metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.136D6, C6.1132+ 1+4, (C4×Q8)⋊22S3, (C4×D12)⋊42C2, C4⋊C4.303D6, (Q8×C12)⋊20C2, (C4×Dic6)⋊42C2, (C2×Q8).208D6, Dic3⋊5D4⋊18C2, C42⋊7S3⋊21C2, D6.D4⋊10C2, C4.19(C4○D12), C2.25(D4○D12), C4⋊D12.10C2, (C2×C6).129C24, C12.123(C4○D4), C12.23D4⋊10C2, (C2×C12).592C23, (C4×C12).181C22, D6⋊C4.145C22, C4.51(Q8⋊3S3), (C2×D12).29C22, (C6×Q8).229C22, (C22×S3).51C23, C4⋊Dic3.401C22, C22.150(S3×C23), (C2×Dic3).59C23, (C4×Dic3).88C22, Dic3⋊C4.116C22, C3⋊2(C22.53C24), (C2×Dic6).244C22, C6.58(C2×C4○D4), C2.68(C2×C4○D12), (S3×C2×C4).207C22, C2.14(C2×Q8⋊3S3), (C3×C4⋊C4).357C22, (C2×C4).291(C22×S3), SmallGroup(192,1144)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.136D6
G = < a,b,c,d | a4=b4=d2=1, c6=a2, ab=ba, cac-1=dad=a-1b2, bc=cb, dbd=b-1, dcd=a2c5 >
Subgroups: 648 in 236 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C4×D4, C4×Q8, C4×Q8, C22.D4, C4.4D4, C4⋊1D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C6×Q8, C22.53C24, C4×Dic6, C4×D12, C4⋊D12, C42⋊7S3, Dic3⋊5D4, D6.D4, C12.23D4, Q8×C12, C42.136D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, C4○D12, Q8⋊3S3, S3×C23, C22.53C24, C2×C4○D12, C2×Q8⋊3S3, D4○D12, C42.136D6
►On 96 points
(1 35 7 29)(2 65 8 71)(3 25 9 31)(4 67 10 61)(5 27 11 33)(6 69 12 63)(13 77 19 83)(14 92 20 86)(15 79 21 73)(16 94 22 88)(17 81 23 75)(18 96 24 90)(26 44 32 38)(28 46 34 40)(30 48 36 42)(37 66 43 72)(39 68 45 62)(41 70 47 64)(49 76 55 82)(50 91 56 85)(51 78 57 84)(52 93 58 87)(53 80 59 74)(54 95 60 89)
(1 13 41 56)(2 14 42 57)(3 15 43 58)(4 16 44 59)(5 17 45 60)(6 18 46 49)(7 19 47 50)(8 20 48 51)(9 21 37 52)(10 22 38 53)(11 23 39 54)(12 24 40 55)(25 79 72 87)(26 80 61 88)(27 81 62 89)(28 82 63 90)(29 83 64 91)(30 84 65 92)(31 73 66 93)(32 74 67 94)(33 75 68 95)(34 76 69 96)(35 77 70 85)(36 78 71 86)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 52)(14 51)(15 50)(16 49)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 72)(36 71)(37 41)(38 40)(42 48)(43 47)(44 46)(73 83)(74 82)(75 81)(76 80)(77 79)(85 87)(88 96)(89 95)(90 94)(91 93)
G:=sub<Sym(96)| (1,35,7,29)(2,65,8,71)(3,25,9,31)(4,67,10,61)(5,27,11,33)(6,69,12,63)(13,77,19,83)(14,92,20,86)(15,79,21,73)(16,94,22,88)(17,81,23,75)(18,96,24,90)(26,44,32,38)(28,46,34,40)(30,48,36,42)(37,66,43,72)(39,68,45,62)(41,70,47,64)(49,76,55,82)(50,91,56,85)(51,78,57,84)(52,93,58,87)(53,80,59,74)(54,95,60,89), (1,13,41,56)(2,14,42,57)(3,15,43,58)(4,16,44,59)(5,17,45,60)(6,18,46,49)(7,19,47,50)(8,20,48,51)(9,21,37,52)(10,22,38,53)(11,23,39,54)(12,24,40,55)(25,79,72,87)(26,80,61,88)(27,81,62,89)(28,82,63,90)(29,83,64,91)(30,84,65,92)(31,73,66,93)(32,74,67,94)(33,75,68,95)(34,76,69,96)(35,77,70,85)(36,78,71,86), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,52)(14,51)(15,50)(16,49)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,72)(36,71)(37,41)(38,40)(42,48)(43,47)(44,46)(73,83)(74,82)(75,81)(76,80)(77,79)(85,87)(88,96)(89,95)(90,94)(91,93)>;
G:=Group( (1,35,7,29)(2,65,8,71)(3,25,9,31)(4,67,10,61)(5,27,11,33)(6,69,12,63)(13,77,19,83)(14,92,20,86)(15,79,21,73)(16,94,22,88)(17,81,23,75)(18,96,24,90)(26,44,32,38)(28,46,34,40)(30,48,36,42)(37,66,43,72)(39,68,45,62)(41,70,47,64)(49,76,55,82)(50,91,56,85)(51,78,57,84)(52,93,58,87)(53,80,59,74)(54,95,60,89), (1,13,41,56)(2,14,42,57)(3,15,43,58)(4,16,44,59)(5,17,45,60)(6,18,46,49)(7,19,47,50)(8,20,48,51)(9,21,37,52)(10,22,38,53)(11,23,39,54)(12,24,40,55)(25,79,72,87)(26,80,61,88)(27,81,62,89)(28,82,63,90)(29,83,64,91)(30,84,65,92)(31,73,66,93)(32,74,67,94)(33,75,68,95)(34,76,69,96)(35,77,70,85)(36,78,71,86), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,52)(14,51)(15,50)(16,49)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,72)(36,71)(37,41)(38,40)(42,48)(43,47)(44,46)(73,83)(74,82)(75,81)(76,80)(77,79)(85,87)(88,96)(89,95)(90,94)(91,93) );
G=PermutationGroup([[(1,35,7,29),(2,65,8,71),(3,25,9,31),(4,67,10,61),(5,27,11,33),(6,69,12,63),(13,77,19,83),(14,92,20,86),(15,79,21,73),(16,94,22,88),(17,81,23,75),(18,96,24,90),(26,44,32,38),(28,46,34,40),(30,48,36,42),(37,66,43,72),(39,68,45,62),(41,70,47,64),(49,76,55,82),(50,91,56,85),(51,78,57,84),(52,93,58,87),(53,80,59,74),(54,95,60,89)], [(1,13,41,56),(2,14,42,57),(3,15,43,58),(4,16,44,59),(5,17,45,60),(6,18,46,49),(7,19,47,50),(8,20,48,51),(9,21,37,52),(10,22,38,53),(11,23,39,54),(12,24,40,55),(25,79,72,87),(26,80,61,88),(27,81,62,89),(28,82,63,90),(29,83,64,91),(30,84,65,92),(31,73,66,93),(32,74,67,94),(33,75,68,95),(34,76,69,96),(35,77,70,85),(36,78,71,86)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,12),(13,52),(14,51),(15,50),(16,49),(17,60),(18,59),(19,58),(20,57),(21,56),(22,55),(23,54),(24,53),(25,70),(26,69),(27,68),(28,67),(29,66),(30,65),(31,64),(32,63),(33,62),(34,61),(35,72),(36,71),(37,41),(38,40),(42,48),(43,47),(44,46),(73,83),(74,82),(75,81),(76,80),(77,79),(85,87),(88,96),(89,95),(90,94),(91,93)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 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 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ 1+4 | Q8⋊3S3 | D4○D12 |
| kernel | C42.136D6 | C4×Dic6 | C4×D12 | C4⋊D12 | C42⋊7S3 | Dic3⋊5D4 | D6.D4 | C12.23D4 | Q8×C12 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 3 | 3 | 1 | 8 | 8 | 1 | 2 | 2 |
Matrix representation of C42.136D6 ►in GL4(𝔽13) generated by
| 5 | 0 | 0 | 0 |
| 6 | 8 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 3 | 6 |
| 0 | 0 | 7 | 10 |
| 1 | 7 | 0 | 0 |
| 9 | 12 | 0 | 0 |
| 0 | 0 | 3 | 10 |
| 0 | 0 | 3 | 6 |
| 1 | 7 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [5,6,0,0,0,8,0,0,0,0,5,0,0,0,0,5],[12,0,0,0,0,12,0,0,0,0,3,7,0,0,6,10],[1,9,0,0,7,12,0,0,0,0,3,3,0,0,10,6],[1,0,0,0,7,12,0,0,0,0,1,0,0,0,1,12] >;
C42.136D6 in GAP, Magma, Sage, TeX
C_4^2._{136}D_6 % in TeX
G:=Group("C4^2.136D6"); // GroupNames label
G:=SmallGroup(192,1144);
// by ID
G=gap.SmallGroup(192,1144);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,184,1571,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^6=a^2,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^2,b*c=c*b,d*b*d=b^-1,d*c*d=a^2*c^5>;
// generators/relations