metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.156D6, C6.972- 1+4, C4⋊C4.211D6, C12⋊2Q8⋊33C2, C42.C2⋊12S3, C42⋊2S3.7C2, (C2×C12).92C23, (C2×C6).242C24, C4.D12.12C2, C2.60(Q8○D12), C4.Dic6⋊37C2, Dic6⋊C4⋊38C2, C12.132(C4○D4), (C4×C12).201C22, D6⋊C4.113C22, C4.21(Q8⋊3S3), C4⋊Dic3.244C22, C22.263(S3×C23), Dic3⋊C4.125C22, (C22×S3).107C23, C3⋊5(C22.35C24), (C2×Dic6).182C22, (C2×Dic3).262C23, (C4×Dic3).147C22, C4⋊C4⋊S3.3C2, C6.119(C2×C4○D4), (S3×C2×C4).132C22, C2.26(C2×Q8⋊3S3), (C3×C42.C2)⋊15C2, (C3×C4⋊C4).197C22, (C2×C4).206(C22×S3), SmallGroup(192,1257)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.156D6
G = < a,b,c,d | a4=b4=1, c6=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=a2b2c5 >
Subgroups: 416 in 192 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22.35C24, C12⋊2Q8, C42⋊2S3, Dic6⋊C4, C4.Dic6, C4.D12, C4⋊C4⋊S3, C3×C42.C2, C42.156D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, Q8⋊3S3, S3×C23, C22.35C24, C2×Q8⋊3S3, Q8○D12, C42.156D6
►On 96 points
Generators in S96
(1 92 59 31)(2 32 60 93)(3 94 49 33)(4 34 50 95)(5 96 51 35)(6 36 52 85)(7 86 53 25)(8 26 54 87)(9 88 55 27)(10 28 56 89)(11 90 57 29)(12 30 58 91)(13 79 61 42)(14 43 62 80)(15 81 63 44)(16 45 64 82)(17 83 65 46)(18 47 66 84)(19 73 67 48)(20 37 68 74)(21 75 69 38)(22 39 70 76)(23 77 71 40)(24 41 72 78)
(1 47 7 41)(2 79 8 73)(3 37 9 43)(4 81 10 75)(5 39 11 45)(6 83 12 77)(13 87 19 93)(14 33 20 27)(15 89 21 95)(16 35 22 29)(17 91 23 85)(18 25 24 31)(26 67 32 61)(28 69 34 63)(30 71 36 65)(38 50 44 56)(40 52 46 58)(42 54 48 60)(49 74 55 80)(51 76 57 82)(53 78 59 84)(62 94 68 88)(64 96 70 90)(66 86 72 92)
(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 59 52)(2 51 60 5)(3 4 49 50)(7 12 53 58)(8 57 54 11)(9 10 55 56)(13 16 61 64)(14 63 62 15)(17 24 65 72)(18 71 66 23)(19 22 67 70)(20 69 68 21)(25 30 86 91)(26 90 87 29)(27 28 88 89)(31 36 92 85)(32 96 93 35)(33 34 94 95)(37 75 74 38)(39 73 76 48)(40 47 77 84)(41 83 78 46)(42 45 79 82)(43 81 80 44)
G:=sub<Sym(96)| (1,92,59,31)(2,32,60,93)(3,94,49,33)(4,34,50,95)(5,96,51,35)(6,36,52,85)(7,86,53,25)(8,26,54,87)(9,88,55,27)(10,28,56,89)(11,90,57,29)(12,30,58,91)(13,79,61,42)(14,43,62,80)(15,81,63,44)(16,45,64,82)(17,83,65,46)(18,47,66,84)(19,73,67,48)(20,37,68,74)(21,75,69,38)(22,39,70,76)(23,77,71,40)(24,41,72,78), (1,47,7,41)(2,79,8,73)(3,37,9,43)(4,81,10,75)(5,39,11,45)(6,83,12,77)(13,87,19,93)(14,33,20,27)(15,89,21,95)(16,35,22,29)(17,91,23,85)(18,25,24,31)(26,67,32,61)(28,69,34,63)(30,71,36,65)(38,50,44,56)(40,52,46,58)(42,54,48,60)(49,74,55,80)(51,76,57,82)(53,78,59,84)(62,94,68,88)(64,96,70,90)(66,86,72,92), (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), (1,6,59,52)(2,51,60,5)(3,4,49,50)(7,12,53,58)(8,57,54,11)(9,10,55,56)(13,16,61,64)(14,63,62,15)(17,24,65,72)(18,71,66,23)(19,22,67,70)(20,69,68,21)(25,30,86,91)(26,90,87,29)(27,28,88,89)(31,36,92,85)(32,96,93,35)(33,34,94,95)(37,75,74,38)(39,73,76,48)(40,47,77,84)(41,83,78,46)(42,45,79,82)(43,81,80,44)>;
G:=Group( (1,92,59,31)(2,32,60,93)(3,94,49,33)(4,34,50,95)(5,96,51,35)(6,36,52,85)(7,86,53,25)(8,26,54,87)(9,88,55,27)(10,28,56,89)(11,90,57,29)(12,30,58,91)(13,79,61,42)(14,43,62,80)(15,81,63,44)(16,45,64,82)(17,83,65,46)(18,47,66,84)(19,73,67,48)(20,37,68,74)(21,75,69,38)(22,39,70,76)(23,77,71,40)(24,41,72,78), (1,47,7,41)(2,79,8,73)(3,37,9,43)(4,81,10,75)(5,39,11,45)(6,83,12,77)(13,87,19,93)(14,33,20,27)(15,89,21,95)(16,35,22,29)(17,91,23,85)(18,25,24,31)(26,67,32,61)(28,69,34,63)(30,71,36,65)(38,50,44,56)(40,52,46,58)(42,54,48,60)(49,74,55,80)(51,76,57,82)(53,78,59,84)(62,94,68,88)(64,96,70,90)(66,86,72,92), (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), (1,6,59,52)(2,51,60,5)(3,4,49,50)(7,12,53,58)(8,57,54,11)(9,10,55,56)(13,16,61,64)(14,63,62,15)(17,24,65,72)(18,71,66,23)(19,22,67,70)(20,69,68,21)(25,30,86,91)(26,90,87,29)(27,28,88,89)(31,36,92,85)(32,96,93,35)(33,34,94,95)(37,75,74,38)(39,73,76,48)(40,47,77,84)(41,83,78,46)(42,45,79,82)(43,81,80,44) );
G=PermutationGroup([[(1,92,59,31),(2,32,60,93),(3,94,49,33),(4,34,50,95),(5,96,51,35),(6,36,52,85),(7,86,53,25),(8,26,54,87),(9,88,55,27),(10,28,56,89),(11,90,57,29),(12,30,58,91),(13,79,61,42),(14,43,62,80),(15,81,63,44),(16,45,64,82),(17,83,65,46),(18,47,66,84),(19,73,67,48),(20,37,68,74),(21,75,69,38),(22,39,70,76),(23,77,71,40),(24,41,72,78)], [(1,47,7,41),(2,79,8,73),(3,37,9,43),(4,81,10,75),(5,39,11,45),(6,83,12,77),(13,87,19,93),(14,33,20,27),(15,89,21,95),(16,35,22,29),(17,91,23,85),(18,25,24,31),(26,67,32,61),(28,69,34,63),(30,71,36,65),(38,50,44,56),(40,52,46,58),(42,54,48,60),(49,74,55,80),(51,76,57,82),(53,78,59,84),(62,94,68,88),(64,96,70,90),(66,86,72,92)], [(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)], [(1,6,59,52),(2,51,60,5),(3,4,49,50),(7,12,53,58),(8,57,54,11),(9,10,55,56),(13,16,61,64),(14,63,62,15),(17,24,65,72),(18,71,66,23),(19,22,67,70),(20,69,68,21),(25,30,86,91),(26,90,87,29),(27,28,88,89),(31,36,92,85),(32,96,93,35),(33,34,94,95),(37,75,74,38),(39,73,76,48),(40,47,77,84),(41,83,78,46),(42,45,79,82),(43,81,80,44)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4Q | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | 2- 1+4 | Q8⋊3S3 | Q8○D12 |
| kernel | C42.156D6 | C12⋊2Q8 | C42⋊2S3 | Dic6⋊C4 | C4.Dic6 | C4.D12 | C4⋊C4⋊S3 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 1 | 1 | 1 | 6 | 4 | 2 | 2 | 4 |
Matrix representation of C42.156D6 ►in GL8(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 6 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 11 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 | 7 | 12 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 8 | 2 |
| 0 | 0 | 0 | 0 | 8 | 1 | 0 | 5 |
| 8 | 5 | 6 | 7 | 0 | 0 | 0 | 0 |
| 8 | 3 | 6 | 12 | 0 | 0 | 0 | 0 |
| 6 | 7 | 5 | 8 | 0 | 0 | 0 | 0 |
| 6 | 12 | 5 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 | 0 | 3 |
| 0 | 0 | 0 | 0 | 8 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 8 | 8 | 1 |
| 5 | 8 | 7 | 6 | 0 | 0 | 0 | 0 |
| 3 | 8 | 12 | 6 | 0 | 0 | 0 | 0 |
| 7 | 6 | 8 | 5 | 0 | 0 | 0 | 0 |
| 12 | 6 | 10 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 | 1 | 2 | 10 |
| 0 | 0 | 0 | 0 | 8 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 5 | 0 | 12 |
G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,6,11,1,0,0,0,0,0,1,0,11,0,0,0,0,1,0,12,7,0,0,0,0,0,1,0,12],[0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,8,0,12,8,0,0,0,0,2,5,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,0,2,5],[8,8,6,6,0,0,0,0,5,3,7,12,0,0,0,0,6,6,5,5,0,0,0,0,7,12,8,10,0,0,0,0,0,0,0,0,12,5,8,12,0,0,0,0,0,12,0,8,0,0,0,0,3,0,1,8,0,0,0,0,0,3,0,1],[5,3,7,12,0,0,0,0,8,8,6,6,0,0,0,0,7,12,8,10,0,0,0,0,6,6,5,5,0,0,0,0,0,0,0,0,12,3,8,1,0,0,0,0,0,1,0,5,0,0,0,0,3,2,1,0,0,0,0,0,0,10,0,12] >;
C42.156D6 in GAP, Magma, Sage, TeX
C_4^2._{156}D_6 % in TeX
G:=Group("C4^2.156D6"); // GroupNames label
G:=SmallGroup(192,1257);
// by ID
G=gap.SmallGroup(192,1257);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,268,675,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=a^2*b^2*c^5>;
// generators/relations