metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.3Q8, C4.3Dic6, C4⋊C4.6S3, C6.6(C2×Q8), (C2×C4).43D6, C4⋊Dic3.7C2, C2.8(C2×Dic6), C3⋊3(C42.C2), C6.25(C4○D4), Dic3⋊C4.3C2, (C2×C6).31C23, (C4×Dic3).2C2, (C2×C12).22C22, C2.4(Q8⋊3S3), C2.12(D4⋊2S3), C22.48(C22×S3), (C2×Dic3).10C22, (C3×C4⋊C4).7C2, SmallGroup(96,97)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.Dic6
G = < a,b,c | a12=b4=1, c2=b2, bab-1=a7, cac-1=a5, cbc-1=a6b-1 >
Subgroups: 106 in 56 conjugacy classes, 33 normal (19 characteristic)
C1, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C42.C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C3×C4⋊C4, C4.Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, Dic6, C22×S3, C42.C2, C2×Dic6, D4⋊2S3, Q8⋊3S3, C4.Dic6
Character table of C4.Dic6
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | √3 | -√3 | -1 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ16 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -√3 | √3 | -1 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ17 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | √3 | √3 | 1 | -√3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ18 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | -√3 | -√3 | 1 | √3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ19 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ21 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
| ρ24 | 4 | -4 | 4 | -4 | -2 | 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 90 78 70)(2 85 79 65)(3 92 80 72)(4 87 81 67)(5 94 82 62)(6 89 83 69)(7 96 84 64)(8 91 73 71)(9 86 74 66)(10 93 75 61)(11 88 76 68)(12 95 77 63)(13 33 37 54)(14 28 38 49)(15 35 39 56)(16 30 40 51)(17 25 41 58)(18 32 42 53)(19 27 43 60)(20 34 44 55)(21 29 45 50)(22 36 46 57)(23 31 47 52)(24 26 48 59)
(1 22 78 46)(2 15 79 39)(3 20 80 44)(4 13 81 37)(5 18 82 42)(6 23 83 47)(7 16 84 40)(8 21 73 45)(9 14 74 38)(10 19 75 43)(11 24 76 48)(12 17 77 41)(25 89 58 69)(26 94 59 62)(27 87 60 67)(28 92 49 72)(29 85 50 65)(30 90 51 70)(31 95 52 63)(32 88 53 68)(33 93 54 61)(34 86 55 66)(35 91 56 71)(36 96 57 64)
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), (1,90,78,70)(2,85,79,65)(3,92,80,72)(4,87,81,67)(5,94,82,62)(6,89,83,69)(7,96,84,64)(8,91,73,71)(9,86,74,66)(10,93,75,61)(11,88,76,68)(12,95,77,63)(13,33,37,54)(14,28,38,49)(15,35,39,56)(16,30,40,51)(17,25,41,58)(18,32,42,53)(19,27,43,60)(20,34,44,55)(21,29,45,50)(22,36,46,57)(23,31,47,52)(24,26,48,59), (1,22,78,46)(2,15,79,39)(3,20,80,44)(4,13,81,37)(5,18,82,42)(6,23,83,47)(7,16,84,40)(8,21,73,45)(9,14,74,38)(10,19,75,43)(11,24,76,48)(12,17,77,41)(25,89,58,69)(26,94,59,62)(27,87,60,67)(28,92,49,72)(29,85,50,65)(30,90,51,70)(31,95,52,63)(32,88,53,68)(33,93,54,61)(34,86,55,66)(35,91,56,71)(36,96,57,64)>;
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), (1,90,78,70)(2,85,79,65)(3,92,80,72)(4,87,81,67)(5,94,82,62)(6,89,83,69)(7,96,84,64)(8,91,73,71)(9,86,74,66)(10,93,75,61)(11,88,76,68)(12,95,77,63)(13,33,37,54)(14,28,38,49)(15,35,39,56)(16,30,40,51)(17,25,41,58)(18,32,42,53)(19,27,43,60)(20,34,44,55)(21,29,45,50)(22,36,46,57)(23,31,47,52)(24,26,48,59), (1,22,78,46)(2,15,79,39)(3,20,80,44)(4,13,81,37)(5,18,82,42)(6,23,83,47)(7,16,84,40)(8,21,73,45)(9,14,74,38)(10,19,75,43)(11,24,76,48)(12,17,77,41)(25,89,58,69)(26,94,59,62)(27,87,60,67)(28,92,49,72)(29,85,50,65)(30,90,51,70)(31,95,52,63)(32,88,53,68)(33,93,54,61)(34,86,55,66)(35,91,56,71)(36,96,57,64) );
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)], [(1,90,78,70),(2,85,79,65),(3,92,80,72),(4,87,81,67),(5,94,82,62),(6,89,83,69),(7,96,84,64),(8,91,73,71),(9,86,74,66),(10,93,75,61),(11,88,76,68),(12,95,77,63),(13,33,37,54),(14,28,38,49),(15,35,39,56),(16,30,40,51),(17,25,41,58),(18,32,42,53),(19,27,43,60),(20,34,44,55),(21,29,45,50),(22,36,46,57),(23,31,47,52),(24,26,48,59)], [(1,22,78,46),(2,15,79,39),(3,20,80,44),(4,13,81,37),(5,18,82,42),(6,23,83,47),(7,16,84,40),(8,21,73,45),(9,14,74,38),(10,19,75,43),(11,24,76,48),(12,17,77,41),(25,89,58,69),(26,94,59,62),(27,87,60,67),(28,92,49,72),(29,85,50,65),(30,90,51,70),(31,95,52,63),(32,88,53,68),(33,93,54,61),(34,86,55,66),(35,91,56,71),(36,96,57,64)]])
C4.Dic6 is a maximal subgroup of
D4.Dic6 C4⋊C4.D6 D4.2Dic6 (C2×C8).200D6 Q8.3Dic6 (C2×Q8).36D6 Q8.4Dic6 Q8⋊C4⋊S3 C24⋊3Q8 Dic6.Q8 C8.8Dic6 D12.Q8 C24⋊4Q8 Dic6.2Q8 C8.6Dic6 D12.2Q8 C6.72+ 1+4 C6.52- 1+4 C6.112+ 1+4 C42.88D6 C42.90D6 C42.94D6 C42.95D6 D4⋊5Dic6 D4⋊6Dic6 C42.229D6 C42.116D6 Q8⋊6Dic6 Q8⋊7Dic6 C42.131D6 C42.134D6 C4⋊C4.178D6 C6.432+ 1+4 C6.1152+ 1+4 C6.472+ 1+4 C4⋊C4.187D6 C6.152- 1+4 C6.212- 1+4 C6.772- 1+4 C6.782- 1+4 C6.252- 1+4 C6.802- 1+4 C6.632+ 1+4 C6.852- 1+4 C6.692+ 1+4 Dic6⋊7Q8 C42.147D6 S3×C42.C2 C42.148D6 D12⋊7Q8 C42.152D6 C42.154D6 C42.156D6 C42.159D6 C42.161D6 C42.162D6 C42.165D6 Dic6⋊8Q8 C42.241D6 C42.174D6 D12⋊9Q8 C36.3Q8 Dic3.Dic6 C62.16C23 C62.39C23 C62.42C23 C62.234C23 Dic5.1Dic6 Dic5.2Dic6 C60.6Q8 C12.Dic10 C4.Dic30
C4.Dic6 is a maximal quotient of
C2.(C4×Dic6) Dic3⋊C4⋊C4 (C2×C4).Dic6 (C22×C4).85D6 C12⋊(C4⋊C4) (C4×Dic3)⋊9C4 (C2×C12).54D4 C4⋊C4⋊6Dic3 (C2×C12).55D4 C36.3Q8 Dic3.Dic6 C62.16C23 C62.39C23 C62.42C23 C62.234C23 Dic5.1Dic6 Dic5.2Dic6 C60.6Q8 C12.Dic10 C4.Dic30
Matrix representation of C4.Dic6 ►in GL4(𝔽13) generated by
| 4 | 0 | 0 | 0 |
| 3 | 10 | 0 | 0 |
| 0 | 0 | 8 | 3 |
| 0 | 0 | 0 | 5 |
| 8 | 0 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 0 | 0 | 12 | 11 |
| 0 | 0 | 1 | 1 |
| 12 | 11 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [4,3,0,0,0,10,0,0,0,0,8,0,0,0,3,5],[8,5,0,0,0,5,0,0,0,0,12,1,0,0,11,1],[12,1,0,0,11,1,0,0,0,0,5,0,0,0,0,5] >;
C4.Dic6 in GAP, Magma, Sage, TeX
C_4.{\rm Dic}_6 % in TeX
G:=Group("C4.Dic6"); // GroupNames label
G:=SmallGroup(96,97);
// by ID
G=gap.SmallGroup(96,97);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,103,506,188,50,2309]);
// Polycyclic
G:=Group<a,b,c|a^12=b^4=1,c^2=b^2,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^6*b^-1>;
// generators/relations
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