non-abelian, soluble, monomial
Aliases: C33⋊2Q16, C6.18S3≀C2, C33⋊4C8.C2, C32⋊2Q8.S3, C3⋊2(C32⋊Q16), C3⋊Dic3.13D6, (C32×C6).12D4, C33⋊5Q8.2C2, C32⋊3(C3⋊Q16), C2.7(C33⋊D4), (C3×C6).18(C3⋊D4), (C3×C32⋊2Q8).1C2, (C3×C3⋊Dic3).10C22, SmallGroup(432,585)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32 — C3×C3⋊Dic3 — C33⋊Q16 |
| C33 — C32×C6 — C3×C3⋊Dic3 — C33⋊Q16 |
Generators and relations for C33⋊Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, dad-1=b-1, eae-1=a-1, bc=cb, dbd-1=a, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >
Subgroups: 412 in 72 conjugacy classes, 15 normal (all characteristic)
C1, C2, C3, C3, C4, C6, C6, C8, Q8, C32, C32, Dic3, C12, Q16, C3×C6, C3×C6, C3⋊C8, Dic6, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3⋊Q16, C32×C6, C32⋊2C8, C32⋊2Q8, C32⋊2Q8, C3×Dic6, C32×Dic3, C3×C3⋊Dic3, C3×C3⋊Dic3, C32⋊Q16, C33⋊4C8, C3×C32⋊2Q8, C33⋊5Q8, C33⋊Q16
Quotients: C1, C2, C22, S3, D4, D6, Q16, C3⋊D4, C3⋊Q16, S3≀C2, C32⋊Q16, C33⋊D4, C33⋊Q16
Character table of C33⋊Q16
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 8 | 12 | 18 | 36 | 2 | 4 | 4 | 4 | 4 | 8 | 54 | 54 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 36 | 36 | 36 | |
| 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 | 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 | -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 | -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 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | -1 | 2 | -1 | 2 | -1 | -1 | 2 | 2 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 0 | -1 | 0 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | -1 | 2 | -1 | 2 | -1 | -1 | -2 | 2 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | -2 | 1 | 1 | -2 | 1 | 1 | 1 | 1 | 0 | -1 | 0 | orthogonal lifted from D6 |
| ρ8 | 2 | -2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | √2 | -√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ9 | 2 | -2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | -√2 | √2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q16, Schur index 2 |
| ρ10 | 2 | 2 | -1 | 2 | -1 | 2 | -1 | -1 | 0 | -2 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | -√-3 | √-3 | 0 | -√-3 | √-3 | -√-3 | √-3 | 0 | 1 | 0 | complex lifted from C3⋊D4 |
| ρ11 | 2 | 2 | -1 | 2 | -1 | 2 | -1 | -1 | 0 | -2 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | √-3 | -√-3 | 0 | √-3 | -√-3 | √-3 | -√-3 | 0 | 1 | 0 | complex lifted from C3⋊D4 |
| ρ12 | 4 | 4 | 4 | -2 | 1 | 1 | 1 | -2 | -2 | 0 | 0 | 4 | 1 | -2 | 1 | 1 | -2 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | -2 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ13 | 4 | 4 | 4 | -2 | 1 | 1 | 1 | -2 | 2 | 0 | 0 | 4 | 1 | -2 | 1 | 1 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | 0 | orthogonal lifted from S3≀C2 |
| ρ14 | 4 | 4 | 4 | 1 | -2 | -2 | -2 | 1 | 0 | 0 | 2 | 4 | -2 | 1 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | -1 | orthogonal lifted from S3≀C2 |
| ρ15 | 4 | 4 | 4 | 1 | -2 | -2 | -2 | 1 | 0 | 0 | -2 | 4 | -2 | 1 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | orthogonal lifted from S3≀C2 |
| ρ16 | 4 | -4 | -2 | 4 | -2 | 4 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | -4 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C3⋊Q16, Schur index 2 |
| ρ17 | 4 | -4 | 4 | -2 | 1 | 1 | 1 | -2 | 0 | 0 | 0 | -4 | -1 | 2 | -1 | -1 | 2 | 0 | 0 | √3 | √3 | √3 | -√3 | -√3 | -√3 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊Q16, Schur index 2 |
| ρ18 | 4 | -4 | 4 | -2 | 1 | 1 | 1 | -2 | 0 | 0 | 0 | -4 | -1 | 2 | -1 | -1 | 2 | 0 | 0 | -√3 | -√3 | -√3 | √3 | √3 | √3 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊Q16, Schur index 2 |
| ρ19 | 4 | -4 | 4 | 1 | -2 | -2 | -2 | 1 | 0 | 0 | 0 | -4 | 2 | -1 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√3 | 0 | √3 | symplectic lifted from C32⋊Q16, Schur index 2 |
| ρ20 | 4 | -4 | 4 | 1 | -2 | -2 | -2 | 1 | 0 | 0 | 0 | -4 | 2 | -1 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √3 | 0 | -√3 | symplectic lifted from C32⋊Q16, Schur index 2 |
| ρ21 | 4 | 4 | -2 | -2 | -1-3√-3/2 | 1 | -1+3√-3/2 | 1 | -2 | 0 | 0 | -2 | -1-3√-3/2 | -2 | 1 | -1+3√-3/2 | 1 | 0 | 0 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | complex lifted from C33⋊D4 |
| ρ22 | 4 | 4 | -2 | -2 | -1+3√-3/2 | 1 | -1-3√-3/2 | 1 | -2 | 0 | 0 | -2 | -1+3√-3/2 | -2 | 1 | -1-3√-3/2 | 1 | 0 | 0 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | complex lifted from C33⋊D4 |
| ρ23 | 4 | -4 | -2 | -2 | -1+3√-3/2 | 1 | -1-3√-3/2 | 1 | 0 | 0 | 0 | 2 | 1-3√-3/2 | 2 | -1 | 1+3√-3/2 | -1 | 0 | 0 | -√3 | ζ4ζ32+2ζ4 | ζ43ζ3+2ζ43 | √3 | ζ43ζ32+2ζ43 | ζ4ζ3+2ζ4 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | 4 | -2 | -2 | -1-3√-3/2 | 1 | -1+3√-3/2 | 1 | 2 | 0 | 0 | -2 | -1-3√-3/2 | -2 | 1 | -1+3√-3/2 | 1 | 0 | 0 | -1 | ζ65 | ζ6 | -1 | ζ65 | ζ6 | -1+√-3 | -1-√-3 | 0 | 0 | 0 | complex lifted from C33⋊D4 |
| ρ25 | 4 | -4 | -2 | -2 | -1-3√-3/2 | 1 | -1+3√-3/2 | 1 | 0 | 0 | 0 | 2 | 1+3√-3/2 | 2 | -1 | 1-3√-3/2 | -1 | 0 | 0 | -√3 | ζ43ζ3+2ζ43 | ζ4ζ32+2ζ4 | √3 | ζ4ζ3+2ζ4 | ζ43ζ32+2ζ43 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ26 | 4 | 4 | -2 | -2 | -1+3√-3/2 | 1 | -1-3√-3/2 | 1 | 2 | 0 | 0 | -2 | -1+3√-3/2 | -2 | 1 | -1-3√-3/2 | 1 | 0 | 0 | -1 | ζ6 | ζ65 | -1 | ζ6 | ζ65 | -1-√-3 | -1+√-3 | 0 | 0 | 0 | complex lifted from C33⋊D4 |
| ρ27 | 4 | -4 | -2 | -2 | -1+3√-3/2 | 1 | -1-3√-3/2 | 1 | 0 | 0 | 0 | 2 | 1-3√-3/2 | 2 | -1 | 1+3√-3/2 | -1 | 0 | 0 | √3 | ζ43ζ32+2ζ43 | ζ4ζ3+2ζ4 | -√3 | ζ4ζ32+2ζ4 | ζ43ζ3+2ζ43 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ28 | 4 | -4 | -2 | -2 | -1-3√-3/2 | 1 | -1+3√-3/2 | 1 | 0 | 0 | 0 | 2 | 1+3√-3/2 | 2 | -1 | 1-3√-3/2 | -1 | 0 | 0 | √3 | ζ4ζ3+2ζ4 | ζ43ζ32+2ζ43 | -√3 | ζ43ζ3+2ζ43 | ζ4ζ32+2ζ4 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ29 | 8 | 8 | -4 | 2 | 2 | -4 | 2 | -1 | 0 | 0 | 0 | -4 | 2 | 2 | -4 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C33⋊D4 |
| ρ30 | 8 | -8 | -4 | 2 | 2 | -4 | 2 | -1 | 0 | 0 | 0 | 4 | -2 | -2 | 4 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(1 31 46)(2 47 32)(3 48 25)(4 26 41)(5 27 42)(6 43 28)(7 44 29)(8 30 45)(9 33 24)(10 34 17)(11 18 35)(12 19 36)(13 37 20)(14 38 21)(15 22 39)(16 23 40)
(1 31 46)(2 32 47)(3 48 25)(4 41 26)(5 27 42)(6 28 43)(7 44 29)(8 45 30)(9 24 33)(10 34 17)(11 35 18)(12 19 36)(13 20 37)(14 38 21)(15 39 22)(16 23 40)
(1 46 31)(2 32 47)(3 48 25)(4 26 41)(5 42 27)(6 28 43)(7 44 29)(8 30 45)(9 24 33)(10 34 17)(11 18 35)(12 36 19)(13 20 37)(14 38 21)(15 22 39)(16 40 23)
(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 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 47 21 43)(18 46 22 42)(19 45 23 41)(20 44 24 48)(25 37 29 33)(26 36 30 40)(27 35 31 39)(28 34 32 38)
G:=sub<Sym(48)| (1,31,46)(2,47,32)(3,48,25)(4,26,41)(5,27,42)(6,43,28)(7,44,29)(8,30,45)(9,33,24)(10,34,17)(11,18,35)(12,19,36)(13,37,20)(14,38,21)(15,22,39)(16,23,40), (1,31,46)(2,32,47)(3,48,25)(4,41,26)(5,27,42)(6,28,43)(7,44,29)(8,45,30)(9,24,33)(10,34,17)(11,35,18)(12,19,36)(13,20,37)(14,38,21)(15,39,22)(16,23,40), (1,46,31)(2,32,47)(3,48,25)(4,26,41)(5,42,27)(6,28,43)(7,44,29)(8,30,45)(9,24,33)(10,34,17)(11,18,35)(12,36,19)(13,20,37)(14,38,21)(15,22,39)(16,40,23), (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), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,47,21,43)(18,46,22,42)(19,45,23,41)(20,44,24,48)(25,37,29,33)(26,36,30,40)(27,35,31,39)(28,34,32,38)>;
G:=Group( (1,31,46)(2,47,32)(3,48,25)(4,26,41)(5,27,42)(6,43,28)(7,44,29)(8,30,45)(9,33,24)(10,34,17)(11,18,35)(12,19,36)(13,37,20)(14,38,21)(15,22,39)(16,23,40), (1,31,46)(2,32,47)(3,48,25)(4,41,26)(5,27,42)(6,28,43)(7,44,29)(8,45,30)(9,24,33)(10,34,17)(11,35,18)(12,19,36)(13,20,37)(14,38,21)(15,39,22)(16,23,40), (1,46,31)(2,32,47)(3,48,25)(4,26,41)(5,42,27)(6,28,43)(7,44,29)(8,30,45)(9,24,33)(10,34,17)(11,18,35)(12,36,19)(13,20,37)(14,38,21)(15,22,39)(16,40,23), (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), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,47,21,43)(18,46,22,42)(19,45,23,41)(20,44,24,48)(25,37,29,33)(26,36,30,40)(27,35,31,39)(28,34,32,38) );
G=PermutationGroup([[(1,31,46),(2,47,32),(3,48,25),(4,26,41),(5,27,42),(6,43,28),(7,44,29),(8,30,45),(9,33,24),(10,34,17),(11,18,35),(12,19,36),(13,37,20),(14,38,21),(15,22,39),(16,23,40)], [(1,31,46),(2,32,47),(3,48,25),(4,41,26),(5,27,42),(6,28,43),(7,44,29),(8,45,30),(9,24,33),(10,34,17),(11,35,18),(12,19,36),(13,20,37),(14,38,21),(15,39,22),(16,23,40)], [(1,46,31),(2,32,47),(3,48,25),(4,26,41),(5,42,27),(6,28,43),(7,44,29),(8,30,45),(9,24,33),(10,34,17),(11,18,35),(12,36,19),(13,20,37),(14,38,21),(15,22,39),(16,40,23)], [(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)], [(1,15,5,11),(2,14,6,10),(3,13,7,9),(4,12,8,16),(17,47,21,43),(18,46,22,42),(19,45,23,41),(20,44,24,48),(25,37,29,33),(26,36,30,40),(27,35,31,39),(28,34,32,38)]])
Matrix representation of C33⋊Q16 ►in GL8(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 72 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 0 | 72 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 72 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 41 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 41 | 0 | 0 | 0 | 0 |
| 16 | 0 | 41 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 41 | 0 | 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 41 | 0 | 13 | 0 | 0 | 0 | 0 |
| 11 | 0 | 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,72,0,0,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,72],[0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,41,0,41,0,0,0,0,41,0,41,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,72,0,0],[41,0,11,0,0,0,0,0,0,41,0,11,0,0,0,0,13,0,32,0,0,0,0,0,0,13,0,32,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C33⋊Q16 in GAP, Magma, Sage, TeX
C_3^3\rtimes Q_{16} % in TeX
G:=Group("C3^3:Q16"); // GroupNames label
G:=SmallGroup(432,585);
// by ID
G=gap.SmallGroup(432,585);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,85,64,254,135,58,1684,571,298,677,1027,14118]);
// 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=b^-1,e*a*e^-1=a^-1,b*c=c*b,d*b*d^-1=a,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
Export