metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.2Q8, C6.4SD16, C4.2Dic6, C3⋊C8⋊2C4, C4⋊C4.2S3, C3⋊1(C4.Q8), C6.3(C4⋊C4), C4.12(C4×S3), C12.2(C2×C4), (C2×C4).34D6, (C2×C6).29D4, C4⋊Dic3.9C2, C2.1(D4.S3), (C2×C12).9C22, C2.4(Dic3⋊C4), C2.1(Q8⋊2S3), C22.13(C3⋊D4), (C2×C3⋊C8).2C2, (C3×C4⋊C4).2C2, SmallGroup(96,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.Q8
G = < a,b,c | a4=b12=1, c2=ab6, bab-1=a-1, ac=ca, cbc-1=a-1b-1 >
Character table of C12.Q8
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 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 | i | -i | -i | i | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | -i | 1 | -1 | i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | i | -i | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | i | 1 | -1 | -i | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | i | -i | -1 | 1 | -1 | 1 | -1 | -1 | 1 | i | -i | -i | 1 | -1 | i | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | -i | i | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -i | i | i | 1 | -1 | -i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ13 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -√3 | -√3 | √3 | 1 | -1 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ14 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | √3 | √3 | -√3 | 1 | -1 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 2i | -2i | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -i | i | i | -1 | 1 | -i | complex lifted from C4×S3 |
| ρ16 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | -2i | 2i | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | i | -i | -i | -1 | 1 | i | complex lifted from C4×S3 |
| ρ17 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | √-3 | -√-3 | √-3 | 1 | 1 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -√-3 | √-3 | -√-3 | 1 | 1 | √-3 | complex lifted from C3⋊D4 |
| ρ19 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | √-2 | √-2 | -√-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | √-2 | -√-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ21 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -√-2 | -√-2 | √-2 | √-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ22 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -√-2 | √-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ23 | 4 | -4 | -4 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊2S3 |
| ρ24 | 4 | -4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4.S3, Schur index 2 |
►Regular action on 96 points
(1 81 13 57)(2 58 14 82)(3 83 15 59)(4 60 16 84)(5 73 17 49)(6 50 18 74)(7 75 19 51)(8 52 20 76)(9 77 21 53)(10 54 22 78)(11 79 23 55)(12 56 24 80)(25 46 90 72)(26 61 91 47)(27 48 92 62)(28 63 93 37)(29 38 94 64)(30 65 95 39)(31 40 96 66)(32 67 85 41)(33 42 86 68)(34 69 87 43)(35 44 88 70)(36 71 89 45)
(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 61 75 85 13 47 51 32)(2 90 52 66 14 25 76 40)(3 71 77 95 15 45 53 30)(4 88 54 64 16 35 78 38)(5 69 79 93 17 43 55 28)(6 86 56 62 18 33 80 48)(7 67 81 91 19 41 57 26)(8 96 58 72 20 31 82 46)(9 65 83 89 21 39 59 36)(10 94 60 70 22 29 84 44)(11 63 73 87 23 37 49 34)(12 92 50 68 24 27 74 42)
G:=sub<Sym(96)| (1,81,13,57)(2,58,14,82)(3,83,15,59)(4,60,16,84)(5,73,17,49)(6,50,18,74)(7,75,19,51)(8,52,20,76)(9,77,21,53)(10,54,22,78)(11,79,23,55)(12,56,24,80)(25,46,90,72)(26,61,91,47)(27,48,92,62)(28,63,93,37)(29,38,94,64)(30,65,95,39)(31,40,96,66)(32,67,85,41)(33,42,86,68)(34,69,87,43)(35,44,88,70)(36,71,89,45), (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), (1,61,75,85,13,47,51,32)(2,90,52,66,14,25,76,40)(3,71,77,95,15,45,53,30)(4,88,54,64,16,35,78,38)(5,69,79,93,17,43,55,28)(6,86,56,62,18,33,80,48)(7,67,81,91,19,41,57,26)(8,96,58,72,20,31,82,46)(9,65,83,89,21,39,59,36)(10,94,60,70,22,29,84,44)(11,63,73,87,23,37,49,34)(12,92,50,68,24,27,74,42)>;
G:=Group( (1,81,13,57)(2,58,14,82)(3,83,15,59)(4,60,16,84)(5,73,17,49)(6,50,18,74)(7,75,19,51)(8,52,20,76)(9,77,21,53)(10,54,22,78)(11,79,23,55)(12,56,24,80)(25,46,90,72)(26,61,91,47)(27,48,92,62)(28,63,93,37)(29,38,94,64)(30,65,95,39)(31,40,96,66)(32,67,85,41)(33,42,86,68)(34,69,87,43)(35,44,88,70)(36,71,89,45), (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), (1,61,75,85,13,47,51,32)(2,90,52,66,14,25,76,40)(3,71,77,95,15,45,53,30)(4,88,54,64,16,35,78,38)(5,69,79,93,17,43,55,28)(6,86,56,62,18,33,80,48)(7,67,81,91,19,41,57,26)(8,96,58,72,20,31,82,46)(9,65,83,89,21,39,59,36)(10,94,60,70,22,29,84,44)(11,63,73,87,23,37,49,34)(12,92,50,68,24,27,74,42) );
G=PermutationGroup([[(1,81,13,57),(2,58,14,82),(3,83,15,59),(4,60,16,84),(5,73,17,49),(6,50,18,74),(7,75,19,51),(8,52,20,76),(9,77,21,53),(10,54,22,78),(11,79,23,55),(12,56,24,80),(25,46,90,72),(26,61,91,47),(27,48,92,62),(28,63,93,37),(29,38,94,64),(30,65,95,39),(31,40,96,66),(32,67,85,41),(33,42,86,68),(34,69,87,43),(35,44,88,70),(36,71,89,45)], [(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)], [(1,61,75,85,13,47,51,32),(2,90,52,66,14,25,76,40),(3,71,77,95,15,45,53,30),(4,88,54,64,16,35,78,38),(5,69,79,93,17,43,55,28),(6,86,56,62,18,33,80,48),(7,67,81,91,19,41,57,26),(8,96,58,72,20,31,82,46),(9,65,83,89,21,39,59,36),(10,94,60,70,22,29,84,44),(11,63,73,87,23,37,49,34),(12,92,50,68,24,27,74,42)]])
C12.Q8 is a maximal subgroup of
Dic3⋊6SD16 D4⋊Dic6 D4.2Dic6 D6.SD16 D6⋊SD16 C3⋊C8⋊D4 C24⋊1C4⋊C2 D4⋊S3⋊C4 Dic3⋊7SD16 C3⋊Q16⋊C4 Q8⋊2Dic6 Q8.3Dic6 D6.1SD16 D6⋊2SD16 D6⋊C8.C2 C3⋊C8.D4 Dic6⋊Q8 C24⋊5Q8 C24⋊3Q8 S3×C4.Q8 C8⋊(C4×S3) D6.2SD16 D6.4SD16 D12⋊Q8 C24⋊4Q8 Dic6.2Q8 C8.6Dic6 C8.27(C4×S3) C8⋊S3⋊C4 C2.D8⋊S3 C2.D8⋊7S3 D12.2Q8 C4⋊C4.225D6 C4⋊C4.228D6 C4⋊C4.231D6 C4⋊C4.232D6 C4⋊C4.233D6 C4⋊C4.234D6 C4⋊C4.236D6 C12.38SD16 D4.3Dic6 C42.48D6 C4×D4.S3 Q8⋊4Dic6 Q8.5Dic6 C4×Q8⋊2S3 C42.59D6 C4⋊D4.S3 C6.Q16⋊C2 C4⋊D4⋊S3 C3⋊C8⋊23D4 (C2×Q8).49D6 (C2×Q8).51D6 C3⋊C8⋊24D4 C3⋊C8.6D4 Dic6.4Q8 C42.68D6 C42.215D6 D12.4Q8 C12.SD16 C42.76D6 D12⋊5Q8 Dic6⋊6Q8 C4.Dic18 C12.Dic6 C12.6Dic6 C12.10Dic6 C30.SD16 C60.Q8 C60.2Q8 Dic5.Dic6
C12.Q8 is a maximal quotient of
C12.39SD16 C8.Dic6 C24.6Q8 C12.C42 C4.Dic18 C12.Dic6 C12.6Dic6 C12.10Dic6 C30.SD16 C60.Q8 C60.2Q8 Dic5.Dic6
Matrix representation of C12.Q8 ►in GL5(𝔽73)
| 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 2 | 0 | 0 |
| 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 27 | 0 | 0 | 0 | 0 |
| 0 | 58 | 39 | 0 | 0 |
| 0 | 41 | 15 | 0 | 0 |
| 0 | 0 | 0 | 30 | 30 |
| 0 | 0 | 0 | 43 | 60 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 67 | 12 | 0 | 0 |
| 0 | 0 | 0 | 5 | 54 |
| 0 | 0 | 0 | 59 | 68 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,72,0,0,0,2,1,0,0,0,0,0,72,0,0,0,0,0,72],[27,0,0,0,0,0,58,41,0,0,0,39,15,0,0,0,0,0,30,43,0,0,0,30,60],[72,0,0,0,0,0,0,67,0,0,0,12,12,0,0,0,0,0,5,59,0,0,0,54,68] >;
C12.Q8 in GAP, Magma, Sage, TeX
C_{12}.Q_8 % in TeX
G:=Group("C12.Q8"); // GroupNames label
G:=SmallGroup(96,15);
// by ID
G=gap.SmallGroup(96,15);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,313,31,297,69,2309]);
// Polycyclic
G:=Group<a,b,c|a^4=b^12=1,c^2=a*b^6,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^-1>;
// generators/relations
Export
Subgroup lattice of C12.Q8 in TeX
Character table of C12.Q8 in TeX