non-abelian, soluble, monomial
Aliases: Dic3⋊2S4, C3⋊S4⋊C4, C3⋊1(C4×S4), A4⋊C4⋊2S3, A4⋊1(C4×S3), C2.2(S3×S4), C23.3S32, C6.11(C2×S4), (C2×A4).3D6, (Dic3×A4)⋊3C2, (C22×C6).3D6, (C6×A4).3C22, C22⋊(C6.D6), (C22×Dic3)⋊2S3, (C2×C3⋊S4).C2, (C2×C6)⋊2(C4×S3), (C3×A4⋊C4)⋊2C2, (C3×A4)⋊2(C2×C4), SmallGroup(288,854)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×A4 — Dic3⋊2S4 |
Generators and relations for Dic3⋊2S4
G = < a,b,c,d,e,f | a6=c2=d2=e3=f2=1, b2=a3, bab-1=faf=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >
Subgroups: 694 in 122 conjugacy classes, 21 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, C3⋊D4, C2×C12, S4, C2×A4, C2×A4, C22×S3, C22×C6, C4×D4, C3×Dic3, C3×A4, C2×C3⋊S3, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, A4⋊C4, C4×A4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C2×S4, C6.D6, C3⋊S4, C6×A4, Dic3⋊4D4, C4×S4, C3×A4⋊C4, Dic3×A4, C2×C3⋊S4, Dic3⋊2S4
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, C4×S3, S4, S32, C2×S4, C6.D6, C4×S4, S3×S4, Dic3⋊2S4
Character table of Dic3⋊2S4
| 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 | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 3 | 3 | 18 | 18 | 2 | 8 | 16 | 3 | 3 | 6 | 6 | 6 | 6 | 9 | 9 | 18 | 18 | 2 | 6 | 6 | 8 | 16 | 12 | 12 | 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 | -i | i | i | -i | i | -i | -1 | 1 | -1 | 1 | -1 | -1 | -1 | i | -i | -i | i | i | -i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -i | i | -i | i | i | -i | -i | i | 1 | -1 | -1 | 1 | -1 | -1 | -1 | i | -i | -i | i | -i | i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | i | i | -i | -i | i | -i | i | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -i | i | i | -i | -i | i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | i | -i | i | -i | -i | i | i | -i | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -i | i | i | -i | i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 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 | 0 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | -2 | 2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | i | -i | complex lifted from C4×S3 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -2 | 1 | -i | i | i | -i | 0 | 0 | complex lifted from C4×S3 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -2 | 1 | i | -i | -i | i | 0 | 0 | complex lifted from C4×S3 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | -2 | 2 | -2 | 1 | 1 | 0 | 0 | 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 | 0 | 1 | -1 | 1 | -1 | 0 | 0 | orthogonal lifted from C2×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 | 0 | -1 | 1 | -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 | 0 | 1 | -1 | 1 | -1 | 0 | 0 | orthogonal lifted from 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 | 0 | -1 | 1 | -1 | 1 | 0 | 0 | orthogonal lifted from S4 |
| ρ21 | 3 | -3 | -1 | 1 | -1 | 1 | 3 | 0 | 0 | 3i | -3i | i | -i | i | -i | -i | i | 1 | -1 | -3 | -1 | 1 | 0 | 0 | -i | -i | i | i | 0 | 0 | complex lifted from C4×S4 |
| ρ22 | 3 | -3 | -1 | 1 | 1 | -1 | 3 | 0 | 0 | 3i | -3i | -i | i | -i | i | -i | i | -1 | 1 | -3 | -1 | 1 | 0 | 0 | i | i | -i | -i | 0 | 0 | complex lifted from C4×S4 |
| ρ23 | 3 | -3 | -1 | 1 | 1 | -1 | 3 | 0 | 0 | -3i | 3i | i | -i | i | -i | i | -i | -1 | 1 | -3 | -1 | 1 | 0 | 0 | -i | -i | i | i | 0 | 0 | complex lifted from C4×S4 |
| ρ24 | 3 | -3 | -1 | 1 | -1 | 1 | 3 | 0 | 0 | -3i | 3i | -i | i | -i | i | i | -i | 1 | -1 | -3 | -1 | 1 | 0 | 0 | i | i | -i | -i | 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 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C6.D6 |
| ρ26 | 4 | 4 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ27 | 6 | 6 | -2 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | -1 | 1 | -1 | 1 | 0 | 0 | orthogonal lifted from S3×S4 |
| ρ28 | 6 | 6 | -2 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | 1 | -1 | 1 | -1 | 0 | 0 | orthogonal lifted from S3×S4 |
| ρ29 | 6 | -6 | -2 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 3 | 1 | -1 | 0 | 0 | i | i | -i | -i | 0 | 0 | complex faithful |
| ρ30 | 6 | -6 | -2 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 3 | 1 | -1 | 0 | 0 | -i | -i | i | i | 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 20 4 23)(2 19 5 22)(3 24 6 21)(7 36 10 33)(8 35 11 32)(9 34 12 31)(13 30 16 27)(14 29 17 26)(15 28 18 25)
(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 4)(2 3)(5 6)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(19 24)(20 23)(21 22)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)
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), (1,20,4,23)(2,19,5,22)(3,24,6,21)(7,36,10,33)(8,35,11,32)(9,34,12,31)(13,30,16,27)(14,29,17,26)(15,28,18,25), (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), (1,4)(2,3)(5,6)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(19,24)(20,23)(21,22)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)>;
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), (1,20,4,23)(2,19,5,22)(3,24,6,21)(7,36,10,33)(8,35,11,32)(9,34,12,31)(13,30,16,27)(14,29,17,26)(15,28,18,25), (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), (1,4)(2,3)(5,6)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(19,24)(20,23)(21,22)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31) );
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)], [(1,20,4,23),(2,19,5,22),(3,24,6,21),(7,36,10,33),(8,35,11,32),(9,34,12,31),(13,30,16,27),(14,29,17,26),(15,28,18,25)], [(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)], [(1,4),(2,3),(5,6),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13),(19,24),(20,23),(21,22),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31)]])
Matrix representation of Dic3⋊2S4 ►in GL5(𝔽13)
| 0 | 12 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 5 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 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 | 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 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 12 | 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))| [0,1,0,0,0,12,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[5,8,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8],[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],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[1,12,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0] >;
Dic3⋊2S4 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes_2S_4 % in TeX
G:=Group("Dic3:2S4"); // GroupNames label
G:=SmallGroup(288,854);
// by ID
G=gap.SmallGroup(288,854);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,28,36,234,1684,3036,782,1777,1350]);
// 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=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*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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