non-abelian, soluble, monomial
Aliases: C33⋊3Q16, C32⋊2Dic12, C6.11S3≀C2, (C3×C6).12D12, C3⋊1(C32⋊Q16), C3⋊Dic3.16D6, C32⋊2C8.2S3, (C32×C6).17D4, C33⋊5Q8.1C2, C2.5(C32⋊2D12), (C3×C32⋊2C8).2C2, (C3×C3⋊Dic3).3C22, SmallGroup(432,590)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32 — C3×C3⋊Dic3 — C33⋊3Q16 |
| C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊3Q16 |
Generators and relations for C33⋊3Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, dad-1=eae-1=b, bc=cb, dbd-1=a-1, ebe-1=a, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 464 in 72 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C3, C3, C4, C6, C6, C8, Q8, C32, C32, Dic3, C12, Q16, C3×C6, C3×C6, C24, Dic6, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, Dic12, C32×C6, C32⋊2C8, C32⋊2Q8, C3×C3⋊Dic3, C3×C3⋊Dic3, C32⋊Q16, C3×C32⋊2C8, C33⋊5Q8, C33⋊3Q16
Quotients: C1, C2, C22, S3, D4, D6, Q16, D12, Dic12, S3≀C2, C32⋊Q16, C32⋊2D12, C33⋊3Q16
Character table of C33⋊3Q16
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 4 | 4 | 8 | 8 | 18 | 36 | 36 | 2 | 4 | 4 | 8 | 8 | 18 | 18 | 18 | 18 | 36 | 36 | 36 | 36 | 18 | 18 | 18 | 18 | |
| 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 | 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 | -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 | 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 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | -1 | 2 | 2 | -1 | -1 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -1 | 2 | 2 | -1 | -1 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ8 | 2 | 2 | -1 | 2 | 2 | -1 | -1 | -2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ9 | 2 | 2 | -1 | 2 | 2 | -1 | -1 | -2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ10 | 2 | -2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | symplectic lifted from Q16, Schur index 2 |
| ρ11 | 2 | -2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | symplectic lifted from Q16, Schur index 2 |
| ρ12 | 2 | -2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | 1 | -√2 | √2 | √3 | -√3 | 0 | 0 | 0 | 0 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | symplectic lifted from Dic12, Schur index 2 |
| ρ13 | 2 | -2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | 1 | √2 | -√2 | -√3 | √3 | 0 | 0 | 0 | 0 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | symplectic lifted from Dic12, Schur index 2 |
| ρ14 | 2 | -2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | 1 | -√2 | √2 | -√3 | √3 | 0 | 0 | 0 | 0 | ζ83ζ3+ζ83+ζ8ζ3 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ32+ζ87+ζ85ζ32 | symplectic lifted from Dic12, Schur index 2 |
| ρ15 | 2 | -2 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 1 | -2 | -2 | 1 | 1 | √2 | -√2 | √3 | -√3 | 0 | 0 | 0 | 0 | ζ87ζ32+ζ87+ζ85ζ32 | ζ83ζ32+ζ8ζ32+ζ8 | ζ87ζ3+ζ85ζ3+ζ85 | ζ83ζ3+ζ83+ζ8ζ3 | symplectic lifted from Dic12, Schur index 2 |
| ρ16 | 4 | 4 | 4 | 1 | -2 | 1 | -2 | 0 | 0 | -2 | 4 | 1 | -2 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ17 | 4 | 4 | 4 | -2 | 1 | -2 | 1 | 0 | -2 | 0 | 4 | -2 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ18 | 4 | 4 | 4 | 1 | -2 | 1 | -2 | 0 | 0 | 2 | 4 | 1 | -2 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ19 | 4 | 4 | 4 | -2 | 1 | -2 | 1 | 0 | 2 | 0 | 4 | -2 | 1 | -2 | 1 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ20 | 4 | -4 | 4 | -2 | 1 | -2 | 1 | 0 | 0 | 0 | -4 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | -√3 | 0 | 0 | √3 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊Q16, Schur index 2 |
| ρ21 | 4 | -4 | 4 | -2 | 1 | -2 | 1 | 0 | 0 | 0 | -4 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | √3 | 0 | 0 | -√3 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊Q16, Schur index 2 |
| ρ22 | 4 | -4 | 4 | 1 | -2 | 1 | -2 | 0 | 0 | 0 | -4 | -1 | 2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | -√3 | √3 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊Q16, Schur index 2 |
| ρ23 | 4 | -4 | 4 | 1 | -2 | 1 | -2 | 0 | 0 | 0 | -4 | -1 | 2 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | √3 | -√3 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊Q16, Schur index 2 |
| ρ24 | 8 | 8 | -4 | -4 | 2 | 2 | -1 | 0 | 0 | 0 | -4 | -4 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊2D12 |
| ρ25 | 8 | 8 | -4 | 2 | -4 | -1 | 2 | 0 | 0 | 0 | -4 | 2 | -4 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊2D12 |
| ρ26 | 8 | -8 | -4 | 2 | -4 | -1 | 2 | 0 | 0 | 0 | 4 | -2 | 4 | 1 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ27 | 8 | -8 | -4 | -4 | 2 | 2 | -1 | 0 | 0 | 0 | 4 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(1 35 25)(3 27 37)(5 39 29)(7 31 33)(9 43 17)(11 19 45)(13 47 21)(15 23 41)
(2 26 36)(4 38 28)(6 30 40)(8 34 32)(10 18 44)(12 46 20)(14 22 48)(16 42 24)
(1 25 35)(2 26 36)(3 27 37)(4 28 38)(5 29 39)(6 30 40)(7 31 33)(8 32 34)(9 43 17)(10 44 18)(11 45 19)(12 46 20)(13 47 21)(14 48 22)(15 41 23)(16 42 24)
(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 12 5 16)(2 11 6 15)(3 10 7 14)(4 9 8 13)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(33 48 37 44)(34 47 38 43)(35 46 39 42)(36 45 40 41)
G:=sub<Sym(48)| (1,35,25)(3,27,37)(5,39,29)(7,31,33)(9,43,17)(11,19,45)(13,47,21)(15,23,41), (2,26,36)(4,38,28)(6,30,40)(8,34,32)(10,18,44)(12,46,20)(14,22,48)(16,42,24), (1,25,35)(2,26,36)(3,27,37)(4,28,38)(5,29,39)(6,30,40)(7,31,33)(8,32,34)(9,43,17)(10,44,18)(11,45,19)(12,46,20)(13,47,21)(14,48,22)(15,41,23)(16,42,24), (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), (1,12,5,16)(2,11,6,15)(3,10,7,14)(4,9,8,13)(17,32,21,28)(18,31,22,27)(19,30,23,26)(20,29,24,25)(33,48,37,44)(34,47,38,43)(35,46,39,42)(36,45,40,41)>;
G:=Group( (1,35,25)(3,27,37)(5,39,29)(7,31,33)(9,43,17)(11,19,45)(13,47,21)(15,23,41), (2,26,36)(4,38,28)(6,30,40)(8,34,32)(10,18,44)(12,46,20)(14,22,48)(16,42,24), (1,25,35)(2,26,36)(3,27,37)(4,28,38)(5,29,39)(6,30,40)(7,31,33)(8,32,34)(9,43,17)(10,44,18)(11,45,19)(12,46,20)(13,47,21)(14,48,22)(15,41,23)(16,42,24), (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), (1,12,5,16)(2,11,6,15)(3,10,7,14)(4,9,8,13)(17,32,21,28)(18,31,22,27)(19,30,23,26)(20,29,24,25)(33,48,37,44)(34,47,38,43)(35,46,39,42)(36,45,40,41) );
G=PermutationGroup([[(1,35,25),(3,27,37),(5,39,29),(7,31,33),(9,43,17),(11,19,45),(13,47,21),(15,23,41)], [(2,26,36),(4,38,28),(6,30,40),(8,34,32),(10,18,44),(12,46,20),(14,22,48),(16,42,24)], [(1,25,35),(2,26,36),(3,27,37),(4,28,38),(5,29,39),(6,30,40),(7,31,33),(8,32,34),(9,43,17),(10,44,18),(11,45,19),(12,46,20),(13,47,21),(14,48,22),(15,41,23),(16,42,24)], [(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)], [(1,12,5,16),(2,11,6,15),(3,10,7,14),(4,9,8,13),(17,32,21,28),(18,31,22,27),(19,30,23,26),(20,29,24,25),(33,48,37,44),(34,47,38,43),(35,46,39,42),(36,45,40,41)]])
Matrix representation of C33⋊3Q16 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 72 | 1 | 72 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 1 | 72 | 1 | 72 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 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 | 1 |
| 23 | 68 | 0 | 0 | 0 | 0 |
| 5 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 18 | 53 | 0 | 0 | 0 | 0 |
| 71 | 55 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,1,0,72,0,0,72,72,1,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,72,72,1,1,0,0,1,0,72,0],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[23,5,0,0,0,0,68,18,0,0,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,1,0,0],[18,71,0,0,0,0,53,55,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C33⋊3Q16 in GAP, Magma, Sage, TeX
C_3^3\rtimes_3Q_{16} % in TeX
G:=Group("C3^3:3Q16"); // GroupNames label
G:=SmallGroup(432,590);
// by ID
G=gap.SmallGroup(432,590);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,92,254,58,1684,1691,298,677,348,1027,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=b,b*c=c*b,d*b*d^-1=a^-1,e*b*e^-1=a,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations
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