Aliases: C4oD4:1A4, C22:(C4.A4), Q8.1(C2xA4), Q8:A4:3C2, (C22xC4).9A4, C4.1(C22:A4), C23.23(C2xA4), (C22xQ8).7C6, (C22xC4oD4):3C3, C2.3(C2xC22:A4), SmallGroup(192,1507)
Series: Derived ►Chief ►Lower central ►Upper central
C22xQ8 — C4oD4:A4 |
Generators and relations for C4oD4:A4
G = < a,b,c,d,e,f | a4=c2=d2=e2=f3=1, b2=a2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=a2b, bd=db, be=eb, fbf-1=a-1bc, cd=dc, ce=ec, fcf-1=a-1b, fdf-1=de=ed, fef-1=d >
Subgroups: 599 in 175 conjugacy classes, 18 normal (10 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C2xC4, D4, Q8, Q8, C23, C23, C12, A4, C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C4oD4, C24, SL2(F3), C2xA4, C23xC4, C22xD4, C22xQ8, C2xC4oD4, C4xA4, C4.A4, C22xC4oD4, Q8:A4, C4oD4:A4
Quotients: C1, C2, C3, C6, A4, C2xA4, C4.A4, C22:A4, C2xC22:A4, C4oD4:A4
Character table of C4oD4:A4
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 12A | 12B | 12C | 12D | |
size | 1 | 1 | 3 | 3 | 6 | 6 | 6 | 6 | 16 | 16 | 1 | 1 | 3 | 3 | 6 | 6 | 6 | 6 | 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 | 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 | linear of order 2 |
ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ζ32 | ζ3 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | linear of order 6 |
ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ζ3 | ζ32 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | linear of order 6 |
ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
ρ7 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 1 | 1 | -i | -i | i | i | complex lifted from C4.A4 |
ρ8 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 1 | 1 | i | i | -i | -i | complex lifted from C4.A4 |
ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | ζ32 | ζ3 | ζ43ζ3 | ζ43ζ32 | ζ4ζ3 | ζ4ζ32 | complex lifted from C4.A4 |
ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | ζ3 | ζ32 | ζ43ζ32 | ζ43ζ3 | ζ4ζ32 | ζ4ζ3 | complex lifted from C4.A4 |
ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | ζ3 | ζ32 | ζ4ζ32 | ζ4ζ3 | ζ43ζ32 | ζ43ζ3 | complex lifted from C4.A4 |
ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | ζ32 | ζ3 | ζ4ζ3 | ζ4ζ32 | ζ43ζ3 | ζ43ζ32 | complex lifted from C4.A4 |
ρ13 | 3 | 3 | -1 | -1 | 1 | 1 | -3 | 1 | 0 | 0 | -3 | -3 | 1 | 1 | -1 | -1 | -1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xA4 |
ρ14 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
ρ15 | 3 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
ρ16 | 3 | 3 | -1 | -1 | -3 | 1 | 1 | 1 | 0 | 0 | -3 | -3 | 1 | 1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xA4 |
ρ17 | 3 | 3 | -1 | -1 | 1 | 1 | 1 | -3 | 0 | 0 | -3 | -3 | 1 | 1 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xA4 |
ρ18 | 3 | 3 | -1 | -1 | 1 | -3 | 1 | 1 | 0 | 0 | -3 | -3 | 1 | 1 | -1 | -1 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xA4 |
ρ19 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | 3 | 3 | -1 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
ρ20 | 3 | 3 | -1 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | 3 | 3 | -1 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
ρ21 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | -1 | 0 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
ρ22 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 0 | 0 | -3 | -3 | -3 | -3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xA4 |
ρ23 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 6i | -6i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
ρ24 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -6i | 6i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
(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 10 3 12)(2 11 4 9)(5 8 7 6)(13 18 15 20)(14 19 16 17)(21 22 23 24)
(1 12)(2 9)(3 10)(4 11)(5 21)(6 22)(7 23)(8 24)(17 19)(18 20)
(1 3)(2 4)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(21 23)(22 24)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 18 10)(6 19 11)(7 20 12)(8 17 9)
G:=sub<Sym(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), (1,10,3,12)(2,11,4,9)(5,8,7,6)(13,18,15,20)(14,19,16,17)(21,22,23,24), (1,12)(2,9)(3,10)(4,11)(5,21)(6,22)(7,23)(8,24)(17,19)(18,20), (1,3)(2,4)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(21,23)(22,24), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,18,10)(6,19,11)(7,20,12)(8,17,9)>;
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), (1,10,3,12)(2,11,4,9)(5,8,7,6)(13,18,15,20)(14,19,16,17)(21,22,23,24), (1,12)(2,9)(3,10)(4,11)(5,21)(6,22)(7,23)(8,24)(17,19)(18,20), (1,3)(2,4)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(21,23)(22,24), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,18,10)(6,19,11)(7,20,12)(8,17,9) );
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)], [(1,10,3,12),(2,11,4,9),(5,8,7,6),(13,18,15,20),(14,19,16,17),(21,22,23,24)], [(1,12),(2,9),(3,10),(4,11),(5,21),(6,22),(7,23),(8,24),(17,19),(18,20)], [(1,3),(2,4),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(21,23),(22,24)], [(1,24,16),(2,21,13),(3,22,14),(4,23,15),(5,18,10),(6,19,11),(7,20,12),(8,17,9)]])
G:=TransitiveGroup(24,302);
Matrix representation of C4oD4:A4 ►in GL5(F13)
5 | 0 | 0 | 0 | 0 |
0 | 5 | 0 | 0 | 0 |
0 | 0 | 12 | 0 | 0 |
0 | 0 | 0 | 12 | 0 |
0 | 0 | 0 | 0 | 12 |
4 | 3 | 0 | 0 | 0 |
3 | 9 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 12 |
0 | 0 | 1 | 0 | 12 |
0 | 0 | 0 | 0 | 12 |
11 | 7 | 0 | 0 | 0 |
7 | 2 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 | 12 |
0 | 0 | 1 | 12 | 0 |
1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 12 | 0 | 0 |
0 | 0 | 12 | 0 | 1 |
0 | 0 | 12 | 1 | 0 |
1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 12 | 1 |
0 | 0 | 0 | 12 | 0 |
0 | 0 | 1 | 12 | 0 |
1 | 0 | 0 | 0 | 0 |
3 | 9 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(13))| [5,0,0,0,0,0,5,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[4,3,0,0,0,3,9,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,12,12,12],[11,7,0,0,0,7,2,0,0,0,0,0,1,1,1,0,0,0,0,12,0,0,0,12,0],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,12,12,0,0,1,0,0],[1,3,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0] >;
C4oD4:A4 in GAP, Magma, Sage, TeX
C_4\circ D_4\rtimes A_4
% in TeX
G:=Group("C4oD4:A4");
// GroupNames label
G:=SmallGroup(192,1507);
// by ID
G=gap.SmallGroup(192,1507);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,135,262,851,172,1524,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=c^2=d^2=e^2=f^3=1,b^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=a^2*b,b*d=d*b,b*e=e*b,f*b*f^-1=a^-1*b*c,c*d=d*c,c*e=e*c,f*c*f^-1=a^-1*b,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations
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