non-abelian, soluble, monomial
Aliases: D6⋊1S4, (C6×S4)⋊3C2, (C2×S4)⋊2S3, (C3×A4)⋊2D4, C23.6S32, C2.14(S3×S4), C6.14(C2×S4), (C2×A4).6D6, A4⋊2(C3⋊D4), C6.7S4⋊3C2, (S3×C23)⋊1S3, C3⋊2(A4⋊D4), (C22×C6).6D6, C22⋊(D6⋊S3), (C6×A4).6C22, (C2×S3×A4)⋊1C2, (C2×C6)⋊1(C3⋊D4), SmallGroup(288,857)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊S4
G = < a,b,c,d,e,f | a6=b2=c2=d2=e3=f2=1, bab=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: 790 in 130 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, A4, A4, D6, D6, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, C3×S3, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, S4, C2×A4, C2×A4, C22×S3, C22×C6, C22×C6, C22≀C2, C3⋊Dic3, C3×A4, S3×C6, D6⋊C4, C6.D4, A4⋊C4, C2×C3⋊D4, C6×D4, C2×S4, C22×A4, S3×C23, D6⋊S3, C3×S4, S3×A4, C6×A4, C23⋊2D6, A4⋊D4, C6.7S4, C6×S4, C2×S3×A4, D6⋊S4
Quotients: C1, C2, C22, S3, D4, D6, C3⋊D4, S4, S32, C2×S4, D6⋊S3, A4⋊D4, S3×S4, D6⋊S4
Character table of D6⋊S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 12A | 12B | |
| size | 1 | 1 | 3 | 3 | 6 | 12 | 18 | 2 | 8 | 16 | 12 | 36 | 36 | 2 | 6 | 6 | 8 | 12 | 12 | 16 | 24 | 24 | 12 | 12 | |
| 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 | 0 | 2 | 0 | -1 | 2 | -1 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ7 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ8 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | -1 | 2 | -1 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 1 | 1 | -1 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ9 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | -1 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ10 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | -2 | √-3 | -√-3 | 1 | 0 | 0 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ11 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | -2 | 2 | -2 | 1 | 0 | 0 | 1 | -√-3 | √-3 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ12 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | -2 | 2 | -2 | 1 | 0 | 0 | 1 | √-3 | -√-3 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | -2 | -√-3 | √-3 | 1 | 0 | 0 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ14 | 3 | 3 | -1 | -1 | 3 | -1 | -1 | 3 | 0 | 0 | 1 | 1 | -1 | 3 | -1 | -1 | 0 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from S4 |
| ρ15 | 3 | 3 | -1 | -1 | -3 | 1 | 1 | 3 | 0 | 0 | -1 | 1 | -1 | 3 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from C2×S4 |
| ρ16 | 3 | 3 | -1 | -1 | 3 | 1 | -1 | 3 | 0 | 0 | -1 | -1 | 1 | 3 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S4 |
| ρ17 | 3 | 3 | -1 | -1 | -3 | -1 | 1 | 3 | 0 | 0 | 1 | -1 | 1 | 3 | -1 | -1 | 0 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from C2×S4 |
| ρ18 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ19 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 2 | -2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | symplectic lifted from D6⋊S3, Schur index 2 |
| ρ20 | 6 | 6 | -2 | -2 | 0 | 2 | 0 | -3 | 0 | 0 | -2 | 0 | 0 | -3 | 1 | 1 | 0 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from S3×S4 |
| ρ21 | 6 | 6 | -2 | -2 | 0 | -2 | 0 | -3 | 0 | 0 | 2 | 0 | 0 | -3 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3×S4 |
| ρ22 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4⋊D4 |
| ρ23 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3 | 1 | -1 | 0 | -√-3 | √-3 | 0 | 0 | 0 | -√-3 | √-3 | complex faithful |
| ρ24 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3 | 1 | -1 | 0 | √-3 | -√-3 | 0 | 0 | 0 | √-3 | -√-3 | 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 6)(2 5)(3 4)(7 12)(8 11)(9 10)(13 18)(14 17)(15 16)(19 21)(22 24)(25 27)(28 30)(31 33)(34 36)
(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(31 34)(32 35)(33 36)
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 31 25)(20 32 26)(21 33 27)(22 34 28)(23 35 29)(24 36 30)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(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), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,18)(14,17)(15,16)(19,21)(22,24)(25,27)(28,30)(31,33)(34,36), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(31,34)(32,35)(33,36), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(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), (1,6)(2,5)(3,4)(7,12)(8,11)(9,10)(13,18)(14,17)(15,16)(19,21)(22,24)(25,27)(28,30)(31,33)(34,36), (7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(31,34)(32,35)(33,36), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,35,29)(24,36,30), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(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)], [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,18),(14,17),(15,16),(19,21),(22,24),(25,27),(28,30),(31,33),(34,36)], [(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,28),(26,29),(27,30)], [(1,4),(2,5),(3,6),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(31,34),(32,35),(33,36)], [(1,7,13),(2,8,14),(3,9,15),(4,10,16),(5,11,17),(6,12,18),(19,31,25),(20,32,26),(21,33,27),(22,34,28),(23,35,29),(24,36,30)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(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 D6⋊S4 ►in GL5(𝔽13)
| 4 | 0 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 3 | 3 | 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 |
| 1 | 0 | 0 | 0 | 0 |
| 11 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(5,GF(13))| [4,6,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[3,6,0,0,0,3,10,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[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],[1,11,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0] >;
D6⋊S4 in GAP, Magma, Sage, TeX
D_6\rtimes S_4
% in TeX
G:=Group("D6:S4"); // GroupNames label
G:=SmallGroup(288,857);
// by ID
G=gap.SmallGroup(288,857);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,85,234,1684,3036,782,1777,1350]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^6=b^2=c^2=d^2=e^3=f^2=1,b*a*b=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
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