metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.126D6, C6.102- (1+4), C6.1092+ (1+4), (Q8×C12)⋊9C2, Q8⋊10(C4×S3), (C4×Q8)⋊11S3, D12⋊15(C2×C4), (C4×D12)⋊37C2, C4⋊C4.325D6, Q8⋊3S3⋊5C4, (Q8×Dic3)⋊9C2, C2.4(D4○D12), C6.27(C23×C4), (C2×Q8).226D6, Dic3⋊5D4⋊17C2, C42⋊2S3⋊16C2, C12.37(C22×C4), (C2×C6).118C24, D6.11(C22×C4), (C4×C12).170C22, (C2×C12).497C23, D6⋊C4.163C22, C22.37(S3×C23), (C6×Q8).218C22, (C2×D12).262C22, C4⋊Dic3.368C22, C2.3(Q8.15D6), Dic3.20(C22×C4), (C4×Dic3).85C22, Dic3⋊C4.138C22, (C22×S3).177C23, C3⋊4(C23.33C23), (C2×Dic3).214C23, C4.37(S3×C2×C4), (S3×C4⋊C4)⋊17C2, (C4×S3)⋊5(C2×C4), (C3×Q8)⋊12(C2×C4), C2.29(S3×C22×C4), (S3×C2×C4).70C22, (C2×Q8⋊3S3).6C2, (C3×C4⋊C4).346C22, (C2×C4).654(C22×S3), SmallGroup(192,1133)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 664 in 294 conjugacy classes, 151 normal (22 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×6], C4 [×10], C22, C22 [×12], S3 [×6], C6 [×3], C2×C4, C2×C4 [×6], C2×C4 [×23], D4 [×12], Q8 [×4], C23 [×3], Dic3 [×2], Dic3 [×4], C12 [×6], C12 [×4], D6 [×6], D6 [×6], C2×C6, C42 [×3], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×7], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C4×S3 [×12], C4×S3 [×6], D12 [×12], C2×Dic3 [×2], C2×Dic3 [×3], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3 [×3], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8, C4×Q8, C2×C4○D4, C4×Dic3 [×3], Dic3⋊C4, Dic3⋊C4 [×3], C4⋊Dic3 [×3], D6⋊C4 [×6], C4×C12 [×3], C3×C4⋊C4 [×3], S3×C2×C4 [×9], C2×D12 [×3], Q8⋊3S3 [×8], C6×Q8, C23.33C23, C42⋊2S3 [×3], C4×D12 [×3], S3×C4⋊C4 [×3], Dic3⋊5D4 [×3], Q8×Dic3, Q8×C12, C2×Q8⋊3S3, C42.126D6
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, Q8.15D6, D4○D12, C42.126D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c5 >
►On 96 points
(1 27 61 89)(2 90 62 28)(3 29 63 91)(4 92 64 30)(5 31 65 93)(6 94 66 32)(7 33 67 95)(8 96 68 34)(9 35 69 85)(10 86 70 36)(11 25 71 87)(12 88 72 26)(13 47 49 78)(14 79 50 48)(15 37 51 80)(16 81 52 38)(17 39 53 82)(18 83 54 40)(19 41 55 84)(20 73 56 42)(21 43 57 74)(22 75 58 44)(23 45 59 76)(24 77 60 46)
(1 20 67 50)(2 21 68 51)(3 22 69 52)(4 23 70 53)(5 24 71 54)(6 13 72 55)(7 14 61 56)(8 15 62 57)(9 16 63 58)(10 17 64 59)(11 18 65 60)(12 19 66 49)(25 83 93 46)(26 84 94 47)(27 73 95 48)(28 74 96 37)(29 75 85 38)(30 76 86 39)(31 77 87 40)(32 78 88 41)(33 79 89 42)(34 80 90 43)(35 81 91 44)(36 82 92 45)
(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 58 7 52)(2 51 8 57)(3 56 9 50)(4 49 10 55)(5 54 11 60)(6 59 12 53)(13 70 19 64)(14 63 20 69)(15 68 21 62)(16 61 22 67)(17 66 23 72)(18 71 24 65)(25 77 31 83)(26 82 32 76)(27 75 33 81)(28 80 34 74)(29 73 35 79)(30 78 36 84)(37 96 43 90)(38 89 44 95)(39 94 45 88)(40 87 46 93)(41 92 47 86)(42 85 48 91)
G:=sub<Sym(96)| (1,27,61,89)(2,90,62,28)(3,29,63,91)(4,92,64,30)(5,31,65,93)(6,94,66,32)(7,33,67,95)(8,96,68,34)(9,35,69,85)(10,86,70,36)(11,25,71,87)(12,88,72,26)(13,47,49,78)(14,79,50,48)(15,37,51,80)(16,81,52,38)(17,39,53,82)(18,83,54,40)(19,41,55,84)(20,73,56,42)(21,43,57,74)(22,75,58,44)(23,45,59,76)(24,77,60,46), (1,20,67,50)(2,21,68,51)(3,22,69,52)(4,23,70,53)(5,24,71,54)(6,13,72,55)(7,14,61,56)(8,15,62,57)(9,16,63,58)(10,17,64,59)(11,18,65,60)(12,19,66,49)(25,83,93,46)(26,84,94,47)(27,73,95,48)(28,74,96,37)(29,75,85,38)(30,76,86,39)(31,77,87,40)(32,78,88,41)(33,79,89,42)(34,80,90,43)(35,81,91,44)(36,82,92,45), (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), (1,58,7,52)(2,51,8,57)(3,56,9,50)(4,49,10,55)(5,54,11,60)(6,59,12,53)(13,70,19,64)(14,63,20,69)(15,68,21,62)(16,61,22,67)(17,66,23,72)(18,71,24,65)(25,77,31,83)(26,82,32,76)(27,75,33,81)(28,80,34,74)(29,73,35,79)(30,78,36,84)(37,96,43,90)(38,89,44,95)(39,94,45,88)(40,87,46,93)(41,92,47,86)(42,85,48,91)>;
G:=Group( (1,27,61,89)(2,90,62,28)(3,29,63,91)(4,92,64,30)(5,31,65,93)(6,94,66,32)(7,33,67,95)(8,96,68,34)(9,35,69,85)(10,86,70,36)(11,25,71,87)(12,88,72,26)(13,47,49,78)(14,79,50,48)(15,37,51,80)(16,81,52,38)(17,39,53,82)(18,83,54,40)(19,41,55,84)(20,73,56,42)(21,43,57,74)(22,75,58,44)(23,45,59,76)(24,77,60,46), (1,20,67,50)(2,21,68,51)(3,22,69,52)(4,23,70,53)(5,24,71,54)(6,13,72,55)(7,14,61,56)(8,15,62,57)(9,16,63,58)(10,17,64,59)(11,18,65,60)(12,19,66,49)(25,83,93,46)(26,84,94,47)(27,73,95,48)(28,74,96,37)(29,75,85,38)(30,76,86,39)(31,77,87,40)(32,78,88,41)(33,79,89,42)(34,80,90,43)(35,81,91,44)(36,82,92,45), (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), (1,58,7,52)(2,51,8,57)(3,56,9,50)(4,49,10,55)(5,54,11,60)(6,59,12,53)(13,70,19,64)(14,63,20,69)(15,68,21,62)(16,61,22,67)(17,66,23,72)(18,71,24,65)(25,77,31,83)(26,82,32,76)(27,75,33,81)(28,80,34,74)(29,73,35,79)(30,78,36,84)(37,96,43,90)(38,89,44,95)(39,94,45,88)(40,87,46,93)(41,92,47,86)(42,85,48,91) );
G=PermutationGroup([(1,27,61,89),(2,90,62,28),(3,29,63,91),(4,92,64,30),(5,31,65,93),(6,94,66,32),(7,33,67,95),(8,96,68,34),(9,35,69,85),(10,86,70,36),(11,25,71,87),(12,88,72,26),(13,47,49,78),(14,79,50,48),(15,37,51,80),(16,81,52,38),(17,39,53,82),(18,83,54,40),(19,41,55,84),(20,73,56,42),(21,43,57,74),(22,75,58,44),(23,45,59,76),(24,77,60,46)], [(1,20,67,50),(2,21,68,51),(3,22,69,52),(4,23,70,53),(5,24,71,54),(6,13,72,55),(7,14,61,56),(8,15,62,57),(9,16,63,58),(10,17,64,59),(11,18,65,60),(12,19,66,49),(25,83,93,46),(26,84,94,47),(27,73,95,48),(28,74,96,37),(29,75,85,38),(30,76,86,39),(31,77,87,40),(32,78,88,41),(33,79,89,42),(34,80,90,43),(35,81,91,44),(36,82,92,45)], [(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)], [(1,58,7,52),(2,51,8,57),(3,56,9,50),(4,49,10,55),(5,54,11,60),(6,59,12,53),(13,70,19,64),(14,63,20,69),(15,68,21,62),(16,61,22,67),(17,66,23,72),(18,71,24,65),(25,77,31,83),(26,82,32,76),(27,75,33,81),(28,80,34,74),(29,73,35,79),(30,78,36,84),(37,96,43,90),(38,89,44,95),(39,94,45,88),(40,87,46,93),(41,92,47,86),(42,85,48,91)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 5 | 3 |
| 0 | 0 | 12 | 7 | 10 | 8 |
| 0 | 0 | 5 | 3 | 7 | 12 |
| 0 | 0 | 10 | 8 | 1 | 6 |
| 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 |
| 5 | 5 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 2 | 2 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,6,12,5,10,0,0,1,7,3,8,0,0,5,10,7,1,0,0,3,8,12,6],[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],[5,8,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,1,0,0,0,0,12,12,0,0],[5,8,0,0,0,0,0,8,0,0,0,0,0,0,11,2,0,0,0,0,4,2,0,0,0,0,0,0,11,2,0,0,0,0,4,2] >;
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3 | 4A | ··· | 4N | 4O | ··· | 4X | 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | D6 | C4×S3 | 2+ (1+4) | 2- (1+4) | Q8.15D6 | D4○D12 |
| kernel | C42.126D6 | C42⋊2S3 | C4×D12 | S3×C4⋊C4 | Dic3⋊5D4 | Q8×Dic3 | Q8×C12 | C2×Q8⋊3S3 | Q8⋊3S3 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | Q8 | C6 | C6 | C2 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 16 | 1 | 3 | 3 | 1 | 8 | 1 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{126}D_6 % in TeX
G:=Group("C4^2.126D6"); // GroupNames label
G:=SmallGroup(192,1133);
// by ID
G=gap.SmallGroup(192,1133);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,675,136,6278]);
// 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*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations