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, (C4×D12)⋊37C2, D12⋊15(C2×C4), 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, (C2×C6).118C24, C12.37(C22×C4), 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
Generators and relations for C42.126D6
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 >
Subgroups: 664 in 294 conjugacy classes, 151 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, S3×C2×C4, C2×D12, Q8⋊3S3, C6×Q8, C23.33C23, C42⋊2S3, C4×D12, S3×C4⋊C4, Dic3⋊5D4, Q8×Dic3, Q8×C12, C2×Q8⋊3S3, C42.126D6
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, Q8.15D6, D4○D12, C42.126D6
►On 96 points
(1 78 59 22)(2 23 60 79)(3 80 49 24)(4 13 50 81)(5 82 51 14)(6 15 52 83)(7 84 53 16)(8 17 54 73)(9 74 55 18)(10 19 56 75)(11 76 57 20)(12 21 58 77)(25 61 95 39)(26 40 96 62)(27 63 85 41)(28 42 86 64)(29 65 87 43)(30 44 88 66)(31 67 89 45)(32 46 90 68)(33 69 91 47)(34 48 92 70)(35 71 93 37)(36 38 94 72)
(1 96 53 32)(2 85 54 33)(3 86 55 34)(4 87 56 35)(5 88 57 36)(6 89 58 25)(7 90 59 26)(8 91 60 27)(9 92 49 28)(10 93 50 29)(11 94 51 30)(12 95 52 31)(13 43 75 71)(14 44 76 72)(15 45 77 61)(16 46 78 62)(17 47 79 63)(18 48 80 64)(19 37 81 65)(20 38 82 66)(21 39 83 67)(22 40 84 68)(23 41 73 69)(24 42 74 70)
(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 28 7 34)(2 33 8 27)(3 26 9 32)(4 31 10 25)(5 36 11 30)(6 29 12 35)(13 45 19 39)(14 38 20 44)(15 43 21 37)(16 48 22 42)(17 41 23 47)(18 46 24 40)(49 96 55 90)(50 89 56 95)(51 94 57 88)(52 87 58 93)(53 92 59 86)(54 85 60 91)(61 81 67 75)(62 74 68 80)(63 79 69 73)(64 84 70 78)(65 77 71 83)(66 82 72 76)
G:=sub<Sym(96)| (1,78,59,22)(2,23,60,79)(3,80,49,24)(4,13,50,81)(5,82,51,14)(6,15,52,83)(7,84,53,16)(8,17,54,73)(9,74,55,18)(10,19,56,75)(11,76,57,20)(12,21,58,77)(25,61,95,39)(26,40,96,62)(27,63,85,41)(28,42,86,64)(29,65,87,43)(30,44,88,66)(31,67,89,45)(32,46,90,68)(33,69,91,47)(34,48,92,70)(35,71,93,37)(36,38,94,72), (1,96,53,32)(2,85,54,33)(3,86,55,34)(4,87,56,35)(5,88,57,36)(6,89,58,25)(7,90,59,26)(8,91,60,27)(9,92,49,28)(10,93,50,29)(11,94,51,30)(12,95,52,31)(13,43,75,71)(14,44,76,72)(15,45,77,61)(16,46,78,62)(17,47,79,63)(18,48,80,64)(19,37,81,65)(20,38,82,66)(21,39,83,67)(22,40,84,68)(23,41,73,69)(24,42,74,70), (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), (1,28,7,34)(2,33,8,27)(3,26,9,32)(4,31,10,25)(5,36,11,30)(6,29,12,35)(13,45,19,39)(14,38,20,44)(15,43,21,37)(16,48,22,42)(17,41,23,47)(18,46,24,40)(49,96,55,90)(50,89,56,95)(51,94,57,88)(52,87,58,93)(53,92,59,86)(54,85,60,91)(61,81,67,75)(62,74,68,80)(63,79,69,73)(64,84,70,78)(65,77,71,83)(66,82,72,76)>;
G:=Group( (1,78,59,22)(2,23,60,79)(3,80,49,24)(4,13,50,81)(5,82,51,14)(6,15,52,83)(7,84,53,16)(8,17,54,73)(9,74,55,18)(10,19,56,75)(11,76,57,20)(12,21,58,77)(25,61,95,39)(26,40,96,62)(27,63,85,41)(28,42,86,64)(29,65,87,43)(30,44,88,66)(31,67,89,45)(32,46,90,68)(33,69,91,47)(34,48,92,70)(35,71,93,37)(36,38,94,72), (1,96,53,32)(2,85,54,33)(3,86,55,34)(4,87,56,35)(5,88,57,36)(6,89,58,25)(7,90,59,26)(8,91,60,27)(9,92,49,28)(10,93,50,29)(11,94,51,30)(12,95,52,31)(13,43,75,71)(14,44,76,72)(15,45,77,61)(16,46,78,62)(17,47,79,63)(18,48,80,64)(19,37,81,65)(20,38,82,66)(21,39,83,67)(22,40,84,68)(23,41,73,69)(24,42,74,70), (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), (1,28,7,34)(2,33,8,27)(3,26,9,32)(4,31,10,25)(5,36,11,30)(6,29,12,35)(13,45,19,39)(14,38,20,44)(15,43,21,37)(16,48,22,42)(17,41,23,47)(18,46,24,40)(49,96,55,90)(50,89,56,95)(51,94,57,88)(52,87,58,93)(53,92,59,86)(54,85,60,91)(61,81,67,75)(62,74,68,80)(63,79,69,73)(64,84,70,78)(65,77,71,83)(66,82,72,76) );
G=PermutationGroup([[(1,78,59,22),(2,23,60,79),(3,80,49,24),(4,13,50,81),(5,82,51,14),(6,15,52,83),(7,84,53,16),(8,17,54,73),(9,74,55,18),(10,19,56,75),(11,76,57,20),(12,21,58,77),(25,61,95,39),(26,40,96,62),(27,63,85,41),(28,42,86,64),(29,65,87,43),(30,44,88,66),(31,67,89,45),(32,46,90,68),(33,69,91,47),(34,48,92,70),(35,71,93,37),(36,38,94,72)], [(1,96,53,32),(2,85,54,33),(3,86,55,34),(4,87,56,35),(5,88,57,36),(6,89,58,25),(7,90,59,26),(8,91,60,27),(9,92,49,28),(10,93,50,29),(11,94,51,30),(12,95,52,31),(13,43,75,71),(14,44,76,72),(15,45,77,61),(16,46,78,62),(17,47,79,63),(18,48,80,64),(19,37,81,65),(20,38,82,66),(21,39,83,67),(22,40,84,68),(23,41,73,69),(24,42,74,70)], [(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)], [(1,28,7,34),(2,33,8,27),(3,26,9,32),(4,31,10,25),(5,36,11,30),(6,29,12,35),(13,45,19,39),(14,38,20,44),(15,43,21,37),(16,48,22,42),(17,41,23,47),(18,46,24,40),(49,96,55,90),(50,89,56,95),(51,94,57,88),(52,87,58,93),(53,92,59,86),(54,85,60,91),(61,81,67,75),(62,74,68,80),(63,79,69,73),(64,84,70,78),(65,77,71,83),(66,82,72,76)]])
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 |
Matrix representation of C42.126D6 ►in 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] >;
C42.126D6 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