metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.20D6, C8⋊C4⋊11S3, C12⋊2Q8⋊3C2, (C2×C4).27D12, (C2×C12).38D4, (C2×C8).160D6, C2.9(C8⋊D6), C6.6(C8⋊C22), (C4×C12).5C22, C42⋊7S3.2C2, C2.Dic12⋊38C2, C2.D24.16C2, C2.8(C8.D6), C6.9(C4.4D4), (C2×D12).8C22, C22.99(C2×D12), C6.4(C8.C22), C12.225(C4○D4), C4.109(C4○D12), (C2×C24).314C22, (C2×C12).735C23, C4⋊Dic3.10C22, (C2×Dic6).8C22, C2.14(C42⋊7S3), C3⋊1(C42.28C22), (C3×C8⋊C4)⋊20C2, (C2×C6).118(C2×D4), (C2×C4).679(C22×S3), SmallGroup(192,273)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.20D6
G = < a,b,c,d | a4=b4=1, c6=a2b, d2=a2, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b-1c5 >
Subgroups: 344 in 100 conjugacy classes, 39 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, C24, Dic6, D12, C2×Dic3, C2×C12, C22×S3, C8⋊C4, D4⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, C2×Dic6, C2×Dic6, C2×D12, C42.28C22, C2.Dic12, C2.D24, C3×C8⋊C4, C12⋊2Q8, C42⋊7S3, C42.20D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4.4D4, C8⋊C22, C8.C22, C2×D12, C4○D12, C42.28C22, C42⋊7S3, C8⋊D6, C8.D6, C42.20D6
►On 96 points
Generators in S96
(1 28 84 68)(2 41 85 57)(3 30 86 70)(4 43 87 59)(5 32 88 72)(6 45 89 61)(7 34 90 50)(8 47 91 63)(9 36 92 52)(10 25 93 65)(11 38 94 54)(12 27 95 67)(13 40 96 56)(14 29 73 69)(15 42 74 58)(16 31 75 71)(17 44 76 60)(18 33 77 49)(19 46 78 62)(20 35 79 51)(21 48 80 64)(22 37 81 53)(23 26 82 66)(24 39 83 55)
(1 90 13 78)(2 91 14 79)(3 92 15 80)(4 93 16 81)(5 94 17 82)(6 95 18 83)(7 96 19 84)(8 73 20 85)(9 74 21 86)(10 75 22 87)(11 76 23 88)(12 77 24 89)(25 71 37 59)(26 72 38 60)(27 49 39 61)(28 50 40 62)(29 51 41 63)(30 52 42 64)(31 53 43 65)(32 54 44 66)(33 55 45 67)(34 56 46 68)(35 57 47 69)(36 58 48 70)
(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 18 84 77)(2 76 85 17)(3 16 86 75)(4 74 87 15)(5 14 88 73)(6 96 89 13)(7 12 90 95)(8 94 91 11)(9 10 92 93)(19 24 78 83)(20 82 79 23)(21 22 80 81)(25 36 65 52)(26 51 66 35)(27 34 67 50)(28 49 68 33)(29 32 69 72)(30 71 70 31)(37 48 53 64)(38 63 54 47)(39 46 55 62)(40 61 56 45)(41 44 57 60)(42 59 58 43)
G:=sub<Sym(96)| (1,28,84,68)(2,41,85,57)(3,30,86,70)(4,43,87,59)(5,32,88,72)(6,45,89,61)(7,34,90,50)(8,47,91,63)(9,36,92,52)(10,25,93,65)(11,38,94,54)(12,27,95,67)(13,40,96,56)(14,29,73,69)(15,42,74,58)(16,31,75,71)(17,44,76,60)(18,33,77,49)(19,46,78,62)(20,35,79,51)(21,48,80,64)(22,37,81,53)(23,26,82,66)(24,39,83,55), (1,90,13,78)(2,91,14,79)(3,92,15,80)(4,93,16,81)(5,94,17,82)(6,95,18,83)(7,96,19,84)(8,73,20,85)(9,74,21,86)(10,75,22,87)(11,76,23,88)(12,77,24,89)(25,71,37,59)(26,72,38,60)(27,49,39,61)(28,50,40,62)(29,51,41,63)(30,52,42,64)(31,53,43,65)(32,54,44,66)(33,55,45,67)(34,56,46,68)(35,57,47,69)(36,58,48,70), (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), (1,18,84,77)(2,76,85,17)(3,16,86,75)(4,74,87,15)(5,14,88,73)(6,96,89,13)(7,12,90,95)(8,94,91,11)(9,10,92,93)(19,24,78,83)(20,82,79,23)(21,22,80,81)(25,36,65,52)(26,51,66,35)(27,34,67,50)(28,49,68,33)(29,32,69,72)(30,71,70,31)(37,48,53,64)(38,63,54,47)(39,46,55,62)(40,61,56,45)(41,44,57,60)(42,59,58,43)>;
G:=Group( (1,28,84,68)(2,41,85,57)(3,30,86,70)(4,43,87,59)(5,32,88,72)(6,45,89,61)(7,34,90,50)(8,47,91,63)(9,36,92,52)(10,25,93,65)(11,38,94,54)(12,27,95,67)(13,40,96,56)(14,29,73,69)(15,42,74,58)(16,31,75,71)(17,44,76,60)(18,33,77,49)(19,46,78,62)(20,35,79,51)(21,48,80,64)(22,37,81,53)(23,26,82,66)(24,39,83,55), (1,90,13,78)(2,91,14,79)(3,92,15,80)(4,93,16,81)(5,94,17,82)(6,95,18,83)(7,96,19,84)(8,73,20,85)(9,74,21,86)(10,75,22,87)(11,76,23,88)(12,77,24,89)(25,71,37,59)(26,72,38,60)(27,49,39,61)(28,50,40,62)(29,51,41,63)(30,52,42,64)(31,53,43,65)(32,54,44,66)(33,55,45,67)(34,56,46,68)(35,57,47,69)(36,58,48,70), (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), (1,18,84,77)(2,76,85,17)(3,16,86,75)(4,74,87,15)(5,14,88,73)(6,96,89,13)(7,12,90,95)(8,94,91,11)(9,10,92,93)(19,24,78,83)(20,82,79,23)(21,22,80,81)(25,36,65,52)(26,51,66,35)(27,34,67,50)(28,49,68,33)(29,32,69,72)(30,71,70,31)(37,48,53,64)(38,63,54,47)(39,46,55,62)(40,61,56,45)(41,44,57,60)(42,59,58,43) );
G=PermutationGroup([[(1,28,84,68),(2,41,85,57),(3,30,86,70),(4,43,87,59),(5,32,88,72),(6,45,89,61),(7,34,90,50),(8,47,91,63),(9,36,92,52),(10,25,93,65),(11,38,94,54),(12,27,95,67),(13,40,96,56),(14,29,73,69),(15,42,74,58),(16,31,75,71),(17,44,76,60),(18,33,77,49),(19,46,78,62),(20,35,79,51),(21,48,80,64),(22,37,81,53),(23,26,82,66),(24,39,83,55)], [(1,90,13,78),(2,91,14,79),(3,92,15,80),(4,93,16,81),(5,94,17,82),(6,95,18,83),(7,96,19,84),(8,73,20,85),(9,74,21,86),(10,75,22,87),(11,76,23,88),(12,77,24,89),(25,71,37,59),(26,72,38,60),(27,49,39,61),(28,50,40,62),(29,51,41,63),(30,52,42,64),(31,53,43,65),(32,54,44,66),(33,55,45,67),(34,56,46,68),(35,57,47,69),(36,58,48,70)], [(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)], [(1,18,84,77),(2,76,85,17),(3,16,86,75),(4,74,87,15),(5,14,88,73),(6,96,89,13),(7,12,90,95),(8,94,91,11),(9,10,92,93),(19,24,78,83),(20,82,79,23),(21,22,80,81),(25,36,65,52),(26,51,66,35),(27,34,67,50),(28,49,68,33),(29,32,69,72),(30,71,70,31),(37,48,53,64),(38,63,54,47),(39,46,55,62),(40,61,56,45),(41,44,57,60),(42,59,58,43)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 24 | 2 | 2 | 2 | 4 | 4 | 24 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
36 irreducible representations
| dim | 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 | S3 | D4 | D6 | D6 | C4○D4 | D12 | C4○D12 | C8⋊C22 | C8.C22 | C8⋊D6 | C8.D6 |
| kernel | C42.20D6 | C2.Dic12 | C2.D24 | C3×C8⋊C4 | C12⋊2Q8 | C42⋊7S3 | C8⋊C4 | C2×C12 | C42 | C2×C8 | C12 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C42.20D6 ►in GL6(𝔽73)
| 1 | 71 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 58 | 65 |
| 0 | 0 | 0 | 11 | 8 | 66 |
| 0 | 0 | 53 | 2 | 62 | 0 |
| 0 | 0 | 71 | 51 | 0 | 62 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 14 | 0 | 0 |
| 0 | 0 | 59 | 66 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 14 |
| 0 | 0 | 0 | 0 | 59 | 66 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 |
| 0 | 0 | 0 | 0 | 71 | 0 |
| 0 | 0 | 33 | 40 | 0 | 0 |
| 0 | 0 | 33 | 66 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 46 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 59 | 14 |
| 0 | 0 | 0 | 0 | 28 | 14 |
| 0 | 0 | 40 | 33 | 0 | 0 |
| 0 | 0 | 66 | 33 | 0 | 0 |
G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,71,72,0,0,0,0,0,0,11,0,53,71,0,0,0,11,2,51,0,0,58,8,62,0,0,0,65,66,0,62],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,59,0,0,0,0,14,66,0,0,0,0,0,0,7,59,0,0,0,0,14,66],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,33,33,0,0,0,0,40,66,0,0,2,71,0,0,0,0,2,0,0,0],[46,46,0,0,0,0,0,27,0,0,0,0,0,0,0,0,40,66,0,0,0,0,33,33,0,0,59,28,0,0,0,0,14,14,0,0] >;
C42.20D6 in GAP, Magma, Sage, TeX
C_4^2._{20}D_6 % in TeX
G:=Group("C4^2.20D6"); // GroupNames label
G:=SmallGroup(192,273);
// by ID
G=gap.SmallGroup(192,273);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,387,142,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2*b,d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^-1*c^5>;
// generators/relations