non-abelian, soluble, monomial
Aliases: C32:D8, C2.3S3wrC2, (C3xC6).3D4, D6:S3:1C2, C32:2C8:1C2, C3:Dic3.1C22, SmallGroup(144,117)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32:D8
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=b, dad=cbc-1=a-1, bd=db, dcd=c-1 >
Quotients: C1, C2, C22, D4, D8, S3wrC2, C32:D8
Character table of C32:D8
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | |
| size | 1 | 1 | 12 | 12 | 4 | 4 | 18 | 4 | 4 | 12 | 12 | 12 | 12 | 18 | 18 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | √2 | -√2 | orthogonal lifted from D8 |
| ρ7 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | -√2 | √2 | orthogonal lifted from D8 |
| ρ8 | 4 | 4 | 0 | -2 | 1 | -2 | 0 | -2 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from S3wrC2 |
| ρ9 | 4 | 4 | 2 | 0 | -2 | 1 | 0 | 1 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3wrC2 |
| ρ10 | 4 | 4 | -2 | 0 | -2 | 1 | 0 | 1 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3wrC2 |
| ρ11 | 4 | 4 | 0 | 2 | 1 | -2 | 0 | -2 | 1 | 0 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from S3wrC2 |
| ρ12 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | -1 | 2 | -√-3 | √-3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ13 | 4 | -4 | 0 | 0 | -2 | 1 | 0 | -1 | 2 | √-3 | -√-3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ14 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 2 | -1 | 0 | 0 | -√-3 | √-3 | 0 | 0 | complex faithful |
| ρ15 | 4 | -4 | 0 | 0 | 1 | -2 | 0 | 2 | -1 | 0 | 0 | √-3 | -√-3 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T218
Generators in S24
(1 21 10)(2 22 11)(3 12 23)(4 13 24)(5 17 14)(6 18 15)(7 16 19)(8 9 20)
(1 21 10)(2 11 22)(3 12 23)(4 24 13)(5 17 14)(6 15 18)(7 16 19)(8 20 9)
(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 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 24)(18 23)(19 22)(20 21)
G:=sub<Sym(24)| (1,21,10)(2,22,11)(3,12,23)(4,13,24)(5,17,14)(6,18,15)(7,16,19)(8,9,20), (1,21,10)(2,11,22)(3,12,23)(4,24,13)(5,17,14)(6,15,18)(7,16,19)(8,20,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,24)(18,23)(19,22)(20,21)>;
G:=Group( (1,21,10)(2,22,11)(3,12,23)(4,13,24)(5,17,14)(6,18,15)(7,16,19)(8,9,20), (1,21,10)(2,11,22)(3,12,23)(4,24,13)(5,17,14)(6,15,18)(7,16,19)(8,20,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,24)(18,23)(19,22)(20,21) );
G=PermutationGroup([[(1,21,10),(2,22,11),(3,12,23),(4,13,24),(5,17,14),(6,18,15),(7,16,19),(8,9,20)], [(1,21,10),(2,11,22),(3,12,23),(4,24,13),(5,17,14),(6,15,18),(7,16,19),(8,20,9)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14),(17,24),(18,23),(19,22),(20,21)]])
G:=TransitiveGroup(24,218);
►On 24 points - transitive group 24T219
Generators in S24
(1 21 15)(2 16 22)(3 9 23)(4 24 10)(5 17 11)(6 12 18)(7 13 19)(8 20 14)
(1 15 21)(2 16 22)(3 23 9)(4 24 10)(5 11 17)(6 12 18)(7 19 13)(8 20 14)
(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 8)(2 7)(3 6)(4 5)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)
G:=sub<Sym(24)| (1,21,15)(2,16,22)(3,9,23)(4,24,10)(5,17,11)(6,12,18)(7,13,19)(8,20,14), (1,15,21)(2,16,22)(3,23,9)(4,24,10)(5,11,17)(6,12,18)(7,19,13)(8,20,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)>;
G:=Group( (1,21,15)(2,16,22)(3,9,23)(4,24,10)(5,17,11)(6,12,18)(7,13,19)(8,20,14), (1,15,21)(2,16,22)(3,23,9)(4,24,10)(5,11,17)(6,12,18)(7,19,13)(8,20,14), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19) );
G=PermutationGroup([[(1,21,15),(2,16,22),(3,9,23),(4,24,10),(5,17,11),(6,12,18),(7,13,19),(8,20,14)], [(1,15,21),(2,16,22),(3,23,9),(4,24,10),(5,11,17),(6,12,18),(7,19,13),(8,20,14)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,18),(10,17),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19)]])
G:=TransitiveGroup(24,219);
C32:D8 is a maximal subgroup of
C32:D8:5C2 C32:D8:C2 C3:S3:D8 C62.12D4 C62.13D4 C33:D8 C32:2D24
C32:D8 is a maximal quotient of
C32:D16 C32:SD32 C32:Q32 C62.3D4 C62.7D4 He3:D8 C33:D8 C32:2D24
Matrix representation of C32:D8 ►in GL4(F7) generated by
| 3 | 2 | 4 | 3 |
| 4 | 5 | 5 | 6 |
| 3 | 3 | 6 | 1 |
| 0 | 0 | 0 | 1 |
| 6 | 2 | 1 | 1 |
| 2 | 6 | 6 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 3 | 2 | 1 |
| 1 | 1 | 5 | 4 |
| 4 | 3 | 5 | 3 |
| 3 | 3 | 4 | 1 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(7))| [3,4,3,0,2,5,3,0,4,5,6,0,3,6,1,1],[6,2,0,0,2,6,0,0,1,6,1,0,1,1,0,2],[0,1,4,3,3,1,3,3,2,5,5,4,1,4,3,1],[0,1,0,0,1,0,0,0,1,1,6,0,0,0,0,1] >;
C32:D8 in GAP, Magma, Sage, TeX
C_3^2\rtimes D_8
% in TeX
G:=Group("C3^2:D8"); // GroupNames label
G:=SmallGroup(144,117);
// by ID
G=gap.SmallGroup(144,117);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,3,73,218,116,50,964,730,256,299,881]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,c*a*c^-1=b,d*a*d=c*b*c^-1=a^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C32:D8 in TeX
Character table of C32:D8 in TeX