direct product, non-abelian, soluble, monomial
Aliases: C2×C33⋊Q8, C6⋊PSU3(𝔽2), C33⋊3(C2×Q8), C3⋊S3⋊2Dic6, (C3×C6)⋊2Dic6, C32⋊C4.7D6, (C32×C6)⋊2Q8, C32⋊3(C2×Dic6), C3⋊2(C2×PSU3(𝔽2)), C33⋊C4.2C22, (C3×C3⋊S3)⋊3Q8, (C2×C3⋊S3).23D6, (C2×C32⋊C4).6S3, (C6×C32⋊C4).7C2, (C3×C3⋊S3).8C23, C3⋊S3.5(C22×S3), (C6×C3⋊S3).37C22, (C2×C33⋊C4).7C2, (C3×C32⋊C4).8C22, SmallGroup(432,758)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C33⋊Q8
G = < a,b,c,d,e,f | a2=b3=c3=d3=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=c, fbf-1=bc-1, cd=dc, ece-1=b-1, fcf-1=b-1c-1, de=ed, fdf-1=d-1, fef-1=e-1 >
Subgroups: 688 in 90 conjugacy classes, 31 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, Q8, C32, C32, Dic3, C12, D6, C2×C6, C2×Q8, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C33, C32⋊C4, C32⋊C4, S3×C6, C2×C3⋊S3, C2×Dic6, C3×C3⋊S3, C32×C6, PSU3(𝔽2), C2×C32⋊C4, C2×C32⋊C4, C3×C32⋊C4, C33⋊C4, C6×C3⋊S3, C2×PSU3(𝔽2), C33⋊Q8, C6×C32⋊C4, C2×C33⋊C4, C2×C33⋊Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, Dic6, C22×S3, C2×Dic6, PSU3(𝔽2), C2×PSU3(𝔽2), C33⋊Q8, C2×C33⋊Q8
Character table of C2×C33⋊Q8
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 9 | 9 | 2 | 8 | 8 | 8 | 18 | 18 | 54 | 54 | 54 | 54 | 2 | 8 | 8 | 8 | 18 | 18 | 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 | 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 | 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 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | -2 | -2 | 2 | -1 | -1 | 2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | -2 | -2 | 2 | -1 | -1 | 2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ13 | 2 | -2 | 2 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | 2 | -2 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ15 | 2 | 2 | -2 | -2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 1 | 1 | -√3 | -√3 | √3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ16 | 2 | 2 | -2 | -2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 1 | 1 | √3 | √3 | -√3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ17 | 2 | -2 | 2 | -2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | -1 | 1 | -√3 | √3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ18 | 2 | -2 | 2 | -2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | -1 | 1 | √3 | -√3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ19 | 8 | -8 | 0 | 0 | 8 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -8 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×PSU3(𝔽2) |
| ρ20 | 8 | 8 | 0 | 0 | 8 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 8 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from PSU3(𝔽2) |
| ρ21 | 8 | 8 | 0 | 0 | -4 | 1-3√-3/2 | -1 | 1+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | -1 | 1-3√-3/2 | 1+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊Q8 |
| ρ22 | 8 | -8 | 0 | 0 | -4 | 1-3√-3/2 | -1 | 1+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 1 | -1+3√-3/2 | -1-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 8 | -8 | 0 | 0 | -4 | 1+3√-3/2 | -1 | 1-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 1 | -1-3√-3/2 | -1+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 8 | 8 | 0 | 0 | -4 | 1+3√-3/2 | -1 | 1-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | -1 | 1+3√-3/2 | 1-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C33⋊Q8 |
►On 48 points
Generators in S48
(1 11)(2 12)(3 9)(4 10)(5 24)(6 21)(7 22)(8 23)(13 41)(14 42)(15 43)(16 44)(17 26)(18 27)(19 28)(20 25)(29 45)(30 46)(31 47)(32 48)(33 37)(34 38)(35 39)(36 40)
(1 19 6)(3 8 17)(9 23 26)(11 28 21)(13 38 47)(14 48 39)(15 45 40)(16 37 46)(29 36 43)(30 44 33)(31 41 34)(32 35 42)
(2 7 20)(4 18 5)(10 27 24)(12 22 25)(13 47 38)(14 48 39)(15 40 45)(16 37 46)(29 43 36)(30 44 33)(31 34 41)(32 35 42)
(1 6 19)(2 7 20)(3 8 17)(4 5 18)(9 23 26)(10 24 27)(11 21 28)(12 22 25)(13 47 38)(14 48 39)(15 45 40)(16 46 37)(29 36 43)(30 33 44)(31 34 41)(32 35 42)
(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 40 3 38)(2 39 4 37)(5 46 7 48)(6 45 8 47)(9 34 11 36)(10 33 12 35)(13 19 15 17)(14 18 16 20)(21 29 23 31)(22 32 24 30)(25 42 27 44)(26 41 28 43)
G:=sub<Sym(48)| (1,11)(2,12)(3,9)(4,10)(5,24)(6,21)(7,22)(8,23)(13,41)(14,42)(15,43)(16,44)(17,26)(18,27)(19,28)(20,25)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40), (1,19,6)(3,8,17)(9,23,26)(11,28,21)(13,38,47)(14,48,39)(15,45,40)(16,37,46)(29,36,43)(30,44,33)(31,41,34)(32,35,42), (2,7,20)(4,18,5)(10,27,24)(12,22,25)(13,47,38)(14,48,39)(15,40,45)(16,37,46)(29,43,36)(30,44,33)(31,34,41)(32,35,42), (1,6,19)(2,7,20)(3,8,17)(4,5,18)(9,23,26)(10,24,27)(11,21,28)(12,22,25)(13,47,38)(14,48,39)(15,45,40)(16,46,37)(29,36,43)(30,33,44)(31,34,41)(32,35,42), (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), (1,40,3,38)(2,39,4,37)(5,46,7,48)(6,45,8,47)(9,34,11,36)(10,33,12,35)(13,19,15,17)(14,18,16,20)(21,29,23,31)(22,32,24,30)(25,42,27,44)(26,41,28,43)>;
G:=Group( (1,11)(2,12)(3,9)(4,10)(5,24)(6,21)(7,22)(8,23)(13,41)(14,42)(15,43)(16,44)(17,26)(18,27)(19,28)(20,25)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40), (1,19,6)(3,8,17)(9,23,26)(11,28,21)(13,38,47)(14,48,39)(15,45,40)(16,37,46)(29,36,43)(30,44,33)(31,41,34)(32,35,42), (2,7,20)(4,18,5)(10,27,24)(12,22,25)(13,47,38)(14,48,39)(15,40,45)(16,37,46)(29,43,36)(30,44,33)(31,34,41)(32,35,42), (1,6,19)(2,7,20)(3,8,17)(4,5,18)(9,23,26)(10,24,27)(11,21,28)(12,22,25)(13,47,38)(14,48,39)(15,45,40)(16,46,37)(29,36,43)(30,33,44)(31,34,41)(32,35,42), (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), (1,40,3,38)(2,39,4,37)(5,46,7,48)(6,45,8,47)(9,34,11,36)(10,33,12,35)(13,19,15,17)(14,18,16,20)(21,29,23,31)(22,32,24,30)(25,42,27,44)(26,41,28,43) );
G=PermutationGroup([[(1,11),(2,12),(3,9),(4,10),(5,24),(6,21),(7,22),(8,23),(13,41),(14,42),(15,43),(16,44),(17,26),(18,27),(19,28),(20,25),(29,45),(30,46),(31,47),(32,48),(33,37),(34,38),(35,39),(36,40)], [(1,19,6),(3,8,17),(9,23,26),(11,28,21),(13,38,47),(14,48,39),(15,45,40),(16,37,46),(29,36,43),(30,44,33),(31,41,34),(32,35,42)], [(2,7,20),(4,18,5),(10,27,24),(12,22,25),(13,47,38),(14,48,39),(15,40,45),(16,37,46),(29,43,36),(30,44,33),(31,34,41),(32,35,42)], [(1,6,19),(2,7,20),(3,8,17),(4,5,18),(9,23,26),(10,24,27),(11,21,28),(12,22,25),(13,47,38),(14,48,39),(15,45,40),(16,46,37),(29,36,43),(30,33,44),(31,34,41),(32,35,42)], [(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)], [(1,40,3,38),(2,39,4,37),(5,46,7,48),(6,45,8,47),(9,34,11,36),(10,33,12,35),(13,19,15,17),(14,18,16,20),(21,29,23,31),(22,32,24,30),(25,42,27,44),(26,41,28,43)]])
Matrix representation of C2×C33⋊Q8 ►in GL8(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 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 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 |
| 12 | 0 | 3 | 3 | 0 | 9 | 4 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 |
| 9 | 9 | 4 | 0 | 9 | 0 | 4 | 3 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 |
| 1 | 1 | 9 | 9 | 0 | 0 | 0 | 9 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 10 | 12 | 12 | 1 | 1 | 12 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 7 | 7 | 7 | 7 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 3 | 3 | 1 | 1 | 12 | 12 | 1 | 5 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 7 | 7 | 0 | 0 | 6 | 6 | 0 | 12 |
G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[9,0,0,0,0,0,0,12,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,3,0,0,0,1,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,9,0,9,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,9,0,1,0,0,0,0,0,9,0,0,9,0,0,0,0,4,0,0,0,3,0,0,0,0,0,0,0,0,9,0,0,9,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9,4,0,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,1,0,3,0,0,0,0,0,1,0,0,3,0,0,0,0,9,0,0,0,3,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9],[0,0,0,1,10,0,0,7,0,0,1,0,10,0,0,7,1,0,0,0,12,0,0,7,0,1,0,0,12,0,0,7,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,8,0,0,12],[0,0,0,3,0,12,0,7,0,0,0,3,12,0,0,7,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,0,12,0,0,12,0,0,0,6,0,12,0,12,0,0,0,6,0,0,12,1,0,0,0,0,0,0,0,5,0,0,0,12] >;
C2×C33⋊Q8 in GAP, Magma, Sage, TeX
C_2\times C_3^3\rtimes Q_8
% in TeX
G:=Group("C2xC3^3:Q8"); // GroupNames label
G:=SmallGroup(432,758);
// by ID
G=gap.SmallGroup(432,758);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,56,141,64,2804,1691,165,2693,348,530,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^3=d^3=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=c,f*b*f^-1=b*c^-1,c*d=d*c,e*c*e^-1=b^-1,f*c*f^-1=b^-1*c^-1,d*e=e*d,f*d*f^-1=d^-1,f*e*f^-1=e^-1>;
// generators/relations
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