metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.116D6, C6.1052+ 1+4, (C4×D4)⋊24S3, (D4×C12)⋊26C2, (C4×D12)⋊33C2, C4⋊C4.287D6, Dic3⋊D4⋊11C2, D6⋊3D4⋊10C2, C12⋊D4⋊16C2, C12⋊7D4⋊12C2, C12⋊3D4⋊10C2, (C2×D4).223D6, C23.9D6⋊9C2, C42⋊2S3⋊15C2, C2.18(D4○D12), C4.46(C4○D12), (C2×C6).106C24, D6⋊C4.88C22, C22⋊C4.118D6, C4.Dic6⋊16C2, (C22×C4).230D6, C12.113(C4○D4), C2.24(D4⋊6D6), (C4×C12).160C22, (C2×C12).164C23, (C6×D4).265C22, C23.28D6⋊4C2, Dic3⋊C4.7C22, (C2×D12).260C22, (C22×S3).40C23, C4⋊Dic3.364C22, C23.113(C22×S3), C22.131(S3×C23), (C22×C12).83C22, (C22×C6).176C23, (C4×Dic3).78C22, (C2×Dic3).47C23, C3⋊2(C22.34C24), C6.D4.16C22, C6.48(C2×C4○D4), C2.55(C2×C4○D12), (S3×C2×C4).203C22, (C3×C4⋊C4).334C22, (C2×C4).581(C22×S3), (C2×C3⋊D4).19C22, (C3×C22⋊C4).129C22, SmallGroup(192,1121)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.116D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=dad-1=a-1b2, bc=cb, dbd-1=a2b-1, dcd-1=a2c-1 >
Subgroups: 680 in 240 conjugacy classes, 95 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C42⋊C2, C4×D4, C4×D4, C4⋊D4, C22.D4, C42.C2, C4⋊1D4, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C2×C3⋊D4, C22×C12, C6×D4, C22.34C24, C42⋊2S3, C4×D12, C23.9D6, Dic3⋊D4, C4.Dic6, C12⋊D4, C23.28D6, C12⋊7D4, D6⋊3D4, C12⋊3D4, D4×C12, C42.116D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, C4○D12, S3×C23, C22.34C24, C2×C4○D12, D4⋊6D6, D4○D12, C42.116D6
►On 96 points
(1 65 37 83)(2 86 38 70)(3 61 39 79)(4 88 40 72)(5 63 41 81)(6 90 42 68)(7 28 60 49)(8 93 55 76)(9 30 56 51)(10 95 57 78)(11 26 58 53)(12 91 59 74)(13 89 21 67)(14 64 22 82)(15 85 23 69)(16 66 24 84)(17 87 19 71)(18 62 20 80)(25 47 52 35)(27 43 54 31)(29 45 50 33)(32 92 44 75)(34 94 46 77)(36 96 48 73)
(1 59 23 31)(2 60 24 32)(3 55 19 33)(4 56 20 34)(5 57 21 35)(6 58 22 36)(7 16 44 38)(8 17 45 39)(9 18 46 40)(10 13 47 41)(11 14 48 42)(12 15 43 37)(25 63 78 67)(26 64 73 68)(27 65 74 69)(28 66 75 70)(29 61 76 71)(30 62 77 72)(49 84 92 86)(50 79 93 87)(51 80 94 88)(52 81 95 89)(53 82 96 90)(54 83 91 85)
(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 53 37 26)(2 25 38 52)(3 51 39 30)(4 29 40 50)(5 49 41 28)(6 27 42 54)(7 67 60 89)(8 88 55 72)(9 71 56 87)(10 86 57 70)(11 69 58 85)(12 90 59 68)(13 75 21 92)(14 91 22 74)(15 73 23 96)(16 95 24 78)(17 77 19 94)(18 93 20 76)(31 64 43 82)(32 81 44 63)(33 62 45 80)(34 79 46 61)(35 66 47 84)(36 83 48 65)
G:=sub<Sym(96)| (1,65,37,83)(2,86,38,70)(3,61,39,79)(4,88,40,72)(5,63,41,81)(6,90,42,68)(7,28,60,49)(8,93,55,76)(9,30,56,51)(10,95,57,78)(11,26,58,53)(12,91,59,74)(13,89,21,67)(14,64,22,82)(15,85,23,69)(16,66,24,84)(17,87,19,71)(18,62,20,80)(25,47,52,35)(27,43,54,31)(29,45,50,33)(32,92,44,75)(34,94,46,77)(36,96,48,73), (1,59,23,31)(2,60,24,32)(3,55,19,33)(4,56,20,34)(5,57,21,35)(6,58,22,36)(7,16,44,38)(8,17,45,39)(9,18,46,40)(10,13,47,41)(11,14,48,42)(12,15,43,37)(25,63,78,67)(26,64,73,68)(27,65,74,69)(28,66,75,70)(29,61,76,71)(30,62,77,72)(49,84,92,86)(50,79,93,87)(51,80,94,88)(52,81,95,89)(53,82,96,90)(54,83,91,85), (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), (1,53,37,26)(2,25,38,52)(3,51,39,30)(4,29,40,50)(5,49,41,28)(6,27,42,54)(7,67,60,89)(8,88,55,72)(9,71,56,87)(10,86,57,70)(11,69,58,85)(12,90,59,68)(13,75,21,92)(14,91,22,74)(15,73,23,96)(16,95,24,78)(17,77,19,94)(18,93,20,76)(31,64,43,82)(32,81,44,63)(33,62,45,80)(34,79,46,61)(35,66,47,84)(36,83,48,65)>;
G:=Group( (1,65,37,83)(2,86,38,70)(3,61,39,79)(4,88,40,72)(5,63,41,81)(6,90,42,68)(7,28,60,49)(8,93,55,76)(9,30,56,51)(10,95,57,78)(11,26,58,53)(12,91,59,74)(13,89,21,67)(14,64,22,82)(15,85,23,69)(16,66,24,84)(17,87,19,71)(18,62,20,80)(25,47,52,35)(27,43,54,31)(29,45,50,33)(32,92,44,75)(34,94,46,77)(36,96,48,73), (1,59,23,31)(2,60,24,32)(3,55,19,33)(4,56,20,34)(5,57,21,35)(6,58,22,36)(7,16,44,38)(8,17,45,39)(9,18,46,40)(10,13,47,41)(11,14,48,42)(12,15,43,37)(25,63,78,67)(26,64,73,68)(27,65,74,69)(28,66,75,70)(29,61,76,71)(30,62,77,72)(49,84,92,86)(50,79,93,87)(51,80,94,88)(52,81,95,89)(53,82,96,90)(54,83,91,85), (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), (1,53,37,26)(2,25,38,52)(3,51,39,30)(4,29,40,50)(5,49,41,28)(6,27,42,54)(7,67,60,89)(8,88,55,72)(9,71,56,87)(10,86,57,70)(11,69,58,85)(12,90,59,68)(13,75,21,92)(14,91,22,74)(15,73,23,96)(16,95,24,78)(17,77,19,94)(18,93,20,76)(31,64,43,82)(32,81,44,63)(33,62,45,80)(34,79,46,61)(35,66,47,84)(36,83,48,65) );
G=PermutationGroup([[(1,65,37,83),(2,86,38,70),(3,61,39,79),(4,88,40,72),(5,63,41,81),(6,90,42,68),(7,28,60,49),(8,93,55,76),(9,30,56,51),(10,95,57,78),(11,26,58,53),(12,91,59,74),(13,89,21,67),(14,64,22,82),(15,85,23,69),(16,66,24,84),(17,87,19,71),(18,62,20,80),(25,47,52,35),(27,43,54,31),(29,45,50,33),(32,92,44,75),(34,94,46,77),(36,96,48,73)], [(1,59,23,31),(2,60,24,32),(3,55,19,33),(4,56,20,34),(5,57,21,35),(6,58,22,36),(7,16,44,38),(8,17,45,39),(9,18,46,40),(10,13,47,41),(11,14,48,42),(12,15,43,37),(25,63,78,67),(26,64,73,68),(27,65,74,69),(28,66,75,70),(29,61,76,71),(30,62,77,72),(49,84,92,86),(50,79,93,87),(51,80,94,88),(52,81,95,89),(53,82,96,90),(54,83,91,85)], [(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)], [(1,53,37,26),(2,25,38,52),(3,51,39,30),(4,29,40,50),(5,49,41,28),(6,27,42,54),(7,67,60,89),(8,88,55,72),(9,71,56,87),(10,86,57,70),(11,69,58,85),(12,90,59,68),(13,75,21,92),(14,91,22,74),(15,73,23,96),(16,95,24,78),(17,77,19,94),(18,93,20,76),(31,64,43,82),(32,81,44,63),(33,62,45,80),(34,79,46,61),(35,66,47,84),(36,83,48,65)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | ··· | 4F | 4G | 4H | 4I | ··· | 4M | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2+ 1+4 | D4⋊6D6 | D4○D12 |
| kernel | C42.116D6 | C42⋊2S3 | C4×D12 | C23.9D6 | Dic3⋊D4 | C4.Dic6 | C12⋊D4 | C23.28D6 | C12⋊7D4 | D6⋊3D4 | C12⋊3D4 | D4×C12 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 2 | 2 | 2 |
Matrix representation of C42.116D6 ►in GL6(𝔽13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 2 | 0 |
| 0 | 0 | 0 | 4 | 0 | 2 |
| 0 | 0 | 11 | 0 | 9 | 0 |
| 0 | 0 | 0 | 11 | 0 | 9 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 4 | 0 | 0 |
| 0 | 0 | 9 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 4 |
| 0 | 0 | 0 | 0 | 9 | 2 |
| 4 | 11 | 0 | 0 | 0 | 0 |
| 2 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 9 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 9 | 5 | 5 |
| 0 | 0 | 5 | 4 | 10 | 8 |
| 0 | 0 | 8 | 8 | 4 | 4 |
| 0 | 0 | 3 | 5 | 8 | 9 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,4,0,11,0,0,0,0,4,0,11,0,0,2,0,9,0,0,0,0,2,0,9],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,11,9,0,0,0,0,4,2,0,0,0,0,0,0,11,9,0,0,0,0,4,2],[4,2,0,0,0,0,11,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,1,0,0,0,0,12,12,0,0],[11,9,0,0,0,0,11,2,0,0,0,0,0,0,9,5,8,3,0,0,9,4,8,5,0,0,5,10,4,8,0,0,5,8,4,9] >;
C42.116D6 in GAP, Magma, Sage, TeX
C_4^2._{116}D_6 % in TeX
G:=Group("C4^2.116D6"); // GroupNames label
G:=SmallGroup(192,1121);
// by ID
G=gap.SmallGroup(192,1121);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,675,192,6278]);
// 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^-1*b^2,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=a^2*c^-1>;
// generators/relations