metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.159D6, C6.982- 1+4, C12⋊Q8⋊39C2, C4⋊C4.116D6, C42⋊2C2.S3, (C4×Dic6)⋊13C2, C22⋊C4.39D6, C12.6Q8⋊8C2, Dic3.Q8⋊36C2, (C4×C12).31C22, (C2×C6).245C24, (C2×C12).93C23, C2.62(Q8○D12), C4.Dic6⋊38C2, Dic6⋊C4⋊39C2, C4⋊Dic3.53C22, C23.61(C22×S3), (C22×C6).59C23, C23.8D6.3C2, Dic3.14(C4○D4), C22.266(S3×C23), Dic3.D4.4C2, C23.16D6.3C2, Dic3⋊C4.126C22, C3⋊6(C22.35C24), (C4×Dic3).217C22, (C2×Dic3).127C23, (C2×Dic6).253C22, C6.D4.61C22, (C22×Dic3).148C22, C2.92(S3×C4○D4), C6.203(C2×C4○D4), (C3×C4⋊C4).200C22, (C3×C42⋊2C2).1C2, (C2×C4).302(C22×S3), (C3×C22⋊C4).70C22, SmallGroup(192,1260)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.159D6
G = < a,b,c,d | a4=b4=1, c6=b2, d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c5 >
Subgroups: 400 in 192 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C22×C6, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C42⋊2C2, C42⋊2C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×Dic3, C22.35C24, C4×Dic6, C12.6Q8, C23.16D6, Dic3.D4, C23.8D6, Dic6⋊C4, C12⋊Q8, Dic3.Q8, C4.Dic6, C3×C42⋊2C2, C42.159D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, S3×C23, C22.35C24, S3×C4○D4, Q8○D12, C42.159D6
►On 96 points
Generators in S96
(1 79 87 20)(2 74 88 15)(3 81 89 22)(4 76 90 17)(5 83 91 24)(6 78 92 19)(7 73 93 14)(8 80 94 21)(9 75 95 16)(10 82 96 23)(11 77 85 18)(12 84 86 13)(25 64 55 42)(26 71 56 37)(27 66 57 44)(28 61 58 39)(29 68 59 46)(30 63 60 41)(31 70 49 48)(32 65 50 43)(33 72 51 38)(34 67 52 45)(35 62 53 40)(36 69 54 47)
(1 76 7 82)(2 24 8 18)(3 78 9 84)(4 14 10 20)(5 80 11 74)(6 16 12 22)(13 89 19 95)(15 91 21 85)(17 93 23 87)(25 45 31 39)(26 62 32 68)(27 47 33 41)(28 64 34 70)(29 37 35 43)(30 66 36 72)(38 60 44 54)(40 50 46 56)(42 52 48 58)(49 61 55 67)(51 63 57 69)(53 65 59 71)(73 96 79 90)(75 86 81 92)(77 88 83 94)
(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 27 87 57)(2 32 88 50)(3 25 89 55)(4 30 90 60)(5 35 91 53)(6 28 92 58)(7 33 93 51)(8 26 94 56)(9 31 95 49)(10 36 96 54)(11 29 85 59)(12 34 86 52)(13 39 84 61)(14 44 73 66)(15 37 74 71)(16 42 75 64)(17 47 76 69)(18 40 77 62)(19 45 78 67)(20 38 79 72)(21 43 80 65)(22 48 81 70)(23 41 82 63)(24 46 83 68)
G:=sub<Sym(96)| (1,79,87,20)(2,74,88,15)(3,81,89,22)(4,76,90,17)(5,83,91,24)(6,78,92,19)(7,73,93,14)(8,80,94,21)(9,75,95,16)(10,82,96,23)(11,77,85,18)(12,84,86,13)(25,64,55,42)(26,71,56,37)(27,66,57,44)(28,61,58,39)(29,68,59,46)(30,63,60,41)(31,70,49,48)(32,65,50,43)(33,72,51,38)(34,67,52,45)(35,62,53,40)(36,69,54,47), (1,76,7,82)(2,24,8,18)(3,78,9,84)(4,14,10,20)(5,80,11,74)(6,16,12,22)(13,89,19,95)(15,91,21,85)(17,93,23,87)(25,45,31,39)(26,62,32,68)(27,47,33,41)(28,64,34,70)(29,37,35,43)(30,66,36,72)(38,60,44,54)(40,50,46,56)(42,52,48,58)(49,61,55,67)(51,63,57,69)(53,65,59,71)(73,96,79,90)(75,86,81,92)(77,88,83,94), (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), (1,27,87,57)(2,32,88,50)(3,25,89,55)(4,30,90,60)(5,35,91,53)(6,28,92,58)(7,33,93,51)(8,26,94,56)(9,31,95,49)(10,36,96,54)(11,29,85,59)(12,34,86,52)(13,39,84,61)(14,44,73,66)(15,37,74,71)(16,42,75,64)(17,47,76,69)(18,40,77,62)(19,45,78,67)(20,38,79,72)(21,43,80,65)(22,48,81,70)(23,41,82,63)(24,46,83,68)>;
G:=Group( (1,79,87,20)(2,74,88,15)(3,81,89,22)(4,76,90,17)(5,83,91,24)(6,78,92,19)(7,73,93,14)(8,80,94,21)(9,75,95,16)(10,82,96,23)(11,77,85,18)(12,84,86,13)(25,64,55,42)(26,71,56,37)(27,66,57,44)(28,61,58,39)(29,68,59,46)(30,63,60,41)(31,70,49,48)(32,65,50,43)(33,72,51,38)(34,67,52,45)(35,62,53,40)(36,69,54,47), (1,76,7,82)(2,24,8,18)(3,78,9,84)(4,14,10,20)(5,80,11,74)(6,16,12,22)(13,89,19,95)(15,91,21,85)(17,93,23,87)(25,45,31,39)(26,62,32,68)(27,47,33,41)(28,64,34,70)(29,37,35,43)(30,66,36,72)(38,60,44,54)(40,50,46,56)(42,52,48,58)(49,61,55,67)(51,63,57,69)(53,65,59,71)(73,96,79,90)(75,86,81,92)(77,88,83,94), (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), (1,27,87,57)(2,32,88,50)(3,25,89,55)(4,30,90,60)(5,35,91,53)(6,28,92,58)(7,33,93,51)(8,26,94,56)(9,31,95,49)(10,36,96,54)(11,29,85,59)(12,34,86,52)(13,39,84,61)(14,44,73,66)(15,37,74,71)(16,42,75,64)(17,47,76,69)(18,40,77,62)(19,45,78,67)(20,38,79,72)(21,43,80,65)(22,48,81,70)(23,41,82,63)(24,46,83,68) );
G=PermutationGroup([[(1,79,87,20),(2,74,88,15),(3,81,89,22),(4,76,90,17),(5,83,91,24),(6,78,92,19),(7,73,93,14),(8,80,94,21),(9,75,95,16),(10,82,96,23),(11,77,85,18),(12,84,86,13),(25,64,55,42),(26,71,56,37),(27,66,57,44),(28,61,58,39),(29,68,59,46),(30,63,60,41),(31,70,49,48),(32,65,50,43),(33,72,51,38),(34,67,52,45),(35,62,53,40),(36,69,54,47)], [(1,76,7,82),(2,24,8,18),(3,78,9,84),(4,14,10,20),(5,80,11,74),(6,16,12,22),(13,89,19,95),(15,91,21,85),(17,93,23,87),(25,45,31,39),(26,62,32,68),(27,47,33,41),(28,64,34,70),(29,37,35,43),(30,66,36,72),(38,60,44,54),(40,50,46,56),(42,52,48,58),(49,61,55,67),(51,63,57,69),(53,65,59,71),(73,96,79,90),(75,86,81,92),(77,88,83,94)], [(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)], [(1,27,87,57),(2,32,88,50),(3,25,89,55),(4,30,90,60),(5,35,91,53),(6,28,92,58),(7,33,93,51),(8,26,94,56),(9,31,95,49),(10,36,96,54),(11,29,85,59),(12,34,86,52),(13,39,84,61),(14,44,73,66),(15,37,74,71),(16,42,75,64),(17,47,76,69),(18,40,77,62),(19,45,78,67),(20,38,79,72),(21,43,80,65),(22,48,81,70),(23,41,82,63),(24,46,83,68)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 8 | 4 | ··· | 4 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | 2- 1+4 | S3×C4○D4 | Q8○D12 |
| kernel | C42.159D6 | C4×Dic6 | C12.6Q8 | C23.16D6 | Dic3.D4 | C23.8D6 | Dic6⋊C4 | C12⋊Q8 | Dic3.Q8 | C4.Dic6 | C3×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | Dic3 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 4 | 2 | 2 | 4 |
Matrix representation of C42.159D6 ►in GL8(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 12 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 | 12 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 4 | 3 | 9 |
| 0 | 0 | 0 | 0 | 9 | 9 | 9 | 10 |
G:=sub<GL(8,GF(13))| [5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,11,1,1,12,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,12,0,0,0,0,11,1,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,1,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,2,12,12,1],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6,0,0,0,0,0,0,0,0,6,3,7,9,0,0,0,0,5,7,4,9,0,0,0,0,0,0,3,9,0,0,0,0,0,0,9,10] >;
C42.159D6 in GAP, Magma, Sage, TeX
C_4^2._{159}D_6 % in TeX
G:=Group("C4^2.159D6"); // GroupNames label
G:=SmallGroup(192,1260);
// by ID
G=gap.SmallGroup(192,1260);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,219,268,675,570,80,6278]);
// 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*b^2,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations