metabelian, supersoluble, monomial
Aliases: C32:4Q8, C12.3S3, C3:2Dic6, C6.12D6, C4.(C3:S3), (C3xC12).1C2, C3:Dic3.3C2, (C3xC6).11C22, C2.3(C2xC3:S3), SmallGroup(72,31)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32:4Q8
G = < a,b,c,d | a3=b3=c4=1, d2=c2, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Quotients: C1, C2, C22, S3, Q8, D6, C3:S3, Dic6, C2xC3:S3, C32:4Q8
Character table of C32:4Q8
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | -1 | 2 | -1 | -1 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 1 | -2 | 1 | 1 | 1 | -2 | 1 | 1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | -1 | 2 | -1 | -1 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | 2 | 2 | -1 | -1 | -1 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | 1 | 1 | -2 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -1 | -1 | 2 | -1 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -1 | -1 | -1 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | -2 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | -2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | -√3 | -√3 | -√3 | √3 | √3 | √3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ15 | 2 | -2 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | -2 | 1 | -√3 | 0 | √3 | √3 | -√3 | 0 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ16 | 2 | -2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | -√3 | √3 | 0 | √3 | 0 | -√3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ17 | 2 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | 0 | -√3 | √3 | 0 | -√3 | √3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ18 | 2 | -2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | -2 | 1 | 1 | 1 | 0 | √3 | -√3 | 0 | √3 | -√3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ19 | 2 | -2 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | √3 | √3 | √3 | -√3 | -√3 | -√3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ20 | 2 | -2 | -1 | 2 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | -2 | 1 | √3 | 0 | -√3 | -√3 | √3 | 0 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ21 | 2 | -2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | √3 | -√3 | 0 | -√3 | 0 | √3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
►Regular action on 72 points
Generators in S72
(1 40 7)(2 37 8)(3 38 5)(4 39 6)(9 66 46)(10 67 47)(11 68 48)(12 65 45)(13 51 31)(14 52 32)(15 49 29)(16 50 30)(17 42 62)(18 43 63)(19 44 64)(20 41 61)(21 58 33)(22 59 34)(23 60 35)(24 57 36)(25 56 71)(26 53 72)(27 54 69)(28 55 70)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 51)(6 36 52)(7 33 49)(8 34 50)(9 69 43)(10 70 44)(11 71 41)(12 72 42)(13 38 60)(14 39 57)(15 40 58)(16 37 59)(17 45 53)(18 46 54)(19 47 55)(20 48 56)(25 61 68)(26 62 65)(27 63 66)(28 64 67)
(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)
(1 18 3 20)(2 17 4 19)(5 41 7 43)(6 44 8 42)(9 51 11 49)(10 50 12 52)(13 68 15 66)(14 67 16 65)(21 54 23 56)(22 53 24 55)(25 58 27 60)(26 57 28 59)(29 46 31 48)(30 45 32 47)(33 69 35 71)(34 72 36 70)(37 62 39 64)(38 61 40 63)
G:=sub<Sym(72)| (1,40,7)(2,37,8)(3,38,5)(4,39,6)(9,66,46)(10,67,47)(11,68,48)(12,65,45)(13,51,31)(14,52,32)(15,49,29)(16,50,30)(17,42,62)(18,43,63)(19,44,64)(20,41,61)(21,58,33)(22,59,34)(23,60,35)(24,57,36)(25,56,71)(26,53,72)(27,54,69)(28,55,70), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,51)(6,36,52)(7,33,49)(8,34,50)(9,69,43)(10,70,44)(11,71,41)(12,72,42)(13,38,60)(14,39,57)(15,40,58)(16,37,59)(17,45,53)(18,46,54)(19,47,55)(20,48,56)(25,61,68)(26,62,65)(27,63,66)(28,64,67), (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), (1,18,3,20)(2,17,4,19)(5,41,7,43)(6,44,8,42)(9,51,11,49)(10,50,12,52)(13,68,15,66)(14,67,16,65)(21,54,23,56)(22,53,24,55)(25,58,27,60)(26,57,28,59)(29,46,31,48)(30,45,32,47)(33,69,35,71)(34,72,36,70)(37,62,39,64)(38,61,40,63)>;
G:=Group( (1,40,7)(2,37,8)(3,38,5)(4,39,6)(9,66,46)(10,67,47)(11,68,48)(12,65,45)(13,51,31)(14,52,32)(15,49,29)(16,50,30)(17,42,62)(18,43,63)(19,44,64)(20,41,61)(21,58,33)(22,59,34)(23,60,35)(24,57,36)(25,56,71)(26,53,72)(27,54,69)(28,55,70), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,51)(6,36,52)(7,33,49)(8,34,50)(9,69,43)(10,70,44)(11,71,41)(12,72,42)(13,38,60)(14,39,57)(15,40,58)(16,37,59)(17,45,53)(18,46,54)(19,47,55)(20,48,56)(25,61,68)(26,62,65)(27,63,66)(28,64,67), (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), (1,18,3,20)(2,17,4,19)(5,41,7,43)(6,44,8,42)(9,51,11,49)(10,50,12,52)(13,68,15,66)(14,67,16,65)(21,54,23,56)(22,53,24,55)(25,58,27,60)(26,57,28,59)(29,46,31,48)(30,45,32,47)(33,69,35,71)(34,72,36,70)(37,62,39,64)(38,61,40,63) );
G=PermutationGroup([[(1,40,7),(2,37,8),(3,38,5),(4,39,6),(9,66,46),(10,67,47),(11,68,48),(12,65,45),(13,51,31),(14,52,32),(15,49,29),(16,50,30),(17,42,62),(18,43,63),(19,44,64),(20,41,61),(21,58,33),(22,59,34),(23,60,35),(24,57,36),(25,56,71),(26,53,72),(27,54,69),(28,55,70)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,35,51),(6,36,52),(7,33,49),(8,34,50),(9,69,43),(10,70,44),(11,71,41),(12,72,42),(13,38,60),(14,39,57),(15,40,58),(16,37,59),(17,45,53),(18,46,54),(19,47,55),(20,48,56),(25,61,68),(26,62,65),(27,63,66),(28,64,67)], [(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)], [(1,18,3,20),(2,17,4,19),(5,41,7,43),(6,44,8,42),(9,51,11,49),(10,50,12,52),(13,68,15,66),(14,67,16,65),(21,54,23,56),(22,53,24,55),(25,58,27,60),(26,57,28,59),(29,46,31,48),(30,45,32,47),(33,69,35,71),(34,72,36,70),(37,62,39,64),(38,61,40,63)]])
C32:4Q8 is a maximal subgroup of
D12.S3 C32:3Q16 C24:2S3 C32:5Q16 C32:9SD16 C32:7Q16 S3xDic6 D12:5S3 C12.59D6 C12.D6 Q8xC3:S3 He3:3Q8 C12.D9 C33:4Q8 C33:8Q8 A4:Dic6 C12.6S4 C15:Dic6 C12.D15
C32:4Q8 is a maximal quotient of
C6.Dic6 C12:Dic3 C12.D9 He3:4Q8 C33:4Q8 C33:8Q8 A4:Dic6 C15:Dic6 C12.D15
Matrix representation of C32:4Q8 ►in GL4(F13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 5 | 3 |
| 0 | 1 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 8 | 9 |
| 3 | 6 | 0 | 0 |
| 7 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 5 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 |
| 0 | 0 | 2 | 8 |
| 0 | 0 | 11 | 11 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,9,5,0,0,0,3],[0,12,0,0,1,12,0,0,0,0,3,8,0,0,0,9],[3,7,0,0,6,10,0,0,0,0,12,0,0,0,0,12],[5,8,0,0,0,8,0,0,0,0,2,11,0,0,8,11] >;
C32:4Q8 in GAP, Magma, Sage, TeX
C_3^2\rtimes_4Q_8
% in TeX
G:=Group("C3^2:4Q8"); // GroupNames label
G:=SmallGroup(72,31);
// by ID
G=gap.SmallGroup(72,31);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,20,61,26,323,1204]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of C32:4Q8 in TeX
Character table of C32:4Q8 in TeX