metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊5C4, Dic3⋊3Q8, C3⋊2(C4×Q8), C4⋊C4.7S3, C4.4(C4×S3), C2.1(S3×Q8), Dic3○(C4⋊C4), (C2×C4).29D6, C6.10(C2×Q8), C12.10(C2×C4), C6.8(C22×C4), C6.24(C4○D4), Dic3⋊C4.4C2, (C2×C6).28C23, Dic3.2(C2×C4), (C2×Dic6).7C2, (C4×Dic3).8C2, C2.3(D4⋊2S3), (C2×C12).21C22, C22.15(C22×S3), (C2×Dic3).48C22, C2.10(S3×C2×C4), (C3×C4⋊C4).4C2, SmallGroup(96,94)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊C4
G = < a,b,c | a12=c4=1, b2=a6, bab-1=a-1, cac-1=a7, bc=cb >
Subgroups: 122 in 70 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C4×Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, C3×C4⋊C4, C2×Dic6, Dic6⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, C22×C4, C2×Q8, C4○D4, C4×S3, C22×S3, C4×Q8, S3×C2×C4, D4⋊2S3, S3×Q8, Dic6⋊C4
Character table of Dic6⋊C4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 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 | 1 | -1 | 1 | -1 | 1 | 1 | i | -i | -1 | -i | i | i | -i | i | -i | -1 | -1 | i | -i | 1 | 1 | -1 | 1 | -1 | i | -i | -i | 1 | i | -1 | linear of order 4 |
| ρ10 | 1 | -1 | 1 | -1 | 1 | 1 | -i | i | -1 | i | -i | -i | i | -i | i | -1 | -1 | -i | i | 1 | 1 | -1 | 1 | -1 | -i | i | i | 1 | -i | -1 | linear of order 4 |
| ρ11 | 1 | -1 | 1 | -1 | 1 | 1 | i | -i | -1 | -i | i | -i | i | -i | i | 1 | 1 | -i | i | -1 | -1 | -1 | 1 | -1 | i | -i | -i | 1 | i | -1 | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | 1 | -i | i | -1 | i | -i | i | -i | i | -i | 1 | 1 | i | -i | -1 | -1 | -1 | 1 | -1 | -i | i | i | 1 | -i | -1 | linear of order 4 |
| ρ13 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | 1 | i | -i | i | -i | i | -i | 1 | -1 | -i | i | -1 | 1 | -1 | 1 | -1 | i | -i | i | -1 | -i | 1 | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | 1 | -i | i | -i | i | -i | i | 1 | -1 | i | -i | -1 | 1 | -1 | 1 | -1 | -i | i | -i | -1 | i | 1 | linear of order 4 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | 1 | i | -i | -i | i | -i | i | -1 | 1 | i | -i | 1 | -1 | -1 | 1 | -1 | i | -i | i | -1 | -i | 1 | linear of order 4 |
| ρ16 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | 1 | -i | i | i | -i | i | -i | -1 | 1 | -i | i | 1 | -1 | -1 | 1 | -1 | -i | i | -i | -1 | i | 1 | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | -1 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ19 | 2 | 2 | 2 | 2 | -1 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ20 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ21 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ22 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ23 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | -1 | -2 | -2i | 2i | 2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | i | -i | i | 1 | -i | -1 | complex lifted from C4×S3 |
| ρ25 | 2 | -2 | 2 | -2 | -1 | 2 | 2i | -2i | -2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -i | i | i | -1 | -i | 1 | complex lifted from C4×S3 |
| ρ26 | 2 | -2 | 2 | -2 | -1 | -2 | 2i | -2i | 2 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -i | i | -i | 1 | i | -1 | complex lifted from C4×S3 |
| ρ27 | 2 | -2 | 2 | -2 | -1 | 2 | -2i | 2i | -2 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | i | -i | -i | -1 | i | 1 | complex lifted from C4×S3 |
| ρ28 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ29 | 4 | 4 | -4 | -4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Q8, Schur index 2 |
| ρ30 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
►Regular action on 96 points
(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 84 7 78)(2 83 8 77)(3 82 9 76)(4 81 10 75)(5 80 11 74)(6 79 12 73)(13 87 19 93)(14 86 20 92)(15 85 21 91)(16 96 22 90)(17 95 23 89)(18 94 24 88)(25 38 31 44)(26 37 32 43)(27 48 33 42)(28 47 34 41)(29 46 35 40)(30 45 36 39)(49 69 55 63)(50 68 56 62)(51 67 57 61)(52 66 58 72)(53 65 59 71)(54 64 60 70)
(1 51 41 21)(2 58 42 16)(3 53 43 23)(4 60 44 18)(5 55 45 13)(6 50 46 20)(7 57 47 15)(8 52 48 22)(9 59 37 17)(10 54 38 24)(11 49 39 19)(12 56 40 14)(25 94 81 70)(26 89 82 65)(27 96 83 72)(28 91 84 67)(29 86 73 62)(30 93 74 69)(31 88 75 64)(32 95 76 71)(33 90 77 66)(34 85 78 61)(35 92 79 68)(36 87 80 63)
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), (1,84,7,78)(2,83,8,77)(3,82,9,76)(4,81,10,75)(5,80,11,74)(6,79,12,73)(13,87,19,93)(14,86,20,92)(15,85,21,91)(16,96,22,90)(17,95,23,89)(18,94,24,88)(25,38,31,44)(26,37,32,43)(27,48,33,42)(28,47,34,41)(29,46,35,40)(30,45,36,39)(49,69,55,63)(50,68,56,62)(51,67,57,61)(52,66,58,72)(53,65,59,71)(54,64,60,70), (1,51,41,21)(2,58,42,16)(3,53,43,23)(4,60,44,18)(5,55,45,13)(6,50,46,20)(7,57,47,15)(8,52,48,22)(9,59,37,17)(10,54,38,24)(11,49,39,19)(12,56,40,14)(25,94,81,70)(26,89,82,65)(27,96,83,72)(28,91,84,67)(29,86,73,62)(30,93,74,69)(31,88,75,64)(32,95,76,71)(33,90,77,66)(34,85,78,61)(35,92,79,68)(36,87,80,63)>;
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), (1,84,7,78)(2,83,8,77)(3,82,9,76)(4,81,10,75)(5,80,11,74)(6,79,12,73)(13,87,19,93)(14,86,20,92)(15,85,21,91)(16,96,22,90)(17,95,23,89)(18,94,24,88)(25,38,31,44)(26,37,32,43)(27,48,33,42)(28,47,34,41)(29,46,35,40)(30,45,36,39)(49,69,55,63)(50,68,56,62)(51,67,57,61)(52,66,58,72)(53,65,59,71)(54,64,60,70), (1,51,41,21)(2,58,42,16)(3,53,43,23)(4,60,44,18)(5,55,45,13)(6,50,46,20)(7,57,47,15)(8,52,48,22)(9,59,37,17)(10,54,38,24)(11,49,39,19)(12,56,40,14)(25,94,81,70)(26,89,82,65)(27,96,83,72)(28,91,84,67)(29,86,73,62)(30,93,74,69)(31,88,75,64)(32,95,76,71)(33,90,77,66)(34,85,78,61)(35,92,79,68)(36,87,80,63) );
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)], [(1,84,7,78),(2,83,8,77),(3,82,9,76),(4,81,10,75),(5,80,11,74),(6,79,12,73),(13,87,19,93),(14,86,20,92),(15,85,21,91),(16,96,22,90),(17,95,23,89),(18,94,24,88),(25,38,31,44),(26,37,32,43),(27,48,33,42),(28,47,34,41),(29,46,35,40),(30,45,36,39),(49,69,55,63),(50,68,56,62),(51,67,57,61),(52,66,58,72),(53,65,59,71),(54,64,60,70)], [(1,51,41,21),(2,58,42,16),(3,53,43,23),(4,60,44,18),(5,55,45,13),(6,50,46,20),(7,57,47,15),(8,52,48,22),(9,59,37,17),(10,54,38,24),(11,49,39,19),(12,56,40,14),(25,94,81,70),(26,89,82,65),(27,96,83,72),(28,91,84,67),(29,86,73,62),(30,93,74,69),(31,88,75,64),(32,95,76,71),(33,90,77,66),(34,85,78,61),(35,92,79,68),(36,87,80,63)]])
Dic6⋊C4 is a maximal subgroup of
D4.S3⋊C4 Dic3⋊6SD16 Dic6⋊2D4 Dic6.D4 C3⋊Q16⋊C4 Dic3⋊4Q16 Dic3⋊Q16 Dic6.11D4 Dic3⋊8SD16 Dic12⋊9C4 Dic6⋊Q8 Dic6.Q8 Dic3⋊5Q16 Dic3.Q16 Dic6.2Q8 C24⋊C2⋊C4 C6.82+ 1+4 C6.102+ 1+4 C6.52- 1+4 C42.87D6 C42.188D6 C42.94D6 C42.98D6 C4×D4⋊2S3 C42.106D6 C42.108D6 C42.114D6 Dic6⋊10Q8 C42.122D6 C4×S3×Q8 C42.125D6 Dic6⋊19D4 Dic6⋊20D4 C4⋊C4.178D6 C6.712- 1+4 (Q8×Dic3)⋊C2 C6.152- 1+4 Dic6⋊21D4 Dic6⋊22D4 C6.522+ 1+4 C6.222- 1+4 C6.232- 1+4 C4⋊C4.197D6 C6.802- 1+4 C6.652+ 1+4 C6.672+ 1+4 Dic6⋊7Q8 C42.236D6 C42.237D6 C42.151D6 C42.154D6 C42.156D6 C42.159D6 C42.160D6 C42.189D6 Dic6⋊8Q8 Dic6⋊9Q8 C42.241D6 C42.178D6 Dic9⋊3Q8 Dic3⋊5Dic6 C62.8C23 C62.13C23 Dic3⋊6Dic6 C62.231C23 Dic3⋊5Dic10 Dic15⋊5Q8 Dic30⋊14C4 Dic15⋊7Q8 Dic15⋊10Q8 Dic6⋊5F5
Dic6⋊C4 is a maximal quotient of
(C2×C12)⋊Q8 Dic3⋊C42 C6.(C4×D4) C2.(C4×D12) C42.27D6 Dic6⋊C8 C42.198D6 C12⋊(C4⋊C4) C4.(D6⋊C4) Dic3×C4⋊C4 Dic3⋊(C4⋊C4) C6.67(C4×D4) Dic9⋊3Q8 Dic3⋊5Dic6 C62.8C23 C62.13C23 Dic3⋊6Dic6 C62.231C23 Dic3⋊5Dic10 Dic15⋊5Q8 Dic30⋊14C4 Dic15⋊7Q8 Dic15⋊10Q8 Dic6⋊5F5
Matrix representation of Dic6⋊C4 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 1 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 3 | 0 | 0 |
| 0 | 6 | 10 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,12,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,1,0],[1,0,0,0,0,0,3,6,0,0,0,3,10,0,0,0,0,0,8,0,0,0,0,0,5],[5,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1] >;
Dic6⋊C4 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes C_4 % in TeX
G:=Group("Dic6:C4"); // GroupNames label
G:=SmallGroup(96,94);
// by ID
G=gap.SmallGroup(96,94);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,55,116,122,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=c^4=1,b^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,b*c=c*b>;
// generators/relations
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