(1 37 7 43)(2 44 8 38)(3 39 9 45)(4 46 10 40)(5 41 11 47)(6 48 12 42)(13 84 19 78)(14 79 20 73)(15 74 21 80)(16 81 22 75)(17 76 23 82)(18 83 24 77)(25 92 31 86)(26 87 32 93)(27 94 33 88)(28 89 34 95)(29 96 35 90)(30 91 36 85)(49 67 55 61)(50 62 56 68)(51 69 57 63)(52 64 58 70)(53 71 59 65)(54 66 60 72)
(1 51 92 24)(2 52 93 13)(3 53 94 14)(4 54 95 15)(5 55 96 16)(6 56 85 17)(7 57 86 18)(8 58 87 19)(9 59 88 20)(10 60 89 21)(11 49 90 22)(12 50 91 23)(25 83 43 63)(26 84 44 64)(27 73 45 65)(28 74 46 66)(29 75 47 67)(30 76 48 68)(31 77 37 69)(32 78 38 70)(33 79 39 71)(34 80 40 72)(35 81 41 61)(36 82 42 62)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 16 19 22)(14 21 20 15)(17 24 23 18)(25 30 31 36)(26 35 32 29)(27 28 33 34)(37 42 43 48)(38 47 44 41)(39 40 45 46)(49 52 55 58)(50 57 56 51)(53 60 59 54)(61 64 67 70)(62 69 68 63)(65 72 71 66)(73 80 79 74)(75 78 81 84)(76 83 82 77)(85 86 91 92)(87 96 93 90)(88 89 94 95)
G:=sub<Sym(96)| (1,37,7,43)(2,44,8,38)(3,39,9,45)(4,46,10,40)(5,41,11,47)(6,48,12,42)(13,84,19,78)(14,79,20,73)(15,74,21,80)(16,81,22,75)(17,76,23,82)(18,83,24,77)(25,92,31,86)(26,87,32,93)(27,94,33,88)(28,89,34,95)(29,96,35,90)(30,91,36,85)(49,67,55,61)(50,62,56,68)(51,69,57,63)(52,64,58,70)(53,71,59,65)(54,66,60,72), (1,51,92,24)(2,52,93,13)(3,53,94,14)(4,54,95,15)(5,55,96,16)(6,56,85,17)(7,57,86,18)(8,58,87,19)(9,59,88,20)(10,60,89,21)(11,49,90,22)(12,50,91,23)(25,83,43,63)(26,84,44,64)(27,73,45,65)(28,74,46,66)(29,75,47,67)(30,76,48,68)(31,77,37,69)(32,78,38,70)(33,79,39,71)(34,80,40,72)(35,81,41,61)(36,82,42,62), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,30,31,36)(26,35,32,29)(27,28,33,34)(37,42,43,48)(38,47,44,41)(39,40,45,46)(49,52,55,58)(50,57,56,51)(53,60,59,54)(61,64,67,70)(62,69,68,63)(65,72,71,66)(73,80,79,74)(75,78,81,84)(76,83,82,77)(85,86,91,92)(87,96,93,90)(88,89,94,95)>;
G:=Group( (1,37,7,43)(2,44,8,38)(3,39,9,45)(4,46,10,40)(5,41,11,47)(6,48,12,42)(13,84,19,78)(14,79,20,73)(15,74,21,80)(16,81,22,75)(17,76,23,82)(18,83,24,77)(25,92,31,86)(26,87,32,93)(27,94,33,88)(28,89,34,95)(29,96,35,90)(30,91,36,85)(49,67,55,61)(50,62,56,68)(51,69,57,63)(52,64,58,70)(53,71,59,65)(54,66,60,72), (1,51,92,24)(2,52,93,13)(3,53,94,14)(4,54,95,15)(5,55,96,16)(6,56,85,17)(7,57,86,18)(8,58,87,19)(9,59,88,20)(10,60,89,21)(11,49,90,22)(12,50,91,23)(25,83,43,63)(26,84,44,64)(27,73,45,65)(28,74,46,66)(29,75,47,67)(30,76,48,68)(31,77,37,69)(32,78,38,70)(33,79,39,71)(34,80,40,72)(35,81,41,61)(36,82,42,62), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,30,31,36)(26,35,32,29)(27,28,33,34)(37,42,43,48)(38,47,44,41)(39,40,45,46)(49,52,55,58)(50,57,56,51)(53,60,59,54)(61,64,67,70)(62,69,68,63)(65,72,71,66)(73,80,79,74)(75,78,81,84)(76,83,82,77)(85,86,91,92)(87,96,93,90)(88,89,94,95) );
G=PermutationGroup([(1,37,7,43),(2,44,8,38),(3,39,9,45),(4,46,10,40),(5,41,11,47),(6,48,12,42),(13,84,19,78),(14,79,20,73),(15,74,21,80),(16,81,22,75),(17,76,23,82),(18,83,24,77),(25,92,31,86),(26,87,32,93),(27,94,33,88),(28,89,34,95),(29,96,35,90),(30,91,36,85),(49,67,55,61),(50,62,56,68),(51,69,57,63),(52,64,58,70),(53,71,59,65),(54,66,60,72)], [(1,51,92,24),(2,52,93,13),(3,53,94,14),(4,54,95,15),(5,55,96,16),(6,56,85,17),(7,57,86,18),(8,58,87,19),(9,59,88,20),(10,60,89,21),(11,49,90,22),(12,50,91,23),(25,83,43,63),(26,84,44,64),(27,73,45,65),(28,74,46,66),(29,75,47,67),(30,76,48,68),(31,77,37,69),(32,78,38,70),(33,79,39,71),(34,80,40,72),(35,81,41,61),(36,82,42,62)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,16,19,22),(14,21,20,15),(17,24,23,18),(25,30,31,36),(26,35,32,29),(27,28,33,34),(37,42,43,48),(38,47,44,41),(39,40,45,46),(49,52,55,58),(50,57,56,51),(53,60,59,54),(61,64,67,70),(62,69,68,63),(65,72,71,66),(73,80,79,74),(75,78,81,84),(76,83,82,77),(85,86,91,92),(87,96,93,90),(88,89,94,95)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 12 | 0 |
| 0 | 0 | 0 | 1 | 0 | 12 |
| 0 | 0 | 2 | 0 | 12 | 0 |
| 0 | 0 | 0 | 2 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 9 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 8 |
| 0 | 0 | 8 | 5 | 5 | 8 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 5 | 8 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 8 | 0 |
| 0 | 0 | 5 | 8 | 8 | 5 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 | 5 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,2,0,0,0,0,1,0,2,0,0,12,0,12,0,0,0,0,12,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,11,9,0,0,0,0,4,2,0,0,0,0,0,0,11,9,0,0,0,0,4,2],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,5,5,0,0,0,0,0,5,0,5,0,0,8,8,8,8],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,0,8,0,0,0,0,8,8,8,8,0,0,0,5,0,5] >;
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4N | 4O | ··· | 4AB | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | D6 | C4×S3 | 2- (1+4) | Q8.15D6 | Q8○D12 |
| kernel | C42.125D6 | C4×Dic6 | C42⋊2S3 | Dic6⋊C4 | C4⋊C4⋊7S3 | Q8×Dic3 | Q8×C12 | C2×S3×Q8 | S3×Q8 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | Q8 | C6 | C2 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 16 | 1 | 3 | 3 | 1 | 8 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{125}D_6 % in TeX
G:=Group("C4^2.125D6"); // GroupNames label
G:=SmallGroup(192,1131);
// by ID
G=gap.SmallGroup(192,1131);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,1123,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations