non-abelian, soluble, monomial
Aliases: C23.4S4, 2+ (1+4).2S3, C2.4(C22⋊S4), C23⋊A4.2C2, SmallGroup(192,1491)
Series: Derived ►Chief ►Lower central ►Upper central
| C23⋊A4 — C23.S4 |
Subgroups: 357 in 71 conjugacy classes, 8 normal (5 characteristic)
C1, C2, C2 [×3], C3, C4 [×5], C22 [×7], C6, C2×C4 [×9], D4 [×3], Q8, C23 [×3], C23, Dic3, A4 [×3], C22⋊C4 [×6], C4⋊C4 [×3], C22×C4, C2×D4 [×3], C4○D4, SL2(𝔽3), C2×A4 [×3], C23⋊C4 [×3], C22.D4 [×3], 2+ (1+4), A4⋊C4 [×3], C23.7D4, C23⋊A4, C23.S4
Quotients:
C1, C2, S3, S4 [×3], C22⋊S4, C23.S4
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f3=1, g2=c, ab=ba, eae=ac=ca, ad=da, faf-1=gag-1=b, dbd=ebe=bc=cb, fbf-1=abc, gbg-1=a, cd=dc, ce=ec, cf=fc, cg=gc, fdf-1=gdg-1=de=ed, fef-1=d, eg=ge, gfg-1=f-1 >
►On 16 points - transitive group 16T441
(1 6)(2 16)(3 8)(4 14)(5 10)(7 12)(9 13)(11 15)
(1 15)(2 7)(3 13)(4 5)(6 11)(8 9)(10 14)(12 16)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 6)(2 16)(3 8)(4 14)(5 12)(7 10)(9 15)(11 13)
(1 9)(2 10)(3 11)(4 12)(5 14)(6 15)(7 16)(8 13)
(5 14 12)(6 9 15)(7 16 10)(8 11 13)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
G:=sub<Sym(16)| (1,6)(2,16)(3,8)(4,14)(5,10)(7,12)(9,13)(11,15), (1,15)(2,7)(3,13)(4,5)(6,11)(8,9)(10,14)(12,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,6)(2,16)(3,8)(4,14)(5,12)(7,10)(9,15)(11,13), (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13), (5,14,12)(6,9,15)(7,16,10)(8,11,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;
G:=Group( (1,6)(2,16)(3,8)(4,14)(5,10)(7,12)(9,13)(11,15), (1,15)(2,7)(3,13)(4,5)(6,11)(8,9)(10,14)(12,16), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,6)(2,16)(3,8)(4,14)(5,12)(7,10)(9,15)(11,13), (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13), (5,14,12)(6,9,15)(7,16,10)(8,11,13), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );
G=PermutationGroup([(1,6),(2,16),(3,8),(4,14),(5,10),(7,12),(9,13),(11,15)], [(1,15),(2,7),(3,13),(4,5),(6,11),(8,9),(10,14),(12,16)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,6),(2,16),(3,8),(4,14),(5,12),(7,10),(9,15),(11,13)], [(1,9),(2,10),(3,11),(4,12),(5,14),(6,15),(7,16),(8,13)], [(5,14,12),(6,9,15),(7,16,10),(8,11,13)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)])
G:=TransitiveGroup(16,441);
Matrix representation ►G ⊆ GL4(𝔽5) generated by
| 1 | 1 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 1 | 1 | 0 | 1 |
| 4 | 0 | 1 | 0 |
| 1 | 0 | 2 | 4 |
| 3 | 3 | 2 | 3 |
| 1 | 4 | 4 | 4 |
| 2 | 3 | 0 | 2 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 2 | 0 | 4 | 3 |
| 1 | 1 | 4 | 1 |
| 3 | 0 | 3 | 3 |
| 0 | 0 | 0 | 4 |
| 1 | 2 | 1 | 3 |
| 0 | 3 | 1 | 1 |
| 0 | 2 | 2 | 3 |
| 0 | 0 | 0 | 4 |
| 0 | 4 | 3 | 0 |
| 0 | 0 | 2 | 4 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 4 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 3 | 0 |
| 0 | 2 | 1 | 0 |
G:=sub<GL(4,GF(5))| [1,0,1,4,1,4,1,0,0,0,0,1,0,0,1,0],[1,3,1,2,0,3,4,3,2,2,4,0,4,3,4,2],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,1,3,0,0,1,0,0,4,4,3,0,3,1,3,4],[1,0,0,0,2,3,2,0,1,1,2,0,3,1,3,4],[0,0,0,1,4,0,0,0,3,2,1,4,0,4,0,0],[3,0,0,0,0,0,0,2,0,1,3,1,0,2,0,0] >;
Character table of C23.S4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6 | |
| size | 1 | 1 | 6 | 6 | 6 | 32 | 12 | 12 | 12 | 24 | 24 | 24 | 32 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ4 | 3 | 3 | 3 | -1 | -1 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | orthogonal lifted from S4 |
| ρ5 | 3 | 3 | 3 | -1 | -1 | 0 | -1 | 1 | 1 | 1 | -1 | -1 | 0 | orthogonal lifted from S4 |
| ρ6 | 3 | 3 | -1 | -1 | 3 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | 0 | orthogonal lifted from S4 |
| ρ7 | 3 | 3 | -1 | -1 | 3 | 0 | -1 | 1 | 1 | -1 | -1 | 1 | 0 | orthogonal lifted from S4 |
| ρ8 | 3 | 3 | -1 | 3 | -1 | 0 | -1 | 1 | 1 | -1 | 1 | -1 | 0 | orthogonal lifted from S4 |
| ρ9 | 3 | 3 | -1 | 3 | -1 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | 0 | orthogonal lifted from S4 |
| ρ10 | 4 | -4 | 0 | 0 | 0 | 1 | 0 | 2i | 2i | 0 | 0 | 0 | -1 | complex faithful |
| ρ11 | 4 | -4 | 0 | 0 | 0 | 1 | 0 | 2i | 2i | 0 | 0 | 0 | -1 | complex faithful |
| ρ12 | 6 | 6 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C22⋊S4 |
| ρ13 | 8 | -8 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | symplectic faithful, Schur index 2 |
In GAP, Magma, Sage, TeX
C_2^3.S_4
% in TeX
G:=Group("C2^3.S4"); // GroupNames label
G:=SmallGroup(192,1491);
// by ID
G=gap.SmallGroup(192,1491);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,57,254,135,171,262,1684,1271,718,1013,516,530]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^3=1,g^2=c,a*b=b*a,e*a*e=a*c=c*a,a*d=d*a,f*a*f^-1=g*a*g^-1=b,d*b*d=e*b*e=b*c=c*b,f*b*f^-1=a*b*c,g*b*g^-1=a,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*d*f^-1=g*d*g^-1=d*e=e*d,f*e*f^-1=d,e*g=g*e,g*f*g^-1=f^-1>;
// generators/relations