non-abelian, soluble, monomial, rational
Aliases: C23⋊2S4, 2+ 1+4⋊4S3, C23⋊A4⋊2C2, C2.6(C22⋊S4), Aut(C22×C4), SmallGroup(192,1493)
Series: Derived ►Chief ►Lower central ►Upper central
| C23⋊A4 — C23⋊S4 |
Generators and relations for C23⋊S4
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f3=g2=1, ab=ba, eae=ac=ca, ad=da, faf-1=gag=b, dbd=ebe=bc=cb, fbf-1=abc, gbg=a, cd=dc, ce=ec, cf=fc, cg=gc, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >
Subgroups: 605 in 98 conjugacy classes, 8 normal (5 characteristic)
C1, C2, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C23, C23, A4, D6, C22⋊C4, C2×D4, C4○D4, C24, SL2(𝔽3), S4, C2×A4, C23⋊C4, C22≀C2, 2+ 1+4, C2×S4, C2≀C22, C23⋊A4, C23⋊S4
Quotients: C1, C2, S3, S4, C22⋊S4, C23⋊S4
Character table of C23⋊S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 6 | |
| size | 1 | 1 | 6 | 6 | 6 | 12 | 12 | 32 | 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 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | -1 | orthogonal lifted from S3 |
| ρ4 | 3 | 3 | 3 | -1 | -1 | 1 | 1 | 0 | -1 | 1 | -1 | -1 | 0 | orthogonal lifted from S4 |
| ρ5 | 3 | 3 | -1 | 3 | -1 | -1 | -1 | 0 | -1 | 1 | -1 | 1 | 0 | orthogonal lifted from S4 |
| ρ6 | 3 | 3 | -1 | -1 | 3 | 1 | 1 | 0 | -1 | -1 | -1 | 1 | 0 | orthogonal lifted from S4 |
| ρ7 | 3 | 3 | -1 | -1 | 3 | -1 | -1 | 0 | -1 | 1 | 1 | -1 | 0 | orthogonal lifted from S4 |
| ρ8 | 3 | 3 | -1 | 3 | -1 | 1 | 1 | 0 | -1 | -1 | 1 | -1 | 0 | orthogonal lifted from S4 |
| ρ9 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | -1 | -1 | 1 | 1 | 0 | orthogonal lifted from S4 |
| ρ10 | 4 | -4 | 0 | 0 | 0 | 2 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | orthogonal faithful |
| ρ11 | 4 | -4 | 0 | 0 | 0 | -2 | 2 | 1 | 0 | 0 | 0 | 0 | -1 | orthogonal faithful |
| ρ12 | 6 | 6 | -2 | -2 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from C22⋊S4 |
| ρ13 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | orthogonal faithful |
►On 8 points - transitive group 8T39
Generators in S8
(1 4)(2 6)(3 7)(5 8)
(1 3)(2 8)(4 7)(5 6)
(1 2)(3 8)(4 6)(5 7)
(3 8)(5 7)
(3 8)(4 6)
(3 4 5)(6 7 8)
(1 2)(3 6)(4 8)(5 7)
G:=sub<Sym(8)| (1,4)(2,6)(3,7)(5,8), (1,3)(2,8)(4,7)(5,6), (1,2)(3,8)(4,6)(5,7), (3,8)(5,7), (3,8)(4,6), (3,4,5)(6,7,8), (1,2)(3,6)(4,8)(5,7)>;
G:=Group( (1,4)(2,6)(3,7)(5,8), (1,3)(2,8)(4,7)(5,6), (1,2)(3,8)(4,6)(5,7), (3,8)(5,7), (3,8)(4,6), (3,4,5)(6,7,8), (1,2)(3,6)(4,8)(5,7) );
G=PermutationGroup([[(1,4),(2,6),(3,7),(5,8)], [(1,3),(2,8),(4,7),(5,6)], [(1,2),(3,8),(4,6),(5,7)], [(3,8),(5,7)], [(3,8),(4,6)], [(3,4,5),(6,7,8)], [(1,2),(3,6),(4,8),(5,7)]])
G:=TransitiveGroup(8,39);
►On 16 points - transitive group 16T442
Generators in S16
(1 12)(2 8)(3 7)(4 15)(5 10)(6 9)(11 16)(13 14)
(1 11)(2 10)(3 6)(4 14)(5 8)(7 9)(12 16)(13 15)
(1 4)(2 3)(5 9)(6 10)(7 8)(11 14)(12 15)(13 16)
(1 12)(2 8)(3 7)(4 15)(5 6)(9 10)(11 13)(14 16)
(1 13)(2 9)(3 5)(4 16)(6 7)(8 10)(11 12)(14 15)
(5 6 7)(8 9 10)(11 12 13)(14 15 16)
(1 3)(2 4)(5 13)(6 12)(7 11)(8 14)(9 16)(10 15)
G:=sub<Sym(16)| (1,12)(2,8)(3,7)(4,15)(5,10)(6,9)(11,16)(13,14), (1,11)(2,10)(3,6)(4,14)(5,8)(7,9)(12,16)(13,15), (1,4)(2,3)(5,9)(6,10)(7,8)(11,14)(12,15)(13,16), (1,12)(2,8)(3,7)(4,15)(5,6)(9,10)(11,13)(14,16), (1,13)(2,9)(3,5)(4,16)(6,7)(8,10)(11,12)(14,15), (5,6,7)(8,9,10)(11,12,13)(14,15,16), (1,3)(2,4)(5,13)(6,12)(7,11)(8,14)(9,16)(10,15)>;
G:=Group( (1,12)(2,8)(3,7)(4,15)(5,10)(6,9)(11,16)(13,14), (1,11)(2,10)(3,6)(4,14)(5,8)(7,9)(12,16)(13,15), (1,4)(2,3)(5,9)(6,10)(7,8)(11,14)(12,15)(13,16), (1,12)(2,8)(3,7)(4,15)(5,6)(9,10)(11,13)(14,16), (1,13)(2,9)(3,5)(4,16)(6,7)(8,10)(11,12)(14,15), (5,6,7)(8,9,10)(11,12,13)(14,15,16), (1,3)(2,4)(5,13)(6,12)(7,11)(8,14)(9,16)(10,15) );
G=PermutationGroup([[(1,12),(2,8),(3,7),(4,15),(5,10),(6,9),(11,16),(13,14)], [(1,11),(2,10),(3,6),(4,14),(5,8),(7,9),(12,16),(13,15)], [(1,4),(2,3),(5,9),(6,10),(7,8),(11,14),(12,15),(13,16)], [(1,12),(2,8),(3,7),(4,15),(5,6),(9,10),(11,13),(14,16)], [(1,13),(2,9),(3,5),(4,16),(6,7),(8,10),(11,12),(14,15)], [(5,6,7),(8,9,10),(11,12,13),(14,15,16)], [(1,3),(2,4),(5,13),(6,12),(7,11),(8,14),(9,16),(10,15)]])
G:=TransitiveGroup(16,442);
►On 24 points - transitive group 24T333
Generators in S24
(1 15)(2 16)(3 19)(4 22)(5 21)(6 17)(7 18)(8 13)(9 24)(10 20)(11 23)(12 14)
(1 18)(2 21)(3 14)(4 20)(5 16)(6 24)(7 15)(8 23)(9 17)(10 22)(11 13)(12 19)
(1 22)(2 23)(3 24)(4 15)(5 13)(6 14)(7 20)(8 21)(9 19)(10 18)(11 16)(12 17)
(1 15)(2 16)(3 19)(4 22)(5 8)(6 12)(7 10)(9 24)(11 23)(13 21)(14 17)(18 20)
(1 20)(2 13)(3 17)(4 10)(5 23)(6 9)(7 22)(8 11)(12 24)(14 19)(15 18)(16 21)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 22)(2 24)(3 23)(4 18)(5 17)(6 16)(7 20)(8 19)(9 21)(10 15)(11 14)(12 13)
G:=sub<Sym(24)| (1,15)(2,16)(3,19)(4,22)(5,21)(6,17)(7,18)(8,13)(9,24)(10,20)(11,23)(12,14), (1,18)(2,21)(3,14)(4,20)(5,16)(6,24)(7,15)(8,23)(9,17)(10,22)(11,13)(12,19), (1,22)(2,23)(3,24)(4,15)(5,13)(6,14)(7,20)(8,21)(9,19)(10,18)(11,16)(12,17), (1,15)(2,16)(3,19)(4,22)(5,8)(6,12)(7,10)(9,24)(11,23)(13,21)(14,17)(18,20), (1,20)(2,13)(3,17)(4,10)(5,23)(6,9)(7,22)(8,11)(12,24)(14,19)(15,18)(16,21), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,22)(2,24)(3,23)(4,18)(5,17)(6,16)(7,20)(8,19)(9,21)(10,15)(11,14)(12,13)>;
G:=Group( (1,15)(2,16)(3,19)(4,22)(5,21)(6,17)(7,18)(8,13)(9,24)(10,20)(11,23)(12,14), (1,18)(2,21)(3,14)(4,20)(5,16)(6,24)(7,15)(8,23)(9,17)(10,22)(11,13)(12,19), (1,22)(2,23)(3,24)(4,15)(5,13)(6,14)(7,20)(8,21)(9,19)(10,18)(11,16)(12,17), (1,15)(2,16)(3,19)(4,22)(5,8)(6,12)(7,10)(9,24)(11,23)(13,21)(14,17)(18,20), (1,20)(2,13)(3,17)(4,10)(5,23)(6,9)(7,22)(8,11)(12,24)(14,19)(15,18)(16,21), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,22)(2,24)(3,23)(4,18)(5,17)(6,16)(7,20)(8,19)(9,21)(10,15)(11,14)(12,13) );
G=PermutationGroup([[(1,15),(2,16),(3,19),(4,22),(5,21),(6,17),(7,18),(8,13),(9,24),(10,20),(11,23),(12,14)], [(1,18),(2,21),(3,14),(4,20),(5,16),(6,24),(7,15),(8,23),(9,17),(10,22),(11,13),(12,19)], [(1,22),(2,23),(3,24),(4,15),(5,13),(6,14),(7,20),(8,21),(9,19),(10,18),(11,16),(12,17)], [(1,15),(2,16),(3,19),(4,22),(5,8),(6,12),(7,10),(9,24),(11,23),(13,21),(14,17),(18,20)], [(1,20),(2,13),(3,17),(4,10),(5,23),(6,9),(7,22),(8,11),(12,24),(14,19),(15,18),(16,21)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,22),(2,24),(3,23),(4,18),(5,17),(6,16),(7,20),(8,19),(9,21),(10,15),(11,14),(12,13)]])
G:=TransitiveGroup(24,333);
►On 24 points - transitive group 24T431
Generators in S24
(1 7)(2 21)(4 10)(5 11)(6 14)(8 23)(12 17)(13 16)(15 18)(20 22)
(1 20)(3 9)(4 10)(5 13)(6 12)(7 22)(11 16)(14 17)(15 18)(19 24)
(1 22)(2 23)(3 24)(4 15)(5 13)(6 14)(7 20)(8 21)(9 19)(10 18)(11 16)(12 17)
(1 15)(2 13)(4 22)(5 23)(7 18)(8 11)(9 19)(10 20)(12 17)(16 21)
(2 13)(3 14)(5 23)(6 24)(7 20)(8 16)(9 12)(10 18)(11 21)(17 19)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(1 22)(2 24)(3 23)(4 15)(5 14)(6 13)(8 9)(10 18)(11 17)(12 16)(19 21)
G:=sub<Sym(24)| (1,7)(2,21)(4,10)(5,11)(6,14)(8,23)(12,17)(13,16)(15,18)(20,22), (1,20)(3,9)(4,10)(5,13)(6,12)(7,22)(11,16)(14,17)(15,18)(19,24), (1,22)(2,23)(3,24)(4,15)(5,13)(6,14)(7,20)(8,21)(9,19)(10,18)(11,16)(12,17), (1,15)(2,13)(4,22)(5,23)(7,18)(8,11)(9,19)(10,20)(12,17)(16,21), (2,13)(3,14)(5,23)(6,24)(7,20)(8,16)(9,12)(10,18)(11,21)(17,19), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,22)(2,24)(3,23)(4,15)(5,14)(6,13)(8,9)(10,18)(11,17)(12,16)(19,21)>;
G:=Group( (1,7)(2,21)(4,10)(5,11)(6,14)(8,23)(12,17)(13,16)(15,18)(20,22), (1,20)(3,9)(4,10)(5,13)(6,12)(7,22)(11,16)(14,17)(15,18)(19,24), (1,22)(2,23)(3,24)(4,15)(5,13)(6,14)(7,20)(8,21)(9,19)(10,18)(11,16)(12,17), (1,15)(2,13)(4,22)(5,23)(7,18)(8,11)(9,19)(10,20)(12,17)(16,21), (2,13)(3,14)(5,23)(6,24)(7,20)(8,16)(9,12)(10,18)(11,21)(17,19), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,22)(2,24)(3,23)(4,15)(5,14)(6,13)(8,9)(10,18)(11,17)(12,16)(19,21) );
G=PermutationGroup([[(1,7),(2,21),(4,10),(5,11),(6,14),(8,23),(12,17),(13,16),(15,18),(20,22)], [(1,20),(3,9),(4,10),(5,13),(6,12),(7,22),(11,16),(14,17),(15,18),(19,24)], [(1,22),(2,23),(3,24),(4,15),(5,13),(6,14),(7,20),(8,21),(9,19),(10,18),(11,16),(12,17)], [(1,15),(2,13),(4,22),(5,23),(7,18),(8,11),(9,19),(10,20),(12,17),(16,21)], [(2,13),(3,14),(5,23),(6,24),(7,20),(8,16),(9,12),(10,18),(11,21),(17,19)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,22),(2,24),(3,23),(4,15),(5,14),(6,13),(8,9),(10,18),(11,17),(12,16),(19,21)]])
G:=TransitiveGroup(24,431);
Polynomial with Galois group C23⋊S4 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 8T39 | x8+x2+1 | 28·2292 |
Matrix representation of C23⋊S4 ►in GL4(ℤ) generated by
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| -1 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | -1 |
G:=sub<GL(4,Integers())| [0,1,0,0,1,0,0,0,0,0,0,-1,0,0,-1,0],[0,0,1,0,0,0,0,-1,1,0,0,0,0,-1,0,0],[-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[0,0,0,1,0,0,1,0,0,1,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,1,0,0,0,0,1,0],[-1,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,-1] >;
C23⋊S4 in GAP, Magma, Sage, TeX
C_2^3\rtimes S_4
% in TeX
G:=Group("C2^3:S4"); // GroupNames label
G:=SmallGroup(192,1493);
// by ID
G=gap.SmallGroup(192,1493);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,57,254,135,171,262,1684,1271,718,1013,516,530]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^3=g^2=1,a*b=b*a,e*a*e=a*c=c*a,a*d=d*a,f*a*f^-1=g*a*g=b,d*b*d=e*b*e=b*c=c*b,f*b*f^-1=a*b*c,g*b*g=a,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*d*f^-1=g*d*g=d*e=e*d,f*e*f^-1=d,e*g=g*e,g*f*g=f^-1>;
// generators/relations
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