direct product, non-abelian, soluble, monomial
Aliases: Dic3×S4, (C3×S4)⋊C4, C3⋊3(C4×S4), (C2×S4).S3, (C6×S4).C2, C2.1(S3×S4), C23.2S32, C6.10(C2×S4), (C2×A4).2D6, C6.7S4⋊1C2, (Dic3×A4)⋊2C2, A4⋊1(C2×Dic3), (C22×C6).2D6, (C6×A4).2C22, C22⋊1(S3×Dic3), (C22×Dic3)⋊1S3, (C2×C6)⋊1(C4×S3), (C3×A4)⋊1(C2×C4), SmallGroup(288,853)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×A4 — Dic3×S4 |
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, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >
Subgroups: 542 in 118 conjugacy classes, 23 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, Dic3, C12, A4, A4, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C4×S3, C2×Dic3, C2×C12, C3×D4, S4, C2×A4, C2×A4, C22×C6, C22×C6, C4×D4, C3×Dic3, C3⋊Dic3, C3×A4, S3×C6, C4×Dic3, C4⋊Dic3, C6.D4, A4⋊C4, C4×A4, C22×Dic3, C22×Dic3, C6×D4, C2×S4, S3×Dic3, C3×S4, C6×A4, D4×Dic3, C4×S4, C6.7S4, Dic3×A4, C6×S4, Dic3×S4
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C4×S3, C2×Dic3, S4, S32, C2×S4, S3×Dic3, C4×S4, S3×S4, Dic3×S4
Character table of Dic3×S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 3 | 3 | 6 | 6 | 2 | 8 | 16 | 3 | 3 | 6 | 6 | 9 | 9 | 18 | 18 | 18 | 18 | 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 | 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 | -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 | 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 | -1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -i | i | -1 | 1 | i | -i | -i | i | -i | i | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | i | -i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | i | 1 | -1 | i | -i | i | -i | i | -i | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | i | -i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | i | -i | -1 | 1 | -i | i | i | -i | i | -i | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -i | i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | i | -i | 1 | -1 | -i | i | -i | i | -i | i | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -i | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -2 | -2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | -1 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 1 | 1 | -1 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | -2 | 2 | -2 | 2 | -2 | -1 | 2 | -1 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -2 | 1 | -1 | 1 | -1 | 1 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | -2 | 2 | -2 | -2 | 2 | -1 | 2 | -1 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -2 | -1 | 1 | 1 | 1 | -1 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | 2i | -2i | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 1 | 0 | 0 | 1 | 0 | 0 | i | -i | complex lifted from C4×S3 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -2i | 2i | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 1 | 0 | 0 | 1 | 0 | 0 | -i | i | complex lifted from C4×S3 |
| ρ17 | 3 | 3 | -1 | -1 | 1 | 1 | 3 | 0 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 3 | -1 | -1 | 0 | 1 | 1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from S4 |
| ρ18 | 3 | 3 | -1 | -1 | 1 | 1 | 3 | 0 | 0 | -3 | -3 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 3 | -1 | -1 | 0 | 1 | 1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ19 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | -3 | -3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 3 | -1 | -1 | 0 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ20 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | 3 | 3 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 3 | -1 | -1 | 0 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from S4 |
| ρ21 | 3 | -3 | -1 | 1 | 1 | -1 | 3 | 0 | 0 | 3i | -3i | 1 | -1 | i | -i | -i | i | i | -i | -3 | -1 | 1 | 0 | -1 | 1 | 0 | -1 | 1 | 0 | 0 | complex lifted from C4×S4 |
| ρ22 | 3 | -3 | -1 | 1 | 1 | -1 | 3 | 0 | 0 | -3i | 3i | 1 | -1 | -i | i | i | -i | -i | i | -3 | -1 | 1 | 0 | -1 | 1 | 0 | -1 | 1 | 0 | 0 | complex lifted from C4×S4 |
| ρ23 | 3 | -3 | -1 | 1 | -1 | 1 | 3 | 0 | 0 | -3i | 3i | -1 | 1 | -i | i | -i | i | i | -i | -3 | -1 | 1 | 0 | 1 | -1 | 0 | 1 | -1 | 0 | 0 | complex lifted from C4×S4 |
| ρ24 | 3 | -3 | -1 | 1 | -1 | 1 | 3 | 0 | 0 | 3i | -3i | -1 | 1 | i | -i | i | -i | -i | i | -3 | -1 | 1 | 0 | 1 | -1 | 0 | 1 | -1 | 0 | 0 | complex lifted from C4×S4 |
| ρ25 | 4 | 4 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Dic3, Schur index 2 |
| ρ27 | 6 | 6 | -2 | -2 | 2 | 2 | -3 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 1 | 1 | 0 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from S3×S4 |
| ρ28 | 6 | 6 | -2 | -2 | -2 | -2 | -3 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 1 | 1 | 0 | 1 | 1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from S3×S4 |
| ρ29 | 6 | -6 | -2 | 2 | 2 | -2 | -3 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 1 | -1 | 0 | 1 | -1 | 0 | 1 | -1 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ30 | 6 | -6 | -2 | 2 | -2 | 2 | -3 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 1 | -1 | 0 | -1 | 1 | 0 | -1 | 1 | 0 | 0 | symplectic faithful, Schur index 2 |
►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 22 4 19)(2 21 5 24)(3 20 6 23)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 32 16 35)(14 31 17 34)(15 36 18 33)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)
(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 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)
(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 31)(26 32)(27 33)(28 34)(29 35)(30 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), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30), (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)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36), (7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)>;
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), (1,22,4,19)(2,21,5,24)(3,20,6,23)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,32,16,35)(14,31,17,34)(15,36,18,33), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30), (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)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36), (7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36) );
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)], [(1,22,4,19),(2,21,5,24),(3,20,6,23),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,32,16,35),(14,31,17,34),(15,36,18,33)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(19,22),(20,23),(21,24),(25,28),(26,29),(27,30)], [(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),(19,25,31),(20,26,32),(21,27,33),(22,28,34),(23,29,35),(24,30,36)], [(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)]])
Matrix representation of Dic3×S4 ►in GL5(𝔽13)
| 4 | 0 | 0 | 0 | 0 |
| 9 | 10 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 5 | 12 | 0 | 0 | 0 |
| 0 | 8 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(5,GF(13))| [4,9,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,12,8,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,12,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12] >;
Dic3×S4 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times S_4 % in TeX
G:=Group("Dic3xS4"); // GroupNames label
G:=SmallGroup(288,853);
// by ID
G=gap.SmallGroup(288,853);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,28,234,1684,3036,782,1777,1350]);
// 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,b*f=f*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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