metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.7D4, Dic6.7D4, M4(2).6D6, C3⋊C8.27D4, D12.C4.C2, C4.153(S3×D4), (C2×Q8).32D6, C12.100(C2×D4), C4.10D4⋊4S3, C3⋊1(D4.5D4), C8.D6.1C2, C12.53D4⋊4C2, C6.12(C4⋊D4), (C2×C12).12C23, C4○D12.8C22, (C6×Q8).10C22, C12.47D4⋊12C2, C2.15(Dic3⋊D4), Q8.11D6.1C2, C4.Dic3.7C22, C22.16(C4○D12), (C2×Dic6).48C22, (C3×M4(2)).23C22, (C2×C3⋊Q16)⋊1C2, (C2×C3⋊C8).4C22, (C2×C6).33(C4○D4), (C3×C4.10D4)⋊2C2, (C2×C4).12(C22×S3), SmallGroup(192,314)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.7D4
G = < a,b,c,d | a12=b2=1, c4=a6, d2=a9, bab=a-1, cac-1=a7, ad=da, cbc-1=a6b, dbd-1=a3b, dcd-1=a9c3 >
Subgroups: 272 in 100 conjugacy classes, 35 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, C12, D6, C2×C6, C2×C8, M4(2), M4(2), SD16, Q16, C2×Q8, C2×Q8, C4○D4, C3⋊C8, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C4.10D4, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, S3×C8, C8⋊S3, C24⋊C2, Dic12, C2×C3⋊C8, C4.Dic3, Q8⋊2S3, C3⋊Q16, C3×M4(2), C2×Dic6, C4○D12, C6×Q8, D4.5D4, C12.53D4, C12.47D4, C3×C4.10D4, D12.C4, C8.D6, Q8.11D6, C2×C3⋊Q16, D12.7D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4⋊D4, C4○D12, S3×D4, D4.5D4, Dic3⋊D4, D12.7D4
Character table of D12.7D4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 12 | 2 | 2 | 2 | 8 | 12 | 24 | 2 | 4 | 4 | 4 | 6 | 6 | 8 | 12 | 24 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | 2 | 2 | -2 | 2 | 0 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2 | -2 | 0 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ17 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | -2i | 0 | 0 | 2i | complex lifted from C4○D4 |
| ρ18 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2i | 0 | 0 | -2i | complex lifted from C4○D4 |
| ρ19 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -√-3 | √-3 | -i | -√3 | √3 | i | complex lifted from C4○D12 |
| ρ20 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -√-3 | √-3 | i | √3 | -√3 | -i | complex lifted from C4○D12 |
| ρ21 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | √-3 | -√-3 | -i | √3 | -√3 | i | complex lifted from C4○D12 |
| ρ22 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 1 | 1 | √-3 | -√-3 | i | -√3 | √3 | -i | complex lifted from C4○D12 |
| ρ23 | 4 | 4 | -4 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | -4 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.5D4, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.5D4, Schur index 2 |
| ρ27 | 8 | -8 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 96 points
Generators in S96
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 22)(14 21)(15 20)(16 19)(17 18)(23 24)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 45)(38 44)(39 43)(40 42)(46 48)(49 57)(50 56)(51 55)(52 54)(58 60)(61 63)(64 72)(65 71)(66 70)(67 69)(73 82)(74 81)(75 80)(76 79)(77 78)(83 84)(86 96)(87 95)(88 94)(89 93)(90 92)
(1 29 18 84 7 35 24 78)(2 36 19 79 8 30 13 73)(3 31 20 74 9 25 14 80)(4 26 21 81 10 32 15 75)(5 33 22 76 11 27 16 82)(6 28 23 83 12 34 17 77)(37 58 93 61 43 52 87 67)(38 53 94 68 44 59 88 62)(39 60 95 63 45 54 89 69)(40 55 96 70 46 49 90 64)(41 50 85 65 47 56 91 71)(42 57 86 72 48 51 92 66)
(1 70 10 67 7 64 4 61)(2 71 11 68 8 65 5 62)(3 72 12 69 9 66 6 63)(13 50 22 59 19 56 16 53)(14 51 23 60 20 57 17 54)(15 52 24 49 21 58 18 55)(25 95 34 92 31 89 28 86)(26 96 35 93 32 90 29 87)(27 85 36 94 33 91 30 88)(37 75 46 84 43 81 40 78)(38 76 47 73 44 82 41 79)(39 77 48 74 45 83 42 80)
G:=sub<Sym(96)| (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(25,32)(26,31)(27,30)(28,29)(33,36)(34,35)(37,45)(38,44)(39,43)(40,42)(46,48)(49,57)(50,56)(51,55)(52,54)(58,60)(61,63)(64,72)(65,71)(66,70)(67,69)(73,82)(74,81)(75,80)(76,79)(77,78)(83,84)(86,96)(87,95)(88,94)(89,93)(90,92), (1,29,18,84,7,35,24,78)(2,36,19,79,8,30,13,73)(3,31,20,74,9,25,14,80)(4,26,21,81,10,32,15,75)(5,33,22,76,11,27,16,82)(6,28,23,83,12,34,17,77)(37,58,93,61,43,52,87,67)(38,53,94,68,44,59,88,62)(39,60,95,63,45,54,89,69)(40,55,96,70,46,49,90,64)(41,50,85,65,47,56,91,71)(42,57,86,72,48,51,92,66), (1,70,10,67,7,64,4,61)(2,71,11,68,8,65,5,62)(3,72,12,69,9,66,6,63)(13,50,22,59,19,56,16,53)(14,51,23,60,20,57,17,54)(15,52,24,49,21,58,18,55)(25,95,34,92,31,89,28,86)(26,96,35,93,32,90,29,87)(27,85,36,94,33,91,30,88)(37,75,46,84,43,81,40,78)(38,76,47,73,44,82,41,79)(39,77,48,74,45,83,42,80)>;
G:=Group( (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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24)(25,32)(26,31)(27,30)(28,29)(33,36)(34,35)(37,45)(38,44)(39,43)(40,42)(46,48)(49,57)(50,56)(51,55)(52,54)(58,60)(61,63)(64,72)(65,71)(66,70)(67,69)(73,82)(74,81)(75,80)(76,79)(77,78)(83,84)(86,96)(87,95)(88,94)(89,93)(90,92), (1,29,18,84,7,35,24,78)(2,36,19,79,8,30,13,73)(3,31,20,74,9,25,14,80)(4,26,21,81,10,32,15,75)(5,33,22,76,11,27,16,82)(6,28,23,83,12,34,17,77)(37,58,93,61,43,52,87,67)(38,53,94,68,44,59,88,62)(39,60,95,63,45,54,89,69)(40,55,96,70,46,49,90,64)(41,50,85,65,47,56,91,71)(42,57,86,72,48,51,92,66), (1,70,10,67,7,64,4,61)(2,71,11,68,8,65,5,62)(3,72,12,69,9,66,6,63)(13,50,22,59,19,56,16,53)(14,51,23,60,20,57,17,54)(15,52,24,49,21,58,18,55)(25,95,34,92,31,89,28,86)(26,96,35,93,32,90,29,87)(27,85,36,94,33,91,30,88)(37,75,46,84,43,81,40,78)(38,76,47,73,44,82,41,79)(39,77,48,74,45,83,42,80) );
G=PermutationGroup([[(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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,22),(14,21),(15,20),(16,19),(17,18),(23,24),(25,32),(26,31),(27,30),(28,29),(33,36),(34,35),(37,45),(38,44),(39,43),(40,42),(46,48),(49,57),(50,56),(51,55),(52,54),(58,60),(61,63),(64,72),(65,71),(66,70),(67,69),(73,82),(74,81),(75,80),(76,79),(77,78),(83,84),(86,96),(87,95),(88,94),(89,93),(90,92)], [(1,29,18,84,7,35,24,78),(2,36,19,79,8,30,13,73),(3,31,20,74,9,25,14,80),(4,26,21,81,10,32,15,75),(5,33,22,76,11,27,16,82),(6,28,23,83,12,34,17,77),(37,58,93,61,43,52,87,67),(38,53,94,68,44,59,88,62),(39,60,95,63,45,54,89,69),(40,55,96,70,46,49,90,64),(41,50,85,65,47,56,91,71),(42,57,86,72,48,51,92,66)], [(1,70,10,67,7,64,4,61),(2,71,11,68,8,65,5,62),(3,72,12,69,9,66,6,63),(13,50,22,59,19,56,16,53),(14,51,23,60,20,57,17,54),(15,52,24,49,21,58,18,55),(25,95,34,92,31,89,28,86),(26,96,35,93,32,90,29,87),(27,85,36,94,33,91,30,88),(37,75,46,84,43,81,40,78),(38,76,47,73,44,82,41,79),(39,77,48,74,45,83,42,80)]])
Matrix representation of D12.7D4 ►in GL8(𝔽73)
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 72 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 72 |
| 67 | 0 | 16 | 32 | 0 | 0 | 0 | 0 |
| 0 | 67 | 41 | 57 | 0 | 0 | 0 | 0 |
| 57 | 41 | 6 | 0 | 0 | 0 | 0 | 0 |
| 32 | 16 | 0 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 57 | 16 | 0 | 41 |
| 0 | 0 | 0 | 0 | 57 | 16 | 32 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 57 | 16 |
| 0 | 0 | 0 | 0 | 0 | 57 | 57 | 16 |
| 57 | 41 | 6 | 0 | 0 | 0 | 0 | 0 |
| 32 | 16 | 0 | 6 | 0 | 0 | 0 | 0 |
| 67 | 0 | 16 | 32 | 0 | 0 | 0 | 0 |
| 0 | 67 | 41 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 20 | 20 | 6 |
| 0 | 0 | 0 | 0 | 53 | 0 | 67 | 53 |
| 0 | 0 | 0 | 0 | 10 | 3 | 6 | 0 |
| 0 | 0 | 0 | 0 | 70 | 63 | 0 | 67 |
G:=sub<GL(8,GF(73))| [1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,2,72,0,0,0,0,0,2,0,0,72],[1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,72,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72],[67,0,57,32,0,0,0,0,0,67,41,16,0,0,0,0,16,41,6,0,0,0,0,0,32,57,0,6,0,0,0,0,0,0,0,0,57,57,16,0,0,0,0,0,16,16,0,57,0,0,0,0,0,32,57,57,0,0,0,0,41,0,16,16],[57,32,67,0,0,0,0,0,41,16,0,67,0,0,0,0,6,0,16,41,0,0,0,0,0,6,32,57,0,0,0,0,0,0,0,0,0,53,10,70,0,0,0,0,20,0,3,63,0,0,0,0,20,67,6,0,0,0,0,0,6,53,0,67] >;
D12.7D4 in GAP, Magma, Sage, TeX
D_{12}._7D_4 % in TeX
G:=Group("D12.7D4"); // GroupNames label
G:=SmallGroup(192,314);
// by ID
G=gap.SmallGroup(192,314);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,555,184,297,136,1684,851,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^4=a^6,d^2=a^9,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^6*b,d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^3>;
// generators/relations
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