non-abelian, soluble, monomial
Aliases: Dic3:S4, (C2xC6):D12, C3:2(C4:S4), (C2xS4):1S3, (C6xS4):2C2, (C3xA4):1D4, C23.4S32, C6.12(C2xS4), C2.13(S3xS4), (C2xA4).4D6, A4:1(C3:D4), (Dic3xA4):1C2, (C22xC6).4D6, (C6xA4).4C22, (C22xDic3):3S3, C22:1(C3:D12), (C2xC3:S4):1C2, SmallGroup(288,855)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3:S4
G = < a,b,c,d,e,f | a6=c2=d2=e3=f2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=a3b, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >
Subgroups: 786 in 122 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C23, C32, Dic3, Dic3, C12, A4, A4, D6, C2xC6, C2xC6, C22:C4, C4:C4, C22xC4, C2xD4, C3xS3, C3:S3, C3xC6, D12, C2xDic3, C3:D4, C2xC12, C3xD4, S4, C2xA4, C2xA4, C22xS3, C22xC6, C22xC6, C4:D4, C3xDic3, C3xA4, S3xC6, C2xC3:S3, Dic3:C4, D6:C4, C6.D4, C4xA4, C22xDic3, C2xC3:D4, C6xD4, C2xS4, C2xS4, C3:D12, C3xS4, C3:S4, C6xA4, C23.14D6, C4:S4, Dic3xA4, C6xS4, C2xC3:S4, Dic3:S4
Quotients: C1, C2, C22, S3, D4, D6, D12, C3:D4, S4, S32, C2xS4, C3:D12, C4:S4, S3xS4, Dic3:S4
Character table of Dic3:S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 3 | 3 | 12 | 36 | 2 | 8 | 16 | 6 | 12 | 18 | 36 | 2 | 6 | 6 | 8 | 12 | 12 | 16 | 12 | 12 | 24 | 24 | |
| 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 | 2 | 2 | 2 | 2 | -2 | 0 | -1 | 2 | -1 | 0 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 1 | 1 | -1 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ6 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ7 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | 2 | 0 | 2 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -2 | 0 | -2 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | -1 | 0 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 1 | 0 | 0 | 1 | 0 | 0 | -√3 | √3 | orthogonal lifted from D12 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 1 | 0 | 0 | 1 | 0 | 0 | √3 | -√3 | orthogonal lifted from D12 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -2 | √-3 | -√-3 | 1 | √-3 | -√-3 | 0 | 0 | complex lifted from C3:D4 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -2 | -√-3 | √-3 | 1 | -√-3 | √-3 | 0 | 0 | complex lifted from C3:D4 |
| ρ14 | 3 | 3 | -1 | -1 | 1 | -1 | 3 | 0 | 0 | -3 | -1 | 1 | 1 | 3 | -1 | -1 | 0 | 1 | 1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ15 | 3 | 3 | -1 | -1 | 1 | 1 | 3 | 0 | 0 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | 0 | 1 | 1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from S4 |
| ρ16 | 3 | 3 | -1 | -1 | -1 | 1 | 3 | 0 | 0 | -3 | 1 | 1 | -1 | 3 | -1 | -1 | 0 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from C2xS4 |
| ρ17 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | 3 | 1 | -1 | 1 | 3 | -1 | -1 | 0 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from S4 |
| ρ18 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C3:D12 |
| ρ19 | 4 | 4 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ20 | 6 | 6 | -2 | -2 | 2 | 0 | -3 | 0 | 0 | 0 | -2 | 0 | 0 | -3 | 1 | 1 | 0 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from S3xS4 |
| ρ21 | 6 | 6 | -2 | -2 | -2 | 0 | -3 | 0 | 0 | 0 | 2 | 0 | 0 | -3 | 1 | 1 | 0 | 1 | 1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from S3xS4 |
| ρ22 | 6 | -6 | -2 | 2 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4:S4 |
| ρ23 | 6 | -6 | -2 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 1 | -1 | 0 | -√-3 | √-3 | 0 | √-3 | -√-3 | 0 | 0 | complex faithful |
| ρ24 | 6 | -6 | -2 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 1 | -1 | 0 | √-3 | -√-3 | 0 | -√-3 | √-3 | 0 | 0 | complex faithful |
►On 36 points
Generators in S36
(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)
(1 23 4 20)(2 22 5 19)(3 21 6 24)(7 33 10 36)(8 32 11 35)(9 31 12 34)(13 27 16 30)(14 26 17 29)(15 25 18 28)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(1 11 17)(2 12 18)(3 7 13)(4 8 14)(5 9 15)(6 10 16)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)
(1 23)(2 24)(3 19)(4 20)(5 21)(6 22)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)
G:=sub<Sym(36)| (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), (1,23,4,20)(2,22,5,19)(3,21,6,24)(7,33,10,36)(8,32,11,35)(9,31,12,34)(13,27,16,30)(14,26,17,29)(15,25,18,28), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(19,22)(20,23)(21,24)(31,34)(32,35)(33,36), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36), (1,11,17)(2,12,18)(3,7,13)(4,8,14)(5,9,15)(6,10,16)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30), (1,23)(2,24)(3,19)(4,20)(5,21)(6,22)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)>;
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), (1,23,4,20)(2,22,5,19)(3,21,6,24)(7,33,10,36)(8,32,11,35)(9,31,12,34)(13,27,16,30)(14,26,17,29)(15,25,18,28), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(19,22)(20,23)(21,24)(31,34)(32,35)(33,36), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36), (1,11,17)(2,12,18)(3,7,13)(4,8,14)(5,9,15)(6,10,16)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30), (1,23)(2,24)(3,19)(4,20)(5,21)(6,22)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36) );
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)], [(1,23,4,20),(2,22,5,19),(3,21,6,24),(7,33,10,36),(8,32,11,35),(9,31,12,34),(13,27,16,30),(14,26,17,29),(15,25,18,28)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(19,22),(20,23),(21,24),(31,34),(32,35),(33,36)], [(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(25,28),(26,29),(27,30),(31,34),(32,35),(33,36)], [(1,11,17),(2,12,18),(3,7,13),(4,8,14),(5,9,15),(6,10,16),(19,31,25),(20,32,26),(21,33,27),(22,34,28),(23,35,29),(24,36,30)], [(1,23),(2,24),(3,19),(4,20),(5,21),(6,22),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36)]])
Matrix representation of Dic3:S4 ►in GL5(F13)
| 7 | 10 | 0 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 3 | 7 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 12 | 12 |
| 0 | 12 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(13))| [7,10,0,0,0,10,7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[3,6,0,0,0,7,10,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,1,0,12,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,12,0,1,0,0,12,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,12,0,0,0,0,12,0,0,0,1,12],[0,12,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
Dic3:S4 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes S_4 % in TeX
G:=Group("Dic3:S4"); // GroupNames label
G:=SmallGroup(288,855);
// by ID
G=gap.SmallGroup(288,855);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,28,85,234,1684,3036,782,1777,1350]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^6=c^2=d^2=e^3=f^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a^3*b,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations
Export