Aliases: C24.1A4, C23⋊1SL2(𝔽3), C23⋊Q8⋊C3, Q8⋊A4⋊1C2, (C22×Q8)⋊1C6, C23.15(C2×A4), C2.2(C24⋊C6), C22.2(C2×SL2(𝔽3)), SmallGroup(192,195)
Series: Derived ►Chief ►Lower central ►Upper central
| C22×Q8 — C24.A4 |
Generators and relations for C24.A4
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g3=1, e2=f2=d, eae-1=ab=ba, faf-1=ac=ca, ad=da, ag=ga, gbg-1=bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, gcg-1=b, fef-1=de=ed, df=fd, dg=gd, geg-1=def, gfg-1=e >
Subgroups: 299 in 57 conjugacy classes, 11 normal (9 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C2×C4, Q8, C23, C23, C23, A4, C2×C6, C22⋊C4, C22×C4, C2×Q8, C24, SL2(𝔽3), C2×A4, C2.C42, C2×C22⋊C4, C22×Q8, C22×A4, C23⋊Q8, Q8⋊A4, C24.A4
Quotients: C1, C2, C3, C6, A4, SL2(𝔽3), C2×A4, C2×SL2(𝔽3), C24⋊C6, C24.A4
Character table of C24.A4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 16 | 16 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | 1 | -1 | -1 | ζ65 | ζ3 | ζ6 | ζ6 | ζ65 | ζ32 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | 1 | -1 | -1 | ζ6 | ζ32 | ζ65 | ζ65 | ζ6 | ζ3 | linear of order 6 |
| ρ7 | 2 | -2 | 2 | -2 | 2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ8 | 2 | -2 | 2 | -2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ9 | 2 | -2 | 2 | -2 | -2 | 2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | ζ3 | ζ3 | ζ6 | ζ32 | ζ65 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ10 | 2 | -2 | 2 | -2 | -2 | 2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | ζ32 | ζ32 | ζ65 | ζ3 | ζ6 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ11 | 2 | -2 | 2 | -2 | 2 | -2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | ζ65 | ζ3 | ζ32 | ζ6 | ζ3 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ12 | 2 | -2 | 2 | -2 | 2 | -2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | ζ6 | ζ32 | ζ3 | ζ65 | ζ32 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ13 | 3 | 3 | 3 | 3 | -3 | -3 | 0 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ14 | 3 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ15 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊C6 |
| ρ16 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C24⋊C6 |
| ρ17 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ18 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T291
Generators in S24
(2 4)(5 7)(10 12)(17 19)(18 20)(21 23)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)
(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(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 9 3 11)(2 12 4 10)(5 22 7 24)(6 21 8 23)(13 19 15 17)(14 18 16 20)
(1 6 13)(2 21 20)(3 8 15)(4 23 18)(5 17 12)(7 19 10)(9 22 14)(11 24 16)
G:=sub<Sym(24)| (2,4)(5,7)(10,12)(17,19)(18,20)(21,23), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(21,23)(22,24), (5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,9,3,11)(2,12,4,10)(5,22,7,24)(6,21,8,23)(13,19,15,17)(14,18,16,20), (1,6,13)(2,21,20)(3,8,15)(4,23,18)(5,17,12)(7,19,10)(9,22,14)(11,24,16)>;
G:=Group( (2,4)(5,7)(10,12)(17,19)(18,20)(21,23), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(21,23)(22,24), (5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,9,3,11)(2,12,4,10)(5,22,7,24)(6,21,8,23)(13,19,15,17)(14,18,16,20), (1,6,13)(2,21,20)(3,8,15)(4,23,18)(5,17,12)(7,19,10)(9,22,14)(11,24,16) );
G=PermutationGroup([[(2,4),(5,7),(10,12),(17,19),(18,20),(21,23)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(21,23),(22,24)], [(5,7),(6,8),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,9,3,11),(2,12,4,10),(5,22,7,24),(6,21,8,23),(13,19,15,17),(14,18,16,20)], [(1,6,13),(2,21,20),(3,8,15),(4,23,18),(5,17,12),(7,19,10),(9,22,14),(11,24,16)]])
G:=TransitiveGroup(24,291);
►On 24 points - transitive group 24T303
Generators in S24
(1 4)(2 3)(5 24)(6 23)(7 22)(8 21)(9 10)(11 12)(13 18)(14 19)(15 20)(16 17)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)
(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(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 9 3 11)(2 12 4 10)(5 22 7 24)(6 21 8 23)(13 19 15 17)(14 18 16 20)
(1 6 13)(2 21 20)(3 8 15)(4 23 18)(5 17 12)(7 19 10)(9 22 14)(11 24 16)
G:=sub<Sym(24)| (1,4)(2,3)(5,24)(6,23)(7,22)(8,21)(9,10)(11,12)(13,18)(14,19)(15,20)(16,17), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(21,23)(22,24), (5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,9,3,11)(2,12,4,10)(5,22,7,24)(6,21,8,23)(13,19,15,17)(14,18,16,20), (1,6,13)(2,21,20)(3,8,15)(4,23,18)(5,17,12)(7,19,10)(9,22,14)(11,24,16)>;
G:=Group( (1,4)(2,3)(5,24)(6,23)(7,22)(8,21)(9,10)(11,12)(13,18)(14,19)(15,20)(16,17), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(21,23)(22,24), (5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,9,3,11)(2,12,4,10)(5,22,7,24)(6,21,8,23)(13,19,15,17)(14,18,16,20), (1,6,13)(2,21,20)(3,8,15)(4,23,18)(5,17,12)(7,19,10)(9,22,14)(11,24,16) );
G=PermutationGroup([[(1,4),(2,3),(5,24),(6,23),(7,22),(8,21),(9,10),(11,12),(13,18),(14,19),(15,20),(16,17)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(21,23),(22,24)], [(5,7),(6,8),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,9,3,11),(2,12,4,10),(5,22,7,24),(6,21,8,23),(13,19,15,17),(14,18,16,20)], [(1,6,13),(2,21,20),(3,8,15),(4,23,18),(5,17,12),(7,19,10),(9,22,14),(11,24,16)]])
G:=TransitiveGroup(24,303);
Matrix representation of C24.A4 ►in GL6(𝔽13)
| 12 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 12 | 0 | 0 |
| 0 | 9 | 12 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 12 |
| 0 | 3 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 10 | 0 | 0 | 0 | 1 | 0 |
| 10 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 4 | 0 | 1 | 0 | 0 | 0 |
| 4 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 2 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 5 | 9 | 8 | 0 | 0 | 0 |
| 12 | 9 | 0 | 5 | 0 | 0 |
| 6 | 3 | 0 | 0 | 0 | 8 |
| 6 | 3 | 0 | 0 | 8 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 8 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 12 |
| 4 | 0 | 0 | 0 | 1 | 0 |
| 9 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 4 | 0 | 1 | 0 |
| 0 | 0 | 4 | 0 | 0 | 1 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 1 | 10 | 0 | 0 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,2,1,9,9,3,3,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[12,0,0,0,10,10,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,4,4,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,12,5,12,6,6,2,1,9,9,3,3,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[8,8,0,0,7,4,0,5,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[9,0,0,0,0,0,0,0,0,0,0,1,11,12,4,4,10,10,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
C24.A4 in GAP, Magma, Sage, TeX
C_2^4.A_4
% in TeX
G:=Group("C2^4.A4"); // GroupNames label
G:=SmallGroup(192,195);
// by ID
G=gap.SmallGroup(192,195);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,632,135,352,1683,262,521,248,851,1524]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^3=1,e^2=f^2=d,e*a*e^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*g=g*a,g*b*g^-1=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,g*c*g^-1=b,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=d*e*f,g*f*g^-1=e>;
// generators/relations
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