metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8:3Dic3, D4:2Dic3, C12.56D4, C3:3C4wrC2, (C3xD4):2C4, (C3xQ8):2C4, (C2xC6).3D4, C12.9(C2xC4), C4oD4.3S3, (C2xC4).41D6, (C4xDic3):2C2, C4.Dic3:4C2, C4.3(C2xDic3), C4.31(C3:D4), C6.18(C22:C4), (C2xC12).20C22, C22.3(C3:D4), C2.8(C6.D4), (C3xC4oD4).1C2, SmallGroup(96,44)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8:3Dic3
G = < a,b,c,d | a4=c6=1, b2=a2, d2=c3, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >
Character table of Q8:3Dic3
class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | 12C | 12D | 12E | |
size | 1 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 4 | 6 | 6 | 6 | 6 | 2 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 4 | 4 | 4 | |
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 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
ρ4 | 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 |
ρ5 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | 1 | -1 | -1 | -1 | -i | i | -1 | -1 | 1 | 1 | 1 | linear of order 4 |
ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | 1 | -1 | -1 | -1 | i | -i | -1 | -1 | 1 | 1 | 1 | linear of order 4 |
ρ7 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | i | -i | i | -i | 1 | 1 | -1 | 1 | -i | i | -1 | -1 | -1 | 1 | -1 | linear of order 4 |
ρ8 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -i | i | -i | i | 1 | 1 | -1 | 1 | i | -i | -1 | -1 | -1 | 1 | -1 | linear of order 4 |
ρ9 | 2 | 2 | -2 | 0 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | 2 | -2 | -1 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
ρ11 | 2 | 2 | 2 | 0 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | -2 | 0 | orthogonal lifted from D4 |
ρ12 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ13 | 2 | 2 | -2 | -2 | -1 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
ρ14 | 2 | 2 | -2 | 2 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 0 | 0 | 1 | 1 | 1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
ρ15 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | √-3 | 1 | -√-3 | 0 | 0 | -1 | -1 | -√-3 | 1 | √-3 | complex lifted from C3:D4 |
ρ16 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | √-3 | -1 | -√-3 | 0 | 0 | 1 | 1 | √-3 | 1 | -√-3 | complex lifted from C3:D4 |
ρ17 | 2 | 2 | 2 | 0 | -1 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -√-3 | -1 | √-3 | 0 | 0 | 1 | 1 | -√-3 | 1 | √-3 | complex lifted from C3:D4 |
ρ18 | 2 | 2 | -2 | 0 | -1 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -√-3 | 1 | √-3 | 0 | 0 | -1 | -1 | √-3 | 1 | -√-3 | complex lifted from C3:D4 |
ρ19 | 2 | -2 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | 1-i | -1-i | -1+i | 1+i | -2 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | complex lifted from C4wrC2 |
ρ20 | 2 | -2 | 0 | 0 | 2 | -2i | 2i | 0 | 0 | -1+i | 1+i | 1-i | -1-i | -2 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | complex lifted from C4wrC2 |
ρ21 | 2 | -2 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | -1-i | 1-i | 1+i | -1+i | -2 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | complex lifted from C4wrC2 |
ρ22 | 2 | -2 | 0 | 0 | 2 | 2i | -2i | 0 | 0 | 1+i | -1+i | -1-i | 1-i | -2 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | complex lifted from C4wrC2 |
ρ23 | 4 | -4 | 0 | 0 | -2 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | complex faithful |
ρ24 | 4 | -4 | 0 | 0 | -2 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | complex faithful |
(1 10 6 7)(2 11 4 8)(3 12 5 9)(13 22 16 19)(14 23 17 20)(15 24 18 21)
(1 22 6 19)(2 20 4 23)(3 24 5 21)(7 16 10 13)(8 14 11 17)(9 18 12 15)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(2 3)(4 5)(8 9)(11 12)(13 19 16 22)(14 24 17 21)(15 23 18 20)
G:=sub<Sym(24)| (1,10,6,7)(2,11,4,8)(3,12,5,9)(13,22,16,19)(14,23,17,20)(15,24,18,21), (1,22,6,19)(2,20,4,23)(3,24,5,21)(7,16,10,13)(8,14,11,17)(9,18,12,15), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (2,3)(4,5)(8,9)(11,12)(13,19,16,22)(14,24,17,21)(15,23,18,20)>;
G:=Group( (1,10,6,7)(2,11,4,8)(3,12,5,9)(13,22,16,19)(14,23,17,20)(15,24,18,21), (1,22,6,19)(2,20,4,23)(3,24,5,21)(7,16,10,13)(8,14,11,17)(9,18,12,15), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (2,3)(4,5)(8,9)(11,12)(13,19,16,22)(14,24,17,21)(15,23,18,20) );
G=PermutationGroup([[(1,10,6,7),(2,11,4,8),(3,12,5,9),(13,22,16,19),(14,23,17,20),(15,24,18,21)], [(1,22,6,19),(2,20,4,23),(3,24,5,21),(7,16,10,13),(8,14,11,17),(9,18,12,15)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(2,3),(4,5),(8,9),(11,12),(13,19,16,22),(14,24,17,21),(15,23,18,20)]])
G:=TransitiveGroup(24,109);
Q8:3Dic3 is a maximal subgroup of
S3xC4wrC2 C42:3D6 C24.100D4 C24.54D4 D8:5Dic3 D8:4Dic3 D12:18D4 D12.38D4 D12.39D4 D12.40D4 (C6xD4):9C4 2+ 1+4:6S3 2+ 1+4.4S3 2- 1+4:4S3 2- 1+4.2S3 Q8:3Dic9 C12.9S4 D12:4Dic3 D12:2Dic3 C62.39D4 C3:U2(F3) C60.96D4 C60.97D4 Q8:3Dic15 Dic10:Dic3 D20:2Dic3
Q8:3Dic3 is a maximal quotient of
C12.2C42 C12.57D8 C12.26Q16 (C6xD4):C4 (C6xQ8):C4 C42.7D6 C42.8D6 Q8:3Dic9 D12:4Dic3 D12:2Dic3 C62.39D4 C60.96D4 C60.97D4 Q8:3Dic15 Dic10:Dic3 D20:2Dic3
Matrix representation of Q8:3Dic3 ►in GL4(F5) generated by
2 | 0 | 0 | 0 |
0 | 3 | 0 | 0 |
0 | 0 | 3 | 0 |
0 | 0 | 0 | 2 |
0 | 3 | 0 | 0 |
3 | 0 | 0 | 0 |
3 | 0 | 0 | 3 |
0 | 2 | 3 | 0 |
0 | 0 | 0 | 1 |
0 | 1 | 4 | 0 |
0 | 1 | 0 | 0 |
4 | 0 | 0 | 4 |
1 | 0 | 0 | 1 |
0 | 0 | 2 | 0 |
0 | 2 | 0 | 0 |
0 | 0 | 0 | 4 |
G:=sub<GL(4,GF(5))| [2,0,0,0,0,3,0,0,0,0,3,0,0,0,0,2],[0,3,3,0,3,0,0,2,0,0,0,3,0,0,3,0],[0,0,0,4,0,1,1,0,0,4,0,0,1,0,0,4],[1,0,0,0,0,0,2,0,0,2,0,0,1,0,0,4] >;
Q8:3Dic3 in GAP, Magma, Sage, TeX
Q_8\rtimes_3{\rm Dic}_3
% in TeX
G:=Group("Q8:3Dic3");
// GroupNames label
G:=SmallGroup(96,44);
// by ID
G=gap.SmallGroup(96,44);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,86,579,297,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^6=1,b^2=a^2,d^2=c^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of Q8:3Dic3 in TeX
Character table of Q8:3Dic3 in TeX