Aliases: C23.3A4, C22.SL2(F3), C2.(C42:C3), C2.C42:C3, SmallGroup(96,3)
Series: Derived ►Chief ►Lower central ►Upper central
C2.C42 — C23.3A4 |
Generators and relations for C23.3A4
G = < a,b,c,d,e,f | a2=b2=c2=f3=1, d2=faf-1=abc, e2=fbf-1=a, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ce=ec, cf=fc, fdf-1=ade, fef-1=bcd >
Character table of C23.3A4
class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | |
size | 1 | 1 | 3 | 3 | 16 | 16 | 6 | 6 | 6 | 6 | 16 | 16 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | linear of order 3 |
ρ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | linear of order 3 |
ρ4 | 2 | -2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | symplectic lifted from SL2(F3), Schur index 2 |
ρ5 | 2 | -2 | -2 | 2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | complex lifted from SL2(F3) |
ρ6 | 2 | -2 | -2 | 2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | complex lifted from SL2(F3) |
ρ7 | 3 | 3 | 3 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from A4 |
ρ8 | 3 | 3 | -1 | -1 | 0 | 0 | 1 | -1-2i | 1 | -1+2i | 0 | 0 | complex lifted from C42:C3 |
ρ9 | 3 | 3 | -1 | -1 | 0 | 0 | 1 | -1+2i | 1 | -1-2i | 0 | 0 | complex lifted from C42:C3 |
ρ10 | 3 | 3 | -1 | -1 | 0 | 0 | -1+2i | 1 | -1-2i | 1 | 0 | 0 | complex lifted from C42:C3 |
ρ11 | 3 | 3 | -1 | -1 | 0 | 0 | -1-2i | 1 | -1+2i | 1 | 0 | 0 | complex lifted from C42:C3 |
ρ12 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
(1 2)(5 6)
(3 7)(4 8)
(1 2)(3 7)(4 8)(5 6)(9 11)(10 12)
(3 4)(5 6)(7 8)(9 10 11 12)
(1 5 2 6)(3 7)(9 12)(10 11)
(1 7 12)(2 3 10)(4 11 5)(6 8 9)
G:=sub<Sym(12)| (1,2)(5,6), (3,7)(4,8), (1,2)(3,7)(4,8)(5,6)(9,11)(10,12), (3,4)(5,6)(7,8)(9,10,11,12), (1,5,2,6)(3,7)(9,12)(10,11), (1,7,12)(2,3,10)(4,11,5)(6,8,9)>;
G:=Group( (1,2)(5,6), (3,7)(4,8), (1,2)(3,7)(4,8)(5,6)(9,11)(10,12), (3,4)(5,6)(7,8)(9,10,11,12), (1,5,2,6)(3,7)(9,12)(10,11), (1,7,12)(2,3,10)(4,11,5)(6,8,9) );
G=PermutationGroup([[(1,2),(5,6)], [(3,7),(4,8)], [(1,2),(3,7),(4,8),(5,6),(9,11),(10,12)], [(3,4),(5,6),(7,8),(9,10,11,12)], [(1,5,2,6),(3,7),(9,12),(10,11)], [(1,7,12),(2,3,10),(4,11,5),(6,8,9)]])
G:=TransitiveGroup(12,57);
(1 3)(2 4)(5 7)(6 8)
(9 12)(10 11)(13 15)(14 16)
(1 3)(2 4)(5 7)(6 8)(9 12)(10 11)(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 6 3 8)(2 7 4 5)(9 13)(10 16)(11 14)(12 15)(17 24)(18 23)(19 22)(20 21)
(1 9 20)(2 15 24)(3 12 18)(4 13 22)(5 10 21)(6 14 19)(7 11 23)(8 16 17)
G:=sub<Sym(24)| (1,3)(2,4)(5,7)(6,8), (9,12)(10,11)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(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), (1,6,3,8)(2,7,4,5)(9,13)(10,16)(11,14)(12,15)(17,24)(18,23)(19,22)(20,21), (1,9,20)(2,15,24)(3,12,18)(4,13,22)(5,10,21)(6,14,19)(7,11,23)(8,16,17)>;
G:=Group( (1,3)(2,4)(5,7)(6,8), (9,12)(10,11)(13,15)(14,16), (1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(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), (1,6,3,8)(2,7,4,5)(9,13)(10,16)(11,14)(12,15)(17,24)(18,23)(19,22)(20,21), (1,9,20)(2,15,24)(3,12,18)(4,13,22)(5,10,21)(6,14,19)(7,11,23)(8,16,17) );
G=PermutationGroup([[(1,3),(2,4),(5,7),(6,8)], [(9,12),(10,11),(13,15),(14,16)], [(1,3),(2,4),(5,7),(6,8),(9,12),(10,11),(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)], [(1,6,3,8),(2,7,4,5),(9,13),(10,16),(11,14),(12,15),(17,24),(18,23),(19,22),(20,21)], [(1,9,20),(2,15,24),(3,12,18),(4,13,22),(5,10,21),(6,14,19),(7,11,23),(8,16,17)]])
G:=TransitiveGroup(24,179);
(1 2)(3 6)(4 5)(7 8)(13 20)(14 17)(15 18)(16 19)
(1 8)(2 7)(3 4)(5 6)(9 23)(10 24)(11 21)(12 22)
(1 7)(2 8)(3 5)(4 6)(9 21)(10 22)(11 23)(12 24)(13 18)(14 19)(15 20)(16 17)
(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 6 2 3)(4 8 5 7)(9 21)(11 23)(13 16 20 19)(14 18 17 15)
(1 24 15)(2 10 13)(3 11 19)(4 9 16)(5 23 14)(6 21 17)(7 12 20)(8 22 18)
G:=sub<Sym(24)| (1,2)(3,6)(4,5)(7,8)(13,20)(14,17)(15,18)(16,19), (1,8)(2,7)(3,4)(5,6)(9,23)(10,24)(11,21)(12,22), (1,7)(2,8)(3,5)(4,6)(9,21)(10,22)(11,23)(12,24)(13,18)(14,19)(15,20)(16,17), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6,2,3)(4,8,5,7)(9,21)(11,23)(13,16,20,19)(14,18,17,15), (1,24,15)(2,10,13)(3,11,19)(4,9,16)(5,23,14)(6,21,17)(7,12,20)(8,22,18)>;
G:=Group( (1,2)(3,6)(4,5)(7,8)(13,20)(14,17)(15,18)(16,19), (1,8)(2,7)(3,4)(5,6)(9,23)(10,24)(11,21)(12,22), (1,7)(2,8)(3,5)(4,6)(9,21)(10,22)(11,23)(12,24)(13,18)(14,19)(15,20)(16,17), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6,2,3)(4,8,5,7)(9,21)(11,23)(13,16,20,19)(14,18,17,15), (1,24,15)(2,10,13)(3,11,19)(4,9,16)(5,23,14)(6,21,17)(7,12,20)(8,22,18) );
G=PermutationGroup([[(1,2),(3,6),(4,5),(7,8),(13,20),(14,17),(15,18),(16,19)], [(1,8),(2,7),(3,4),(5,6),(9,23),(10,24),(11,21),(12,22)], [(1,7),(2,8),(3,5),(4,6),(9,21),(10,22),(11,23),(12,24),(13,18),(14,19),(15,20),(16,17)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,6,2,3),(4,8,5,7),(9,21),(11,23),(13,16,20,19),(14,18,17,15)], [(1,24,15),(2,10,13),(3,11,19),(4,9,16),(5,23,14),(6,21,17),(7,12,20),(8,22,18)]])
G:=TransitiveGroup(24,180);
C23.3A4 is a maximal subgroup of
C23.7S4 C23.8S4 C42:4C4:C3 C23:2D4:C3 C24.2A4 C24.3A4 C23.19(C2xA4)
C23.3A4 is a maximal quotient of C23.SL2(F3) C2.(C42:C9)
action | f(x) | Disc(f) |
---|---|---|
12T57 | x12-69x10-2091x8-7571x6+134691x4+960267x2+1545049 | 212·328·712·1110·134·10912·1132·1278 |
Matrix representation of C23.3A4 ►in GL5(F13)
12 | 0 | 0 | 0 | 0 |
0 | 12 | 0 | 0 | 0 |
0 | 0 | 12 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 12 |
12 | 0 | 0 | 0 | 0 |
0 | 12 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 12 | 0 |
0 | 0 | 0 | 0 | 12 |
12 | 0 | 0 | 0 | 0 |
0 | 12 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 |
0 | 1 | 0 | 0 | 0 |
12 | 0 | 0 | 0 | 0 |
0 | 0 | 5 | 0 | 0 |
0 | 0 | 0 | 8 | 0 |
0 | 0 | 0 | 0 | 1 |
4 | 3 | 0 | 0 | 0 |
3 | 9 | 0 | 0 | 0 |
0 | 0 | 5 | 0 | 0 |
0 | 0 | 0 | 12 | 0 |
0 | 0 | 0 | 0 | 5 |
1 | 0 | 0 | 0 | 0 |
3 | 9 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,12,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,8,0,0,0,0,0,1],[4,3,0,0,0,3,9,0,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,5],[1,3,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;
C23.3A4 in GAP, Magma, Sage, TeX
C_2^3._3A_4
% in TeX
G:=Group("C2^3.3A4");
// GroupNames label
G:=SmallGroup(96,3);
// by ID
G=gap.SmallGroup(96,3);
# by ID
G:=PCGroup([6,-3,-2,2,-2,2,-2,73,151,596,332,50,1731,2524]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=f^3=1,d^2=f*a*f^-1=a*b*c,e^2=f*b*f^-1=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*d*e^-1=c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=a*d*e,f*e*f^-1=b*c*d>;
// generators/relations
Export
Subgroup lattice of C23.3A4 in TeX
Character table of C23.3A4 in TeX