metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.140D6, C6.892- 1+4, C6.722+ 1+4, (C2×D4).109D6, (C2×Q8).107D6, C22⋊C4.34D6, C4.4D4.9S3, (C2×C6).216C24, (C2×C12).78C23, C2.50(Q8○D12), C12.6Q8⋊28C2, Dic3⋊Q8⋊23C2, C2.74(D4⋊6D6), (C4×C12).221C22, (C6×D4).209C22, C23.8D6⋊38C2, C4⋊Dic3.50C22, (C22×C6).46C23, C23.48(C22×S3), (C6×Q8).125C22, Dic3.D4⋊39C2, Dic3⋊C4.83C22, C22.237(S3×C23), C23.23D6.6C2, C3⋊3(C22.57C24), (C2×Dic3).111C23, (C4×Dic3).132C22, (C2×Dic6).176C22, C6.D4.53C22, (C22×Dic3).141C22, (C3×C4.4D4).7C2, (C2×C4).192(C22×S3), (C3×C22⋊C4).63C22, SmallGroup(192,1231)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.140D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 448 in 196 conjugacy classes, 91 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C2×Dic6, C22×Dic3, C6×D4, C6×Q8, C22.57C24, C12.6Q8, Dic3.D4, C23.8D6, C23.23D6, Dic3⋊Q8, C3×C4.4D4, C42.140D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, 2- 1+4, S3×C23, C22.57C24, D4⋊6D6, Q8○D12, C42.140D6
►On 96 points
Generators in S96
(1 60 10 28)(2 58 11 26)(3 56 12 30)(4 25 7 57)(5 29 8 55)(6 27 9 59)(13 85 18 81)(14 89 16 79)(15 87 17 83)(19 86 23 82)(20 90 24 80)(21 88 22 84)(31 72 34 74)(32 78 35 70)(33 68 36 76)(37 43 40 62)(38 66 41 47)(39 45 42 64)(44 53 63 50)(46 49 65 52)(48 51 61 54)(67 94 75 91)(69 96 77 93)(71 92 73 95)
(1 44 4 47)(2 48 5 45)(3 46 6 43)(7 66 10 63)(8 64 11 61)(9 62 12 65)(13 73 22 76)(14 77 23 74)(15 75 24 78)(16 69 19 72)(17 67 20 70)(18 71 21 68)(25 38 60 53)(26 54 55 39)(27 40 56 49)(28 50 57 41)(29 42 58 51)(30 52 59 37)(31 89 93 82)(32 83 94 90)(33 85 95 84)(34 79 96 86)(35 87 91 80)(36 81 92 88)
(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 15 4 24)(2 14 5 23)(3 13 6 22)(7 20 10 17)(8 19 11 16)(9 21 12 18)(25 90 60 83)(26 89 55 82)(27 88 56 81)(28 87 57 80)(29 86 58 79)(30 85 59 84)(31 42 93 51)(32 41 94 50)(33 40 95 49)(34 39 96 54)(35 38 91 53)(36 37 92 52)(43 71 46 68)(44 70 47 67)(45 69 48 72)(61 74 64 77)(62 73 65 76)(63 78 66 75)
G:=sub<Sym(96)| (1,60,10,28)(2,58,11,26)(3,56,12,30)(4,25,7,57)(5,29,8,55)(6,27,9,59)(13,85,18,81)(14,89,16,79)(15,87,17,83)(19,86,23,82)(20,90,24,80)(21,88,22,84)(31,72,34,74)(32,78,35,70)(33,68,36,76)(37,43,40,62)(38,66,41,47)(39,45,42,64)(44,53,63,50)(46,49,65,52)(48,51,61,54)(67,94,75,91)(69,96,77,93)(71,92,73,95), (1,44,4,47)(2,48,5,45)(3,46,6,43)(7,66,10,63)(8,64,11,61)(9,62,12,65)(13,73,22,76)(14,77,23,74)(15,75,24,78)(16,69,19,72)(17,67,20,70)(18,71,21,68)(25,38,60,53)(26,54,55,39)(27,40,56,49)(28,50,57,41)(29,42,58,51)(30,52,59,37)(31,89,93,82)(32,83,94,90)(33,85,95,84)(34,79,96,86)(35,87,91,80)(36,81,92,88), (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), (1,15,4,24)(2,14,5,23)(3,13,6,22)(7,20,10,17)(8,19,11,16)(9,21,12,18)(25,90,60,83)(26,89,55,82)(27,88,56,81)(28,87,57,80)(29,86,58,79)(30,85,59,84)(31,42,93,51)(32,41,94,50)(33,40,95,49)(34,39,96,54)(35,38,91,53)(36,37,92,52)(43,71,46,68)(44,70,47,67)(45,69,48,72)(61,74,64,77)(62,73,65,76)(63,78,66,75)>;
G:=Group( (1,60,10,28)(2,58,11,26)(3,56,12,30)(4,25,7,57)(5,29,8,55)(6,27,9,59)(13,85,18,81)(14,89,16,79)(15,87,17,83)(19,86,23,82)(20,90,24,80)(21,88,22,84)(31,72,34,74)(32,78,35,70)(33,68,36,76)(37,43,40,62)(38,66,41,47)(39,45,42,64)(44,53,63,50)(46,49,65,52)(48,51,61,54)(67,94,75,91)(69,96,77,93)(71,92,73,95), (1,44,4,47)(2,48,5,45)(3,46,6,43)(7,66,10,63)(8,64,11,61)(9,62,12,65)(13,73,22,76)(14,77,23,74)(15,75,24,78)(16,69,19,72)(17,67,20,70)(18,71,21,68)(25,38,60,53)(26,54,55,39)(27,40,56,49)(28,50,57,41)(29,42,58,51)(30,52,59,37)(31,89,93,82)(32,83,94,90)(33,85,95,84)(34,79,96,86)(35,87,91,80)(36,81,92,88), (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), (1,15,4,24)(2,14,5,23)(3,13,6,22)(7,20,10,17)(8,19,11,16)(9,21,12,18)(25,90,60,83)(26,89,55,82)(27,88,56,81)(28,87,57,80)(29,86,58,79)(30,85,59,84)(31,42,93,51)(32,41,94,50)(33,40,95,49)(34,39,96,54)(35,38,91,53)(36,37,92,52)(43,71,46,68)(44,70,47,67)(45,69,48,72)(61,74,64,77)(62,73,65,76)(63,78,66,75) );
G=PermutationGroup([[(1,60,10,28),(2,58,11,26),(3,56,12,30),(4,25,7,57),(5,29,8,55),(6,27,9,59),(13,85,18,81),(14,89,16,79),(15,87,17,83),(19,86,23,82),(20,90,24,80),(21,88,22,84),(31,72,34,74),(32,78,35,70),(33,68,36,76),(37,43,40,62),(38,66,41,47),(39,45,42,64),(44,53,63,50),(46,49,65,52),(48,51,61,54),(67,94,75,91),(69,96,77,93),(71,92,73,95)], [(1,44,4,47),(2,48,5,45),(3,46,6,43),(7,66,10,63),(8,64,11,61),(9,62,12,65),(13,73,22,76),(14,77,23,74),(15,75,24,78),(16,69,19,72),(17,67,20,70),(18,71,21,68),(25,38,60,53),(26,54,55,39),(27,40,56,49),(28,50,57,41),(29,42,58,51),(30,52,59,37),(31,89,93,82),(32,83,94,90),(33,85,95,84),(34,79,96,86),(35,87,91,80),(36,81,92,88)], [(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)], [(1,15,4,24),(2,14,5,23),(3,13,6,22),(7,20,10,17),(8,19,11,16),(9,21,12,18),(25,90,60,83),(26,89,55,82),(27,88,56,81),(28,87,57,80),(29,86,58,79),(30,85,59,84),(31,42,93,51),(32,41,94,50),(33,40,95,49),(34,39,96,54),(35,38,91,53),(36,37,92,52),(43,71,46,68),(44,70,47,67),(45,69,48,72),(61,74,64,77),(62,73,65,76),(63,78,66,75)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4E | 4F | ··· | 4M | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
33 irreducible representations
| dim | 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 | S3 | D6 | D6 | D6 | D6 | 2+ 1+4 | 2- 1+4 | D4⋊6D6 | Q8○D12 |
| kernel | C42.140D6 | C12.6Q8 | Dic3.D4 | C23.8D6 | C23.23D6 | Dic3⋊Q8 | C3×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 4 | 4 | 2 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of C42.140D6 ►in GL8(𝔽13)
| 3 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 0 | 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 6 | 6 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 | 1 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 11 | 12 | 0 | 0 | 0 | 0 |
| 5 | 5 | 1 | 0 | 0 | 0 | 0 | 0 |
| 4 | 5 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 12 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 10 | 1 | 1 | 0 | 0 | 0 | 0 |
| 8 | 9 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 0 | 4 |
| 9 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 11 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 2 | 4 | 0 | 0 | 0 | 0 |
| 2 | 11 | 2 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 3 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 6 | 10 | 9 | 0 |
G:=sub<GL(8,GF(13))| [3,6,7,0,0,0,0,0,7,10,0,6,0,0,0,0,0,0,10,6,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,12,12,0,1,0,0,0,0,2,1,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0],[0,12,5,4,0,0,0,0,0,12,5,5,0,0,0,0,12,11,1,1,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,1,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,1,9,8,0,0,0,0,12,12,10,9,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,3,3,7,0,0,0,0,0,10,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,4],[9,11,4,2,0,0,0,0,2,4,0,11,0,0,0,0,0,0,2,2,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,4,4,3,6,0,0,0,0,0,0,0,10,0,0,0,0,8,4,9,9,0,0,0,0,0,4,0,0] >;
C42.140D6 in GAP, Magma, Sage, TeX
C_4^2._{140}D_6 % in TeX
G:=Group("C4^2.140D6"); // GroupNames label
G:=SmallGroup(192,1231);
// by ID
G=gap.SmallGroup(192,1231);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,219,184,1571,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations