metabelian, supersoluble, monomial
Aliases: C32⋊3Q16, C3⋊2Dic12, C12.15D6, C6.15D12, Dic6.1S3, C3⋊C8.S3, C4.4S32, (C3×C6).12D4, C3⋊1(C3⋊Q16), C6.4(C3⋊D4), (C3×C12).7C22, (C3×Dic6).2C2, C2.7(C3⋊D12), C32⋊4Q8.2C2, (C3×C3⋊C8).1C2, SmallGroup(144,62)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊3Q16
G = < a,b,c,d | a3=b3=c8=1, d2=c4, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Character table of C32⋊3Q16
class | 1 | 2 | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 24A | 24B | 24C | 24D | |
size | 1 | 1 | 2 | 2 | 4 | 2 | 12 | 36 | 2 | 2 | 4 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 6 | 6 | 6 | 6 | |
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 | 2 | 2 | 2 | -1 | -1 | 2 | 0 | 0 | -1 | 2 | -1 | -2 | -2 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
ρ6 | 2 | 2 | -1 | 2 | -1 | 2 | 2 | 0 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
ρ7 | 2 | 2 | -1 | 2 | -1 | 2 | -2 | 0 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
ρ8 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ9 | 2 | 2 | 2 | -1 | -1 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ10 | 2 | 2 | 2 | -1 | -1 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 1 | 1 | -2 | 1 | 1 | 0 | 0 | -√3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
ρ11 | 2 | 2 | 2 | -1 | -1 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 1 | 1 | -2 | 1 | 1 | 0 | 0 | √3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
ρ12 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
ρ13 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
ρ14 | 2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 1 | -2 | 1 | -√2 | √2 | -√3 | √3 | 0 | -√3 | √3 | 0 | 0 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | symplectic lifted from Dic12, Schur index 2 |
ρ15 | 2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 1 | -2 | 1 | √2 | -√2 | √3 | -√3 | 0 | √3 | -√3 | 0 | 0 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | symplectic lifted from Dic12, Schur index 2 |
ρ16 | 2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 1 | -2 | 1 | √2 | -√2 | -√3 | √3 | 0 | -√3 | √3 | 0 | 0 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | symplectic lifted from Dic12, Schur index 2 |
ρ17 | 2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 1 | -2 | 1 | -√2 | √2 | √3 | -√3 | 0 | √3 | -√3 | 0 | 0 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | symplectic lifted from Dic12, Schur index 2 |
ρ18 | 2 | 2 | -1 | 2 | -1 | -2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | √-3 | -√-3 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
ρ19 | 2 | 2 | -1 | 2 | -1 | -2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | -√-3 | √-3 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
ρ20 | 4 | 4 | -2 | -2 | 1 | 4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
ρ21 | 4 | 4 | -2 | -2 | 1 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3⋊D12 |
ρ22 | 4 | -4 | -2 | 4 | -2 | 0 | 0 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C3⋊Q16, Schur index 2 |
ρ23 | 4 | -4 | -2 | -2 | 1 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 2√3 | -2√3 | 0 | -√3 | √3 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
ρ24 | 4 | -4 | -2 | -2 | 1 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | -2√3 | 2√3 | 0 | √3 | -√3 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
(1 33 25)(2 26 34)(3 35 27)(4 28 36)(5 37 29)(6 30 38)(7 39 31)(8 32 40)(9 46 20)(10 21 47)(11 48 22)(12 23 41)(13 42 24)(14 17 43)(15 44 18)(16 19 45)
(1 25 33)(2 26 34)(3 27 35)(4 28 36)(5 29 37)(6 30 38)(7 31 39)(8 32 40)(9 46 20)(10 47 21)(11 48 22)(12 41 23)(13 42 24)(14 43 17)(15 44 18)(16 45 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)(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 9 5 13)(2 16 6 12)(3 15 7 11)(4 14 8 10)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(33 46 37 42)(34 45 38 41)(35 44 39 48)(36 43 40 47)
G:=sub<Sym(48)| (1,33,25)(2,26,34)(3,35,27)(4,28,36)(5,37,29)(6,30,38)(7,39,31)(8,32,40)(9,46,20)(10,21,47)(11,48,22)(12,23,41)(13,42,24)(14,17,43)(15,44,18)(16,19,45), (1,25,33)(2,26,34)(3,27,35)(4,28,36)(5,29,37)(6,30,38)(7,31,39)(8,32,40)(9,46,20)(10,47,21)(11,48,22)(12,41,23)(13,42,24)(14,43,17)(15,44,18)(16,45,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,9,5,13)(2,16,6,12)(3,15,7,11)(4,14,8,10)(17,32,21,28)(18,31,22,27)(19,30,23,26)(20,29,24,25)(33,46,37,42)(34,45,38,41)(35,44,39,48)(36,43,40,47)>;
G:=Group( (1,33,25)(2,26,34)(3,35,27)(4,28,36)(5,37,29)(6,30,38)(7,39,31)(8,32,40)(9,46,20)(10,21,47)(11,48,22)(12,23,41)(13,42,24)(14,17,43)(15,44,18)(16,19,45), (1,25,33)(2,26,34)(3,27,35)(4,28,36)(5,29,37)(6,30,38)(7,31,39)(8,32,40)(9,46,20)(10,47,21)(11,48,22)(12,41,23)(13,42,24)(14,43,17)(15,44,18)(16,45,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,9,5,13)(2,16,6,12)(3,15,7,11)(4,14,8,10)(17,32,21,28)(18,31,22,27)(19,30,23,26)(20,29,24,25)(33,46,37,42)(34,45,38,41)(35,44,39,48)(36,43,40,47) );
G=PermutationGroup([(1,33,25),(2,26,34),(3,35,27),(4,28,36),(5,37,29),(6,30,38),(7,39,31),(8,32,40),(9,46,20),(10,21,47),(11,48,22),(12,23,41),(13,42,24),(14,17,43),(15,44,18),(16,19,45)], [(1,25,33),(2,26,34),(3,27,35),(4,28,36),(5,29,37),(6,30,38),(7,31,39),(8,32,40),(9,46,20),(10,47,21),(11,48,22),(12,41,23),(13,42,24),(14,43,17),(15,44,18),(16,45,19)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,9,5,13),(2,16,6,12),(3,15,7,11),(4,14,8,10),(17,32,21,28),(18,31,22,27),(19,30,23,26),(20,29,24,25),(33,46,37,42),(34,45,38,41),(35,44,39,48),(36,43,40,47)])
C32⋊3Q16 is a maximal subgroup of
S3×Dic12 C24.3D6 Dic12⋊S3 D6.1D12 D12.27D6 D12.29D6 Dic6.29D6 Dic6.19D6 Dic6.D6 D12.22D6 D12.8D6 S3×C3⋊Q16 Dic6.9D6 D12.24D6 D12.15D6 C3⋊Dic36 C9⋊Dic12 He3⋊2Q16 He3⋊3Q16 C33⋊7Q16 C33⋊8Q16 C33⋊9Q16
C32⋊3Q16 is a maximal quotient of
C6.Dic12 C12.73D12 C6.18D24 C3⋊Dic36 C9⋊Dic12 He3⋊3Q16 C33⋊7Q16 C33⋊8Q16 C33⋊9Q16
Matrix representation of C32⋊3Q16 ►in GL6(𝔽73)
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 72 |
0 | 0 | 0 | 0 | 1 | 72 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 72 | 72 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 41 | 0 | 0 | 0 | 0 |
16 | 41 | 0 | 0 | 0 | 0 |
0 | 0 | 72 | 0 | 0 | 0 |
0 | 0 | 0 | 72 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 1 | 0 |
19 | 13 | 0 | 0 | 0 | 0 |
62 | 54 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 72 | 72 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,41,41,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[19,62,0,0,0,0,13,54,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C32⋊3Q16 in GAP, Magma, Sage, TeX
C_3^2\rtimes_3Q_{16}
% in TeX
G:=Group("C3^2:3Q16");
// GroupNames label
G:=SmallGroup(144,62);
// by ID
G=gap.SmallGroup(144,62);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,79,218,50,490,3461]);
// 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=a^-1,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of C32⋊3Q16 in TeX
Character table of C32⋊3Q16 in TeX