metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊Q8, C4⋊1Dic6, Dic3⋊1Q8, Dic3.2D4, C3⋊2(C4⋊Q8), C4⋊C4.4S3, C2.4(S3×Q8), C6.5(C2×Q8), (C2×C4).42D6, C6.22(C2×D4), C2.11(S3×D4), C2.7(C2×Dic6), Dic3⋊C4.2C2, C4⋊Dic3.11C2, (C2×C6).29C23, (C2×C12).4C22, (C4×Dic3).1C2, (C2×Dic6).3C2, C22.46(C22×S3), (C2×Dic3).8C22, (C3×C4⋊C4).5C2, SmallGroup(96,95)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊Q8
G = < a,b,c | a12=b4=1, c2=b2, bab-1=a7, cac-1=a5, cbc-1=b-1 >
Subgroups: 138 in 68 conjugacy classes, 37 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, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×Dic6, C12⋊Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, Dic6, C22×S3, C4⋊Q8, C2×Dic6, S3×D4, S3×Q8, C12⋊Q8
Character table of C12⋊Q8
| 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 D6 |
| 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 S3 |
| 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 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 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 |
| ρ16 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ18 | 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 |
| ρ19 | 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 |
| ρ20 | 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 |
| ρ21 | 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 |
| ρ22 | 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 |
| 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 S3×D4 |
| 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 S3×Q8, 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 92 18 70)(2 87 19 65)(3 94 20 72)(4 89 21 67)(5 96 22 62)(6 91 23 69)(7 86 24 64)(8 93 13 71)(9 88 14 66)(10 95 15 61)(11 90 16 68)(12 85 17 63)(25 48 74 60)(26 43 75 55)(27 38 76 50)(28 45 77 57)(29 40 78 52)(30 47 79 59)(31 42 80 54)(32 37 81 49)(33 44 82 56)(34 39 83 51)(35 46 84 58)(36 41 73 53)
(1 84 18 35)(2 77 19 28)(3 82 20 33)(4 75 21 26)(5 80 22 31)(6 73 23 36)(7 78 24 29)(8 83 13 34)(9 76 14 27)(10 81 15 32)(11 74 16 25)(12 79 17 30)(37 61 49 95)(38 66 50 88)(39 71 51 93)(40 64 52 86)(41 69 53 91)(42 62 54 96)(43 67 55 89)(44 72 56 94)(45 65 57 87)(46 70 58 92)(47 63 59 85)(48 68 60 90)
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), (1,92,18,70)(2,87,19,65)(3,94,20,72)(4,89,21,67)(5,96,22,62)(6,91,23,69)(7,86,24,64)(8,93,13,71)(9,88,14,66)(10,95,15,61)(11,90,16,68)(12,85,17,63)(25,48,74,60)(26,43,75,55)(27,38,76,50)(28,45,77,57)(29,40,78,52)(30,47,79,59)(31,42,80,54)(32,37,81,49)(33,44,82,56)(34,39,83,51)(35,46,84,58)(36,41,73,53), (1,84,18,35)(2,77,19,28)(3,82,20,33)(4,75,21,26)(5,80,22,31)(6,73,23,36)(7,78,24,29)(8,83,13,34)(9,76,14,27)(10,81,15,32)(11,74,16,25)(12,79,17,30)(37,61,49,95)(38,66,50,88)(39,71,51,93)(40,64,52,86)(41,69,53,91)(42,62,54,96)(43,67,55,89)(44,72,56,94)(45,65,57,87)(46,70,58,92)(47,63,59,85)(48,68,60,90)>;
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), (1,92,18,70)(2,87,19,65)(3,94,20,72)(4,89,21,67)(5,96,22,62)(6,91,23,69)(7,86,24,64)(8,93,13,71)(9,88,14,66)(10,95,15,61)(11,90,16,68)(12,85,17,63)(25,48,74,60)(26,43,75,55)(27,38,76,50)(28,45,77,57)(29,40,78,52)(30,47,79,59)(31,42,80,54)(32,37,81,49)(33,44,82,56)(34,39,83,51)(35,46,84,58)(36,41,73,53), (1,84,18,35)(2,77,19,28)(3,82,20,33)(4,75,21,26)(5,80,22,31)(6,73,23,36)(7,78,24,29)(8,83,13,34)(9,76,14,27)(10,81,15,32)(11,74,16,25)(12,79,17,30)(37,61,49,95)(38,66,50,88)(39,71,51,93)(40,64,52,86)(41,69,53,91)(42,62,54,96)(43,67,55,89)(44,72,56,94)(45,65,57,87)(46,70,58,92)(47,63,59,85)(48,68,60,90) );
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)], [(1,92,18,70),(2,87,19,65),(3,94,20,72),(4,89,21,67),(5,96,22,62),(6,91,23,69),(7,86,24,64),(8,93,13,71),(9,88,14,66),(10,95,15,61),(11,90,16,68),(12,85,17,63),(25,48,74,60),(26,43,75,55),(27,38,76,50),(28,45,77,57),(29,40,78,52),(30,47,79,59),(31,42,80,54),(32,37,81,49),(33,44,82,56),(34,39,83,51),(35,46,84,58),(36,41,73,53)], [(1,84,18,35),(2,77,19,28),(3,82,20,33),(4,75,21,26),(5,80,22,31),(6,73,23,36),(7,78,24,29),(8,83,13,34),(9,76,14,27),(10,81,15,32),(11,74,16,25),(12,79,17,30),(37,61,49,95),(38,66,50,88),(39,71,51,93),(40,64,52,86),(41,69,53,91),(42,62,54,96),(43,67,55,89),(44,72,56,94),(45,65,57,87),(46,70,58,92),(47,63,59,85),(48,68,60,90)]])
C12⋊Q8 is a maximal subgroup of
Dic3.D8 Dic3.SD16 D4⋊Dic6 C12⋊Q8⋊C2 Q8⋊2Dic6 Dic3.1Q16 Q8⋊3Dic6 (C2×C8).D6 Dic6⋊Q8 C24⋊5Q8 C24⋊3Q8 D12⋊Q8 C24⋊2Q8 Dic3.Q16 C24⋊4Q8 D12⋊2Q8 C6.72+ 1+4 C6.2- 1+4 C6.102+ 1+4 C42.88D6 C42.90D6 C42.97D6 C42.98D6 D4×Dic6 D4⋊5Dic6 C42.228D6 C42.115D6 Q8×Dic6 Q8⋊6Dic6 C42.232D6 C42.133D6 C12⋊(C4○D4) C6.712- 1+4 C6.732- 1+4 C6.452+ 1+4 (Q8×Dic3)⋊C2 C6.752- 1+4 C6.162- 1+4 Dic6⋊21D4 C6.1182+ 1+4 C6.232- 1+4 C6.242- 1+4 C6.252- 1+4 C6.792- 1+4 C6.812- 1+4 C6.822- 1+4 C6.652+ 1+4 Dic6⋊7Q8 C42.236D6 C42.148D6 D12⋊7Q8 C42.154D6 C42.157D6 C42.159D6 C42.160D6 C42.164D6 C42.165D6 Dic6⋊9Q8 S3×C4⋊Q8 D12⋊8Q8 C42.174D6 C36⋊Q8 C62.9C23 C62.10C23 C12⋊3Dic6 C12⋊Dic6 C12⋊2Dic6 Dic3⋊Dic10 Dic15⋊Q8 C20⋊4Dic6 C20⋊Dic6 C4⋊Dic30
C12⋊Q8 is a maximal quotient of
(C2×C12)⋊Q8 C6.(C4×Q8) (C2×C4)⋊Dic6 C6.(C4⋊Q8) C24⋊5Q8 C24⋊3Q8 C8.8Dic6 C24⋊2Q8 C24⋊4Q8 C8.6Dic6 C12⋊(C4⋊C4) (C4×Dic3)⋊8C4 (C2×Dic3)⋊Q8 (C2×C12).54D4 C4⋊C4⋊6Dic3 C36⋊Q8 C62.9C23 C62.10C23 C12⋊3Dic6 C12⋊Dic6 C12⋊2Dic6 Dic3⋊Dic10 Dic15⋊Q8 C20⋊4Dic6 C20⋊Dic6 C4⋊Dic30
Matrix representation of C12⋊Q8 ►in GL4(𝔽13) generated by
| 10 | 0 | 0 | 0 |
| 2 | 4 | 0 | 0 |
| 0 | 0 | 7 | 11 |
| 0 | 0 | 12 | 6 |
| 8 | 0 | 0 | 0 |
| 1 | 5 | 0 | 0 |
| 0 | 0 | 9 | 3 |
| 0 | 0 | 3 | 4 |
| 3 | 4 | 0 | 0 |
| 4 | 10 | 0 | 0 |
| 0 | 0 | 6 | 2 |
| 0 | 0 | 1 | 7 |
G:=sub<GL(4,GF(13))| [10,2,0,0,0,4,0,0,0,0,7,12,0,0,11,6],[8,1,0,0,0,5,0,0,0,0,9,3,0,0,3,4],[3,4,0,0,4,10,0,0,0,0,6,1,0,0,2,7] >;
C12⋊Q8 in GAP, Magma, Sage, TeX
C_{12}\rtimes Q_8 % in TeX
G:=Group("C12:Q8"); // GroupNames label
G:=SmallGroup(96,95);
// by ID
G=gap.SmallGroup(96,95);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,103,218,188,50,2309]);
// 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=b^-1>;
// generators/relations
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