direct product, non-abelian, soluble, monomial
Aliases: C2×A4⋊Q8, C23⋊Dic6, C24.7D6, (C2×A4)⋊Q8, A4⋊1(C2×Q8), C4.22(C2×S4), (C2×C4).16S4, C22⋊(C2×Dic6), C2.3(C22×S4), (C23×C4).6S3, A4⋊C4.1C22, (C2×A4).1C23, C22.22(C2×S4), (C22×C4).16D6, (C4×A4).11C22, C23.1(C22×S3), (C22×A4).8C22, (C2×C4×A4).3C2, (C2×A4⋊C4).3C2, SmallGroup(192,1468)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×A4⋊Q8
G = < a,b,c,d,e,f | a2=b2=c2=d3=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd-1=fbf-1=bc=cb, be=eb, dcd-1=b, ce=ec, cf=fc, de=ed, fdf-1=d-1, fef-1=e-1 >
Subgroups: 534 in 165 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C2×C4, C2×C4, Q8, C23, C23, C23, Dic3, C12, A4, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C2×A4, C2×A4, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, A4⋊C4, C4×A4, C2×Dic6, C22×A4, C2×C22⋊Q8, A4⋊Q8, C2×A4⋊C4, C2×C4×A4, C2×A4⋊Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, Dic6, S4, C22×S3, C2×Dic6, C2×S4, A4⋊Q8, C22×S4, C2×A4⋊Q8
Character table of C2×A4⋊Q8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 8 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| 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 | 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 | 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 | 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 | 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 | 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 | -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 | -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 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | -1 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | -1 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ15 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √3 | -√3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ16 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√3 | √3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√3 | √3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ18 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √3 | -√3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ19 | 3 | 3 | -3 | -3 | 1 | 1 | -1 | -1 | 0 | 3 | -3 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ20 | 3 | 3 | -3 | -3 | 1 | 1 | -1 | -1 | 0 | -3 | 3 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ21 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | -3 | -3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ22 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ23 | 3 | 3 | -3 | -3 | 1 | 1 | -1 | -1 | 0 | 3 | -3 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ24 | 3 | 3 | -3 | -3 | 1 | 1 | -1 | -1 | 0 | -3 | 3 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ25 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | -3 | -3 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ26 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ27 | 6 | -6 | 6 | -6 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from A4⋊Q8, Schur index 2 |
| ρ28 | 6 | -6 | -6 | 6 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from A4⋊Q8, Schur index 2 |
►On 48 points
Generators in S48
(1 35)(2 36)(3 33)(4 34)(5 29)(6 30)(7 31)(8 32)(9 43)(10 44)(11 41)(12 42)(13 25)(14 26)(15 27)(16 28)(17 39)(18 40)(19 37)(20 38)(21 45)(22 46)(23 47)(24 48)
(1 5)(2 6)(3 7)(4 8)(9 23)(10 24)(11 21)(12 22)(13 15)(14 16)(17 19)(18 20)(25 27)(26 28)(29 35)(30 36)(31 33)(32 34)(37 39)(38 40)(41 45)(42 46)(43 47)(44 48)
(1 3)(2 4)(5 7)(6 8)(9 21)(10 22)(11 23)(12 24)(13 40)(14 37)(15 38)(16 39)(17 28)(18 25)(19 26)(20 27)(29 31)(30 32)(33 35)(34 36)(41 47)(42 48)(43 45)(44 46)
(1 23 27)(2 24 28)(3 21 25)(4 22 26)(5 11 18)(6 12 19)(7 9 20)(8 10 17)(13 33 45)(14 34 46)(15 35 47)(16 36 48)(29 41 40)(30 42 37)(31 43 38)(32 44 39)
(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 5 3 7)(2 8 4 6)(9 27 11 25)(10 26 12 28)(13 43 15 41)(14 42 16 44)(17 22 19 24)(18 21 20 23)(29 33 31 35)(30 36 32 34)(37 48 39 46)(38 47 40 45)
G:=sub<Sym(48)| (1,35)(2,36)(3,33)(4,34)(5,29)(6,30)(7,31)(8,32)(9,43)(10,44)(11,41)(12,42)(13,25)(14,26)(15,27)(16,28)(17,39)(18,40)(19,37)(20,38)(21,45)(22,46)(23,47)(24,48), (1,5)(2,6)(3,7)(4,8)(9,23)(10,24)(11,21)(12,22)(13,15)(14,16)(17,19)(18,20)(25,27)(26,28)(29,35)(30,36)(31,33)(32,34)(37,39)(38,40)(41,45)(42,46)(43,47)(44,48), (1,3)(2,4)(5,7)(6,8)(9,21)(10,22)(11,23)(12,24)(13,40)(14,37)(15,38)(16,39)(17,28)(18,25)(19,26)(20,27)(29,31)(30,32)(33,35)(34,36)(41,47)(42,48)(43,45)(44,46), (1,23,27)(2,24,28)(3,21,25)(4,22,26)(5,11,18)(6,12,19)(7,9,20)(8,10,17)(13,33,45)(14,34,46)(15,35,47)(16,36,48)(29,41,40)(30,42,37)(31,43,38)(32,44,39), (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), (1,5,3,7)(2,8,4,6)(9,27,11,25)(10,26,12,28)(13,43,15,41)(14,42,16,44)(17,22,19,24)(18,21,20,23)(29,33,31,35)(30,36,32,34)(37,48,39,46)(38,47,40,45)>;
G:=Group( (1,35)(2,36)(3,33)(4,34)(5,29)(6,30)(7,31)(8,32)(9,43)(10,44)(11,41)(12,42)(13,25)(14,26)(15,27)(16,28)(17,39)(18,40)(19,37)(20,38)(21,45)(22,46)(23,47)(24,48), (1,5)(2,6)(3,7)(4,8)(9,23)(10,24)(11,21)(12,22)(13,15)(14,16)(17,19)(18,20)(25,27)(26,28)(29,35)(30,36)(31,33)(32,34)(37,39)(38,40)(41,45)(42,46)(43,47)(44,48), (1,3)(2,4)(5,7)(6,8)(9,21)(10,22)(11,23)(12,24)(13,40)(14,37)(15,38)(16,39)(17,28)(18,25)(19,26)(20,27)(29,31)(30,32)(33,35)(34,36)(41,47)(42,48)(43,45)(44,46), (1,23,27)(2,24,28)(3,21,25)(4,22,26)(5,11,18)(6,12,19)(7,9,20)(8,10,17)(13,33,45)(14,34,46)(15,35,47)(16,36,48)(29,41,40)(30,42,37)(31,43,38)(32,44,39), (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), (1,5,3,7)(2,8,4,6)(9,27,11,25)(10,26,12,28)(13,43,15,41)(14,42,16,44)(17,22,19,24)(18,21,20,23)(29,33,31,35)(30,36,32,34)(37,48,39,46)(38,47,40,45) );
G=PermutationGroup([[(1,35),(2,36),(3,33),(4,34),(5,29),(6,30),(7,31),(8,32),(9,43),(10,44),(11,41),(12,42),(13,25),(14,26),(15,27),(16,28),(17,39),(18,40),(19,37),(20,38),(21,45),(22,46),(23,47),(24,48)], [(1,5),(2,6),(3,7),(4,8),(9,23),(10,24),(11,21),(12,22),(13,15),(14,16),(17,19),(18,20),(25,27),(26,28),(29,35),(30,36),(31,33),(32,34),(37,39),(38,40),(41,45),(42,46),(43,47),(44,48)], [(1,3),(2,4),(5,7),(6,8),(9,21),(10,22),(11,23),(12,24),(13,40),(14,37),(15,38),(16,39),(17,28),(18,25),(19,26),(20,27),(29,31),(30,32),(33,35),(34,36),(41,47),(42,48),(43,45),(44,46)], [(1,23,27),(2,24,28),(3,21,25),(4,22,26),(5,11,18),(6,12,19),(7,9,20),(8,10,17),(13,33,45),(14,34,46),(15,35,47),(16,36,48),(29,41,40),(30,42,37),(31,43,38),(32,44,39)], [(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)], [(1,5,3,7),(2,8,4,6),(9,27,11,25),(10,26,12,28),(13,43,15,41),(14,42,16,44),(17,22,19,24),(18,21,20,23),(29,33,31,35),(30,36,32,34),(37,48,39,46),(38,47,40,45)]])
Matrix representation of C2×A4⋊Q8 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 12 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 |
| 0 | 0 | 1 | 0 | 12 |
| 0 | 8 | 0 | 0 | 0 |
| 8 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 11 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 1 | 12 |
| 10 | 4 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 8 | 12 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,12,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0,0,0,12,0,0,0,0,0,12],[0,8,0,0,0,8,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,11,12,12],[10,4,0,0,0,4,3,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[8,0,0,0,0,12,5,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0] >;
C2×A4⋊Q8 in GAP, Magma, Sage, TeX
C_2\times A_4\rtimes Q_8
% in TeX
G:=Group("C2xA4:Q8"); // GroupNames label
G:=SmallGroup(192,1468);
// by ID
G=gap.SmallGroup(192,1468);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,56,254,58,1124,4037,285,2358,475]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=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,d*b*d^-1=f*b*f^-1=b*c=c*b,b*e=e*b,d*c*d^-1=b,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f^-1=d^-1,f*e*f^-1=e^-1>;
// generators/relations
Export