metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.72D6, (C2×D4).54D6, C4⋊1D4.5S3, (C2×C12).291D4, C12.75(C4○D4), D4⋊Dic3⋊21C2, C6.93(C8⋊C22), C12.6Q8⋊13C2, (C6×D4).70C22, C4.23(D4⋊2S3), (C4×C12).119C22, (C2×C12).389C23, C42.S3⋊12C2, C6.44(C4.4D4), C2.14(D12⋊6C22), C4⋊Dic3.155C22, C2.11(C23.12D6), C3⋊4(C42.29C22), (C2×C6).520(C2×D4), (C3×C4⋊1D4).3C2, (C2×C4).69(C3⋊D4), (C2×C3⋊C8).129C22, (C2×C4).487(C22×S3), C22.193(C2×C3⋊D4), SmallGroup(192,630)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.72D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >
Subgroups: 304 in 110 conjugacy classes, 39 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×C8, C2×D4, C2×D4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C8⋊C4, D4⋊C4, C42.C2, C4⋊1D4, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C4×C12, C6×D4, C6×D4, C42.29C22, C42.S3, D4⋊Dic3, C12.6Q8, C3×C4⋊1D4, C42.72D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4.4D4, C8⋊C22, D4⋊2S3, C2×C3⋊D4, C42.29C22, D12⋊6C22, C23.12D6, C42.72D6
Character table of C42.72D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 4 | 4 | 24 | 24 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | -2 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | -2 | 2 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | -2 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | -2 | 2 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | -√-3 | -√-3 | √-3 | √-3 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | -√-3 | √-3 | √-3 | -√-3 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | √-3 | -√-3 | -√-3 | √-3 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | √-3 | √-3 | -√-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | -2 | 0 | 0 | 2 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | -2 | 0 | 0 | 2 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | -4 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| ρ27 | 4 | 4 | -4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-3 | 2√-3 | 0 | 0 | complex lifted from D12⋊6C22 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-3 | -2√-3 | 0 | 0 | complex lifted from D12⋊6C22 |
| ρ29 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-3 | 0 | 0 | 0 | 0 | -2√-3 | complex lifted from D12⋊6C22 |
| ρ30 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-3 | 0 | 0 | 0 | 0 | 2√-3 | complex lifted from D12⋊6C22 |
►On 96 points
Generators in S96
(1 4 13 16)(2 17 14 5)(3 6 15 18)(7 19 32 71)(8 72 33 20)(9 21 34 67)(10 68 35 22)(11 23 36 69)(12 70 31 24)(25 55 52 64)(26 65 53 56)(27 57 54 66)(28 61 49 58)(29 59 50 62)(30 63 51 60)(37 40 43 46)(38 47 44 41)(39 42 45 48)(73 95 79 89)(74 90 80 96)(75 91 81 85)(76 86 82 92)(77 93 83 87)(78 88 84 94)
(1 23 37 33)(2 34 38 24)(3 19 39 35)(4 36 40 20)(5 21 41 31)(6 32 42 22)(7 48 68 18)(8 13 69 43)(9 44 70 14)(10 15 71 45)(11 46 72 16)(12 17 67 47)(25 95 76 61)(26 62 77 96)(27 91 78 63)(28 64 73 92)(29 93 74 65)(30 66 75 94)(49 55 79 86)(50 87 80 56)(51 57 81 88)(52 89 82 58)(53 59 83 90)(54 85 84 60)
(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 66 13 57)(2 80 14 74)(3 64 15 55)(4 84 16 78)(5 62 17 59)(6 82 18 76)(7 95 32 89)(8 81 33 75)(9 93 34 87)(10 79 35 73)(11 91 36 85)(12 83 31 77)(19 28 71 49)(20 60 72 63)(21 26 67 53)(22 58 68 61)(23 30 69 51)(24 56 70 65)(25 42 52 48)(27 40 54 46)(29 38 50 44)(37 94 43 88)(39 92 45 86)(41 96 47 90)
G:=sub<Sym(96)| (1,4,13,16)(2,17,14,5)(3,6,15,18)(7,19,32,71)(8,72,33,20)(9,21,34,67)(10,68,35,22)(11,23,36,69)(12,70,31,24)(25,55,52,64)(26,65,53,56)(27,57,54,66)(28,61,49,58)(29,59,50,62)(30,63,51,60)(37,40,43,46)(38,47,44,41)(39,42,45,48)(73,95,79,89)(74,90,80,96)(75,91,81,85)(76,86,82,92)(77,93,83,87)(78,88,84,94), (1,23,37,33)(2,34,38,24)(3,19,39,35)(4,36,40,20)(5,21,41,31)(6,32,42,22)(7,48,68,18)(8,13,69,43)(9,44,70,14)(10,15,71,45)(11,46,72,16)(12,17,67,47)(25,95,76,61)(26,62,77,96)(27,91,78,63)(28,64,73,92)(29,93,74,65)(30,66,75,94)(49,55,79,86)(50,87,80,56)(51,57,81,88)(52,89,82,58)(53,59,83,90)(54,85,84,60), (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), (1,66,13,57)(2,80,14,74)(3,64,15,55)(4,84,16,78)(5,62,17,59)(6,82,18,76)(7,95,32,89)(8,81,33,75)(9,93,34,87)(10,79,35,73)(11,91,36,85)(12,83,31,77)(19,28,71,49)(20,60,72,63)(21,26,67,53)(22,58,68,61)(23,30,69,51)(24,56,70,65)(25,42,52,48)(27,40,54,46)(29,38,50,44)(37,94,43,88)(39,92,45,86)(41,96,47,90)>;
G:=Group( (1,4,13,16)(2,17,14,5)(3,6,15,18)(7,19,32,71)(8,72,33,20)(9,21,34,67)(10,68,35,22)(11,23,36,69)(12,70,31,24)(25,55,52,64)(26,65,53,56)(27,57,54,66)(28,61,49,58)(29,59,50,62)(30,63,51,60)(37,40,43,46)(38,47,44,41)(39,42,45,48)(73,95,79,89)(74,90,80,96)(75,91,81,85)(76,86,82,92)(77,93,83,87)(78,88,84,94), (1,23,37,33)(2,34,38,24)(3,19,39,35)(4,36,40,20)(5,21,41,31)(6,32,42,22)(7,48,68,18)(8,13,69,43)(9,44,70,14)(10,15,71,45)(11,46,72,16)(12,17,67,47)(25,95,76,61)(26,62,77,96)(27,91,78,63)(28,64,73,92)(29,93,74,65)(30,66,75,94)(49,55,79,86)(50,87,80,56)(51,57,81,88)(52,89,82,58)(53,59,83,90)(54,85,84,60), (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), (1,66,13,57)(2,80,14,74)(3,64,15,55)(4,84,16,78)(5,62,17,59)(6,82,18,76)(7,95,32,89)(8,81,33,75)(9,93,34,87)(10,79,35,73)(11,91,36,85)(12,83,31,77)(19,28,71,49)(20,60,72,63)(21,26,67,53)(22,58,68,61)(23,30,69,51)(24,56,70,65)(25,42,52,48)(27,40,54,46)(29,38,50,44)(37,94,43,88)(39,92,45,86)(41,96,47,90) );
G=PermutationGroup([[(1,4,13,16),(2,17,14,5),(3,6,15,18),(7,19,32,71),(8,72,33,20),(9,21,34,67),(10,68,35,22),(11,23,36,69),(12,70,31,24),(25,55,52,64),(26,65,53,56),(27,57,54,66),(28,61,49,58),(29,59,50,62),(30,63,51,60),(37,40,43,46),(38,47,44,41),(39,42,45,48),(73,95,79,89),(74,90,80,96),(75,91,81,85),(76,86,82,92),(77,93,83,87),(78,88,84,94)], [(1,23,37,33),(2,34,38,24),(3,19,39,35),(4,36,40,20),(5,21,41,31),(6,32,42,22),(7,48,68,18),(8,13,69,43),(9,44,70,14),(10,15,71,45),(11,46,72,16),(12,17,67,47),(25,95,76,61),(26,62,77,96),(27,91,78,63),(28,64,73,92),(29,93,74,65),(30,66,75,94),(49,55,79,86),(50,87,80,56),(51,57,81,88),(52,89,82,58),(53,59,83,90),(54,85,84,60)], [(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)], [(1,66,13,57),(2,80,14,74),(3,64,15,55),(4,84,16,78),(5,62,17,59),(6,82,18,76),(7,95,32,89),(8,81,33,75),(9,93,34,87),(10,79,35,73),(11,91,36,85),(12,83,31,77),(19,28,71,49),(20,60,72,63),(21,26,67,53),(22,58,68,61),(23,30,69,51),(24,56,70,65),(25,42,52,48),(27,40,54,46),(29,38,50,44),(37,94,43,88),(39,92,45,86),(41,96,47,90)]])
Matrix representation of C42.72D6 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 13 |
| 0 | 0 | 0 | 0 | 60 | 43 |
| 0 | 0 | 43 | 60 | 0 | 0 |
| 0 | 0 | 13 | 30 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 43 |
| 0 | 0 | 0 | 0 | 30 | 30 |
| 0 | 0 | 60 | 43 | 0 | 0 |
| 0 | 0 | 30 | 30 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 30 | 65 | 43 |
| 0 | 0 | 38 | 65 | 35 | 8 |
| 0 | 0 | 65 | 43 | 65 | 43 |
| 0 | 0 | 35 | 8 | 35 | 8 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,0,43,13,0,0,0,0,60,30,0,0,30,60,0,0,0,0,13,43,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,30,0,0,0,0,43,30,0,0,60,30,0,0,0,0,43,30,0,0],[46,0,0,0,0,0,0,27,0,0,0,0,0,0,8,38,65,35,0,0,30,65,43,8,0,0,65,35,65,35,0,0,43,8,43,8] >;
C42.72D6 in GAP, Magma, Sage, TeX
C_4^2._{72}D_6 % in TeX
G:=Group("C4^2.72D6"); // GroupNames label
G:=SmallGroup(192,630);
// by ID
G=gap.SmallGroup(192,630);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,64,590,135,438,102,6278]);
// 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=a^-1,d*a*d^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b*c^-1>;
// generators/relations
Export