non-abelian, soluble, monomial
Aliases: C32⋊2SD16, C2.4S3≀C2, (C3×C6).4D4, D6⋊S3.C2, C32⋊2C8⋊2C2, C32⋊2Q8⋊1C2, C3⋊Dic3.2C22, SmallGroup(144,118)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊2SD16
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=b-1, dad=a-1, cbc-1=a, bd=db, dcd=c3 >
Character table of C32⋊2SD16
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 12A | 12B | |
| size | 1 | 1 | 12 | 4 | 4 | 12 | 18 | 4 | 4 | 12 | 12 | 18 | 18 | 12 | 12 | |
| 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 | 2 | 2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | -2 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | √-2 | -√-2 | 0 | 0 | complex lifted from SD16 |
| ρ7 | 2 | -2 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | -√-2 | √-2 | 0 | 0 | complex lifted from SD16 |
| ρ8 | 4 | 4 | 0 | -2 | 1 | 2 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3≀C2 |
| ρ9 | 4 | 4 | 2 | 1 | -2 | 0 | 0 | -2 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ10 | 4 | 4 | 0 | -2 | 1 | -2 | 0 | 1 | -2 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from S3≀C2 |
| ρ11 | 4 | 4 | -2 | 1 | -2 | 0 | 0 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ12 | 4 | -4 | 0 | -2 | 1 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | √3 | -√3 | symplectic faithful, Schur index 2 |
| ρ13 | 4 | -4 | 0 | -2 | 1 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | -√3 | √3 | symplectic faithful, Schur index 2 |
| ρ14 | 4 | -4 | 0 | 1 | -2 | 0 | 0 | 2 | -1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | complex faithful |
| ρ15 | 4 | -4 | 0 | 1 | -2 | 0 | 0 | 2 | -1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T217
Generators in S24
(2 14 22)(4 24 16)(6 10 18)(8 20 12)
(1 21 13)(3 15 23)(5 17 9)(7 11 19)
(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 4)(3 7)(6 8)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)
G:=sub<Sym(24)| (2,14,22)(4,24,16)(6,10,18)(8,20,12), (1,21,13)(3,15,23)(5,17,9)(7,11,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24)>;
G:=Group( (2,14,22)(4,24,16)(6,10,18)(8,20,12), (1,21,13)(3,15,23)(5,17,9)(7,11,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,4)(3,7)(6,8)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24) );
G=PermutationGroup([[(2,14,22),(4,24,16),(6,10,18),(8,20,12)], [(1,21,13),(3,15,23),(5,17,9),(7,11,19)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,4),(3,7),(6,8),(10,12),(11,15),(14,16),(18,20),(19,23),(22,24)]])
G:=TransitiveGroup(24,217);
►On 24 points - transitive group 24T220
Generators in S24
(1 15 21)(3 23 9)(5 11 17)(7 19 13)
(2 16 22)(4 24 10)(6 12 18)(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)
(2 4)(3 7)(6 8)(9 19)(10 22)(11 17)(12 20)(13 23)(14 18)(15 21)(16 24)
G:=sub<Sym(24)| (1,15,21)(3,23,9)(5,11,17)(7,19,13), (2,16,22)(4,24,10)(6,12,18)(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), (2,4)(3,7)(6,8)(9,19)(10,22)(11,17)(12,20)(13,23)(14,18)(15,21)(16,24)>;
G:=Group( (1,15,21)(3,23,9)(5,11,17)(7,19,13), (2,16,22)(4,24,10)(6,12,18)(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), (2,4)(3,7)(6,8)(9,19)(10,22)(11,17)(12,20)(13,23)(14,18)(15,21)(16,24) );
G=PermutationGroup([[(1,15,21),(3,23,9),(5,11,17),(7,19,13)], [(2,16,22),(4,24,10),(6,12,18),(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)], [(2,4),(3,7),(6,8),(9,19),(10,22),(11,17),(12,20),(13,23),(14,18),(15,21),(16,24)]])
G:=TransitiveGroup(24,220);
C32⋊2SD16 is a maximal subgroup of
C32⋊D8⋊5C2 C32⋊D8⋊C2 C32⋊Q16⋊C2 C3⋊S3⋊2SD16 C62.12D4 C62.13D4 C62.15D4 C33⋊6SD16 C33⋊7SD16 C33⋊8SD16
C32⋊2SD16 is a maximal quotient of
C62.3D4 C62.4D4 C62.6D4 He3⋊2SD16 C33⋊6SD16 C33⋊7SD16 C33⋊8SD16
Matrix representation of C32⋊2SD16 ►in GL4(𝔽7) generated by
| 3 | 0 | 6 | 0 |
| 4 | 6 | 3 | 6 |
| 1 | 1 | 5 | 4 |
| 1 | 0 | 3 | 1 |
| 5 | 1 | 2 | 0 |
| 0 | 5 | 1 | 4 |
| 1 | 2 | 5 | 0 |
| 5 | 2 | 4 | 0 |
| 5 | 6 | 4 | 3 |
| 4 | 0 | 6 | 1 |
| 5 | 5 | 1 | 2 |
| 2 | 2 | 3 | 1 |
| 6 | 6 | 5 | 6 |
| 4 | 6 | 3 | 6 |
| 5 | 4 | 2 | 1 |
| 0 | 1 | 2 | 0 |
G:=sub<GL(4,GF(7))| [3,4,1,1,0,6,1,0,6,3,5,3,0,6,4,1],[5,0,1,5,1,5,2,2,2,1,5,4,0,4,0,0],[5,4,5,2,6,0,5,2,4,6,1,3,3,1,2,1],[6,4,5,0,6,6,4,1,5,3,2,2,6,6,1,0] >;
C32⋊2SD16 in GAP, Magma, Sage, TeX
C_3^2\rtimes_2{\rm SD}_{16} % in TeX
G:=Group("C3^2:2SD16"); // GroupNames label
G:=SmallGroup(144,118);
// by ID
G=gap.SmallGroup(144,118);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,3,73,55,218,116,50,964,730,256,299,881]);
// 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^-1,d*a*d=a^-1,c*b*c^-1=a,b*d=d*b,d*c*d=c^3>;
// generators/relations
Export
Subgroup lattice of C32⋊2SD16 in TeX
Character table of C32⋊2SD16 in TeX