non-abelian, soluble, monomial
Aliases: C32⋊Q16, C2.5S3≀C2, (C3×C6).5D4, C32⋊2Q8.C2, C32⋊2C8.2C2, C3⋊Dic3.3C22, SmallGroup(144,119)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊Q16
G = < a,b,c,d | a3=b3=c8=1, d2=c4, ab=ba, cac-1=dad-1=b, cbc-1=a-1, dbd-1=a, dcd-1=c-1 >
Character table of C32⋊Q16
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 8A | 8B | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 4 | 4 | 12 | 12 | 18 | 4 | 4 | 18 | 18 | 12 | 12 | 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 | 2 | 2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -√2 | √2 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ7 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | √2 | -√2 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ8 | 4 | 4 | -2 | 1 | 0 | 2 | 0 | -2 | 1 | 0 | 0 | -1 | 0 | 0 | -1 | orthogonal lifted from S3≀C2 |
| ρ9 | 4 | 4 | -2 | 1 | 0 | -2 | 0 | -2 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | orthogonal lifted from S3≀C2 |
| ρ10 | 4 | 4 | 1 | -2 | 2 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | -1 | -1 | 0 | orthogonal lifted from S3≀C2 |
| ρ11 | 4 | 4 | 1 | -2 | -2 | 0 | 0 | 1 | -2 | 0 | 0 | 0 | 1 | 1 | 0 | orthogonal lifted from S3≀C2 |
| ρ12 | 4 | -4 | -2 | 1 | 0 | 0 | 0 | 2 | -1 | 0 | 0 | -√3 | 0 | 0 | √3 | symplectic faithful, Schur index 2 |
| ρ13 | 4 | -4 | -2 | 1 | 0 | 0 | 0 | 2 | -1 | 0 | 0 | √3 | 0 | 0 | -√3 | symplectic faithful, Schur index 2 |
| ρ14 | 4 | -4 | 1 | -2 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | √3 | -√3 | 0 | symplectic faithful, Schur index 2 |
| ρ15 | 4 | -4 | 1 | -2 | 0 | 0 | 0 | -1 | 2 | 0 | 0 | 0 | -√3 | √3 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(2 47 32)(4 26 41)(6 43 28)(8 30 45)(9 19 35)(11 37 21)(13 23 39)(15 33 17)
(1 46 31)(3 25 48)(5 42 27)(7 29 44)(10 36 20)(12 22 38)(14 40 24)(16 18 34)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)
(1 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 31 13 27)(10 30 14 26)(11 29 15 25)(12 28 16 32)(33 48 37 44)(34 47 38 43)(35 46 39 42)(36 45 40 41)
G:=sub<Sym(48)| (2,47,32)(4,26,41)(6,43,28)(8,30,45)(9,19,35)(11,37,21)(13,23,39)(15,33,17), (1,46,31)(3,25,48)(5,42,27)(7,29,44)(10,36,20)(12,22,38)(14,40,24)(16,18,34), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,31,13,27)(10,30,14,26)(11,29,15,25)(12,28,16,32)(33,48,37,44)(34,47,38,43)(35,46,39,42)(36,45,40,41)>;
G:=Group( (2,47,32)(4,26,41)(6,43,28)(8,30,45)(9,19,35)(11,37,21)(13,23,39)(15,33,17), (1,46,31)(3,25,48)(5,42,27)(7,29,44)(10,36,20)(12,22,38)(14,40,24)(16,18,34), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,31,13,27)(10,30,14,26)(11,29,15,25)(12,28,16,32)(33,48,37,44)(34,47,38,43)(35,46,39,42)(36,45,40,41) );
G=PermutationGroup([[(2,47,32),(4,26,41),(6,43,28),(8,30,45),(9,19,35),(11,37,21),(13,23,39),(15,33,17)], [(1,46,31),(3,25,48),(5,42,27),(7,29,44),(10,36,20),(12,22,38),(14,40,24),(16,18,34)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,23,5,19),(2,22,6,18),(3,21,7,17),(4,20,8,24),(9,31,13,27),(10,30,14,26),(11,29,15,25),(12,28,16,32),(33,48,37,44),(34,47,38,43),(35,46,39,42),(36,45,40,41)]])
C32⋊Q16 is a maximal subgroup of
C32⋊D8⋊5C2 C32⋊Q16⋊C2 C3⋊S3⋊Q16 C62.13D4 C62.15D4 C33⋊Q16 C33⋊3Q16
C32⋊Q16 is a maximal quotient of C62.4D4 C62.7D4 He3⋊Q16 C33⋊Q16 C33⋊3Q16
Matrix representation of C32⋊Q16 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 72 |
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 14 | 68 |
| 0 | 0 | 54 | 59 |
| 7 | 14 | 0 | 0 |
| 59 | 66 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,72],[0,72,0,0,1,72,0,0,0,0,1,0,0,0,0,1],[0,0,7,59,0,0,14,66,14,54,0,0,68,59,0,0],[0,0,72,0,0,0,0,72,1,0,0,0,0,1,0,0] >;
C32⋊Q16 in GAP, Magma, Sage, TeX
C_3^2\rtimes Q_{16} % in TeX
G:=Group("C3^2:Q16"); // GroupNames label
G:=SmallGroup(144,119);
// by ID
G=gap.SmallGroup(144,119);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,3,48,73,55,218,116,50,964,730,256,299,881]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^8=1,d^2=c^4,a*b=b*a,c*a*c^-1=d*a*d^-1=b,c*b*c^-1=a^-1,d*b*d^-1=a,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of C32⋊Q16 in TeX
Character table of C32⋊Q16 in TeX