metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.143D6, C6.1272+ 1+4, (C4×D12)⋊46C2, (Q8×Dic3)⋊20C2, (C4×Dic6)⋊46C2, (D4×Dic3)⋊31C2, (C2×D4).176D6, C12⋊3D4.9C2, C4.4D4⋊14S3, (C2×Q8).163D6, C22⋊C4.36D6, C23.9D6⋊46C2, C2.51(D4○D12), (C2×C6).225C24, D6⋊C4.37C22, Dic3⋊4D4⋊34C2, C12.126(C4○D4), C12.23D4⋊23C2, C4.16(D4⋊2S3), (C4×C12).188C22, (C2×C12).505C23, (C6×D4).158C22, (C22×C6).55C23, C23.57(C22×S3), (C6×Q8).129C22, Dic3.39(C4○D4), C23.11D6⋊41C2, (C2×D12).266C22, C23.21D6⋊26C2, (C22×S3).97C23, C4⋊Dic3.235C22, C22.246(S3×C23), Dic3⋊C4.142C22, C3⋊4(C22.53C24), (C2×Dic6).250C22, (C4×Dic3).135C22, (C2×Dic3).256C23, C6.D4.58C22, (C22×Dic3).145C22, C2.81(S3×C4○D4), C6.192(C2×C4○D4), (C3×C4.4D4)⋊17C2, C2.57(C2×D4⋊2S3), (S3×C2×C4).216C22, (C2×C4).198(C22×S3), (C2×C3⋊D4).63C22, (C3×C22⋊C4).67C22, SmallGroup(192,1240)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.143D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, dbd-1=b-1, dcd-1=a2c-1 >
Subgroups: 608 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C4×D4, C4×Q8, C22.D4, C4.4D4, C4.4D4, C4⋊1D4, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C6×D4, C6×Q8, C22.53C24, C4×Dic6, C4×D12, Dic3⋊4D4, C23.9D6, C23.11D6, C23.21D6, D4×Dic3, C12⋊3D4, Q8×Dic3, C12.23D4, C3×C4.4D4, C42.143D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, D4⋊2S3, S3×C23, C22.53C24, C2×D4⋊2S3, S3×C4○D4, D4○D12, C42.143D6
►On 96 points
(1 65 37 71)(2 86 38 51)(3 61 39 67)(4 88 40 53)(5 63 41 69)(6 90 42 49)(7 95 57 83)(8 26 58 75)(9 91 59 79)(10 28 60 77)(11 93 55 81)(12 30 56 73)(13 29 45 78)(14 94 46 82)(15 25 47 74)(16 96 48 84)(17 27 43 76)(18 92 44 80)(19 52 33 87)(20 68 34 62)(21 54 35 89)(22 70 36 64)(23 50 31 85)(24 72 32 66)
(1 9 31 17)(2 60 32 44)(3 11 33 13)(4 56 34 46)(5 7 35 15)(6 58 36 48)(8 22 16 42)(10 24 18 38)(12 20 14 40)(19 45 39 55)(21 47 41 57)(23 43 37 59)(25 63 95 89)(26 70 96 49)(27 65 91 85)(28 72 92 51)(29 61 93 87)(30 68 94 53)(50 76 71 79)(52 78 67 81)(54 74 69 83)(62 82 88 73)(64 84 90 75)(66 80 86 77)
(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 37 42)(2 41 38 5)(3 4 39 40)(7 44 57 18)(8 17 58 43)(9 48 59 16)(10 15 60 47)(11 46 55 14)(12 13 56 45)(19 20 33 34)(21 24 35 32)(22 31 36 23)(25 80 74 92)(26 91 75 79)(27 84 76 96)(28 95 77 83)(29 82 78 94)(30 93 73 81)(49 85 90 50)(51 89 86 54)(52 53 87 88)(61 62 67 68)(63 66 69 72)(64 71 70 65)
G:=sub<Sym(96)| (1,65,37,71)(2,86,38,51)(3,61,39,67)(4,88,40,53)(5,63,41,69)(6,90,42,49)(7,95,57,83)(8,26,58,75)(9,91,59,79)(10,28,60,77)(11,93,55,81)(12,30,56,73)(13,29,45,78)(14,94,46,82)(15,25,47,74)(16,96,48,84)(17,27,43,76)(18,92,44,80)(19,52,33,87)(20,68,34,62)(21,54,35,89)(22,70,36,64)(23,50,31,85)(24,72,32,66), (1,9,31,17)(2,60,32,44)(3,11,33,13)(4,56,34,46)(5,7,35,15)(6,58,36,48)(8,22,16,42)(10,24,18,38)(12,20,14,40)(19,45,39,55)(21,47,41,57)(23,43,37,59)(25,63,95,89)(26,70,96,49)(27,65,91,85)(28,72,92,51)(29,61,93,87)(30,68,94,53)(50,76,71,79)(52,78,67,81)(54,74,69,83)(62,82,88,73)(64,84,90,75)(66,80,86,77), (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), (1,6,37,42)(2,41,38,5)(3,4,39,40)(7,44,57,18)(8,17,58,43)(9,48,59,16)(10,15,60,47)(11,46,55,14)(12,13,56,45)(19,20,33,34)(21,24,35,32)(22,31,36,23)(25,80,74,92)(26,91,75,79)(27,84,76,96)(28,95,77,83)(29,82,78,94)(30,93,73,81)(49,85,90,50)(51,89,86,54)(52,53,87,88)(61,62,67,68)(63,66,69,72)(64,71,70,65)>;
G:=Group( (1,65,37,71)(2,86,38,51)(3,61,39,67)(4,88,40,53)(5,63,41,69)(6,90,42,49)(7,95,57,83)(8,26,58,75)(9,91,59,79)(10,28,60,77)(11,93,55,81)(12,30,56,73)(13,29,45,78)(14,94,46,82)(15,25,47,74)(16,96,48,84)(17,27,43,76)(18,92,44,80)(19,52,33,87)(20,68,34,62)(21,54,35,89)(22,70,36,64)(23,50,31,85)(24,72,32,66), (1,9,31,17)(2,60,32,44)(3,11,33,13)(4,56,34,46)(5,7,35,15)(6,58,36,48)(8,22,16,42)(10,24,18,38)(12,20,14,40)(19,45,39,55)(21,47,41,57)(23,43,37,59)(25,63,95,89)(26,70,96,49)(27,65,91,85)(28,72,92,51)(29,61,93,87)(30,68,94,53)(50,76,71,79)(52,78,67,81)(54,74,69,83)(62,82,88,73)(64,84,90,75)(66,80,86,77), (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), (1,6,37,42)(2,41,38,5)(3,4,39,40)(7,44,57,18)(8,17,58,43)(9,48,59,16)(10,15,60,47)(11,46,55,14)(12,13,56,45)(19,20,33,34)(21,24,35,32)(22,31,36,23)(25,80,74,92)(26,91,75,79)(27,84,76,96)(28,95,77,83)(29,82,78,94)(30,93,73,81)(49,85,90,50)(51,89,86,54)(52,53,87,88)(61,62,67,68)(63,66,69,72)(64,71,70,65) );
G=PermutationGroup([[(1,65,37,71),(2,86,38,51),(3,61,39,67),(4,88,40,53),(5,63,41,69),(6,90,42,49),(7,95,57,83),(8,26,58,75),(9,91,59,79),(10,28,60,77),(11,93,55,81),(12,30,56,73),(13,29,45,78),(14,94,46,82),(15,25,47,74),(16,96,48,84),(17,27,43,76),(18,92,44,80),(19,52,33,87),(20,68,34,62),(21,54,35,89),(22,70,36,64),(23,50,31,85),(24,72,32,66)], [(1,9,31,17),(2,60,32,44),(3,11,33,13),(4,56,34,46),(5,7,35,15),(6,58,36,48),(8,22,16,42),(10,24,18,38),(12,20,14,40),(19,45,39,55),(21,47,41,57),(23,43,37,59),(25,63,95,89),(26,70,96,49),(27,65,91,85),(28,72,92,51),(29,61,93,87),(30,68,94,53),(50,76,71,79),(52,78,67,81),(54,74,69,83),(62,82,88,73),(64,84,90,75),(66,80,86,77)], [(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)], [(1,6,37,42),(2,41,38,5),(3,4,39,40),(7,44,57,18),(8,17,58,43),(9,48,59,16),(10,15,60,47),(11,46,55,14),(12,13,56,45),(19,20,33,34),(21,24,35,32),(22,31,36,23),(25,80,74,92),(26,91,75,79),(27,84,76,96),(28,95,77,83),(29,82,78,94),(30,93,73,81),(49,85,90,50),(51,89,86,54),(52,53,87,88),(61,62,67,68),(63,66,69,72),(64,71,70,65)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4O | 4P | 4Q | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | 2+ 1+4 | D4⋊2S3 | S3×C4○D4 | D4○D12 |
| kernel | C42.143D6 | C4×Dic6 | C4×D12 | Dic3⋊4D4 | C23.9D6 | C23.11D6 | C23.21D6 | D4×Dic3 | C12⋊3D4 | Q8×Dic3 | C12.23D4 | C3×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | Dic3 | C12 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of C42.143D6 ►in GL6(𝔽13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 12 | 3 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 8 | 2 | 0 | 0 | 0 | 0 |
| 1 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 5 | 11 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,5],[12,0,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[8,1,0,0,0,0,2,5,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,0,0,0,0,0,11,8,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0] >;
C42.143D6 in GAP, Magma, Sage, TeX
C_4^2._{143}D_6 % in TeX
G:=Group("C4^2.143D6"); // GroupNames label
G:=SmallGroup(192,1240);
// by ID
G=gap.SmallGroup(192,1240);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,1571,297,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,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,d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations