direct product, non-abelian, soluble
Aliases: C2×C6.6S4, C6⋊GL2(𝔽3), SL2(𝔽3)⋊3D6, (C6×Q8)⋊1S3, (C3×Q8)⋊2D6, C6.32(C2×S4), (C2×C6).15S4, C22.5(C3⋊S4), C3⋊2(C2×GL2(𝔽3)), (C2×SL2(𝔽3))⋊2S3, (C6×SL2(𝔽3))⋊1C2, (C3×SL2(𝔽3))⋊3C22, Q8⋊(C2×C3⋊S3), C2.6(C2×C3⋊S4), (C2×Q8)⋊1(C3⋊S3), SmallGroup(288,911)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C3×SL2(𝔽3) — C2×C6.6S4 |
| C3×SL2(𝔽3) — C2×C6.6S4 |
Generators and relations for C2×C6.6S4
G = < a,b,c,d,e,f | a2=b6=e3=f2=1, c2=d2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, dcd-1=b3c, ece-1=b3cd, fcf=cd, ede-1=c, fdf=b3d, fef=e-1 >
Subgroups: 1016 in 144 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C8, C2×C4, D4, Q8, Q8, C23, C32, C12, D6, C2×C6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3⋊S3, C3×C6, C3⋊C8, SL2(𝔽3), D12, C2×C12, C3×Q8, C3×Q8, C22×S3, C2×SD16, C2×C3⋊S3, C62, C2×C3⋊C8, Q8⋊2S3, GL2(𝔽3), C2×SL2(𝔽3), C2×D12, C6×Q8, C3×SL2(𝔽3), C22×C3⋊S3, C2×Q8⋊2S3, C2×GL2(𝔽3), C6.6S4, C6×SL2(𝔽3), C2×C6.6S4
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S4, C2×C3⋊S3, GL2(𝔽3), C2×S4, C3⋊S4, C2×GL2(𝔽3), C6.6S4, C2×C3⋊S4, C2×C6.6S4
Character table of C2×C6.6S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 8A | 8B | 8C | 8D | 12A | 12B | |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 2 | 8 | 8 | 8 | 6 | 6 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 18 | 18 | 18 | 18 | 12 | 12 | |
| 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 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | -2 | 1 | -1 | 1 | -1 | 1 | 1 | -2 | -2 | 1 | 1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | -2 | 1 | -1 | 1 | 2 | 1 | 1 | 1 | 1 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | -2 | -2 | 2 | -2 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 2 | orthogonal lifted from D6 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | -2 | 1 | -1 | 1 | -1 | -2 | -2 | 1 | 1 | 1 | 1 | -1 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | 2 | orthogonal lifted from S3 |
| ρ13 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | -2 | -2 | 2 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -√-2 | -√-2 | √-2 | √-2 | 0 | 0 | complex lifted from GL2(𝔽3) |
| ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | -2 | -2 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | √-2 | -√-2 | √-2 | -√-2 | 0 | 0 | complex lifted from GL2(𝔽3) |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | -2 | -2 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -√-2 | √-2 | -√-2 | √-2 | 0 | 0 | complex lifted from GL2(𝔽3) |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | -2 | -2 | 2 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | √-2 | √-2 | -√-2 | -√-2 | 0 | 0 | complex lifted from GL2(𝔽3) |
| ρ17 | 3 | 3 | -3 | -3 | -1 | 1 | 3 | 0 | 0 | 0 | -1 | 1 | -3 | 3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | 1 | -1 | orthogonal lifted from C2×S4 |
| ρ18 | 3 | 3 | 3 | 3 | -1 | -1 | 3 | 0 | 0 | 0 | -1 | -1 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from S4 |
| ρ19 | 3 | 3 | 3 | 3 | 1 | 1 | 3 | 0 | 0 | 0 | -1 | -1 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ20 | 3 | 3 | -3 | -3 | 1 | -1 | 3 | 0 | 0 | 0 | -1 | 1 | -3 | 3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | -1 | orthogonal lifted from C2×S4 |
| ρ21 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | -2 | 1 | 1 | 0 | 0 | 2 | 2 | -2 | 2 | -1 | 1 | -1 | 1 | 2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C6.6S4 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 1 | 1 | -2 | 0 | 0 | -2 | 2 | 2 | -1 | 1 | -1 | -2 | 2 | 1 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C6.6S4 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 1 | -2 | 1 | 0 | 0 | 2 | 2 | -2 | -1 | 2 | -2 | -1 | 1 | -1 | 1 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C6.6S4 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 4 | 1 | 1 | 1 | 0 | 0 | 4 | -4 | -4 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from GL2(𝔽3) |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 4 | 1 | 1 | 1 | 0 | 0 | -4 | -4 | 4 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from GL2(𝔽3) |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 1 | 0 | 0 | -2 | 2 | 2 | 2 | 1 | -1 | 1 | -1 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C6.6S4 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 1 | -2 | 1 | 0 | 0 | -2 | 2 | 2 | -1 | -2 | 2 | 1 | -1 | 1 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C6.6S4 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 1 | 1 | -2 | 0 | 0 | 2 | 2 | -2 | -1 | -1 | 1 | 2 | -2 | -1 | 1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C6.6S4 |
| ρ29 | 6 | 6 | -6 | -6 | 0 | 0 | -3 | 0 | 0 | 0 | -2 | 2 | 3 | -3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | orthogonal lifted from C2×C3⋊S4 |
| ρ30 | 6 | 6 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | 0 | -2 | -2 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from C3⋊S4 |
►On 48 points
Generators in S48
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)
(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 26 4 29)(2 27 5 30)(3 28 6 25)(7 43 10 46)(8 44 11 47)(9 45 12 48)(13 32 16 35)(14 33 17 36)(15 34 18 31)(19 40 22 37)(20 41 23 38)(21 42 24 39)
(1 41 4 38)(2 42 5 39)(3 37 6 40)(7 17 10 14)(8 18 11 15)(9 13 12 16)(19 28 22 25)(20 29 23 26)(21 30 24 27)(31 44 34 47)(32 45 35 48)(33 46 36 43)
(1 5 3)(2 6 4)(7 47 35)(8 48 36)(9 43 31)(10 44 32)(11 45 33)(12 46 34)(13 17 15)(14 18 16)(19 26 39)(20 27 40)(21 28 41)(22 29 42)(23 30 37)(24 25 38)
(1 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(19 32)(20 31)(21 36)(22 35)(23 34)(24 33)(25 45)(26 44)(27 43)(28 48)(29 47)(30 46)
G:=sub<Sym(48)| (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,36)(26,31)(27,32)(28,33)(29,34)(30,35), (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), (1,26,4,29)(2,27,5,30)(3,28,6,25)(7,43,10,46)(8,44,11,47)(9,45,12,48)(13,32,16,35)(14,33,17,36)(15,34,18,31)(19,40,22,37)(20,41,23,38)(21,42,24,39), (1,41,4,38)(2,42,5,39)(3,37,6,40)(7,17,10,14)(8,18,11,15)(9,13,12,16)(19,28,22,25)(20,29,23,26)(21,30,24,27)(31,44,34,47)(32,45,35,48)(33,46,36,43), (1,5,3)(2,6,4)(7,47,35)(8,48,36)(9,43,31)(10,44,32)(11,45,33)(12,46,34)(13,17,15)(14,18,16)(19,26,39)(20,27,40)(21,28,41)(22,29,42)(23,30,37)(24,25,38), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(19,32)(20,31)(21,36)(22,35)(23,34)(24,33)(25,45)(26,44)(27,43)(28,48)(29,47)(30,46)>;
G:=Group( (1,18)(2,13)(3,14)(4,15)(5,16)(6,17)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,36)(26,31)(27,32)(28,33)(29,34)(30,35), (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), (1,26,4,29)(2,27,5,30)(3,28,6,25)(7,43,10,46)(8,44,11,47)(9,45,12,48)(13,32,16,35)(14,33,17,36)(15,34,18,31)(19,40,22,37)(20,41,23,38)(21,42,24,39), (1,41,4,38)(2,42,5,39)(3,37,6,40)(7,17,10,14)(8,18,11,15)(9,13,12,16)(19,28,22,25)(20,29,23,26)(21,30,24,27)(31,44,34,47)(32,45,35,48)(33,46,36,43), (1,5,3)(2,6,4)(7,47,35)(8,48,36)(9,43,31)(10,44,32)(11,45,33)(12,46,34)(13,17,15)(14,18,16)(19,26,39)(20,27,40)(21,28,41)(22,29,42)(23,30,37)(24,25,38), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(19,32)(20,31)(21,36)(22,35)(23,34)(24,33)(25,45)(26,44)(27,43)(28,48)(29,47)(30,46) );
G=PermutationGroup([[(1,18),(2,13),(3,14),(4,15),(5,16),(6,17),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,36),(26,31),(27,32),(28,33),(29,34),(30,35)], [(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)], [(1,26,4,29),(2,27,5,30),(3,28,6,25),(7,43,10,46),(8,44,11,47),(9,45,12,48),(13,32,16,35),(14,33,17,36),(15,34,18,31),(19,40,22,37),(20,41,23,38),(21,42,24,39)], [(1,41,4,38),(2,42,5,39),(3,37,6,40),(7,17,10,14),(8,18,11,15),(9,13,12,16),(19,28,22,25),(20,29,23,26),(21,30,24,27),(31,44,34,47),(32,45,35,48),(33,46,36,43)], [(1,5,3),(2,6,4),(7,47,35),(8,48,36),(9,43,31),(10,44,32),(11,45,33),(12,46,34),(13,17,15),(14,18,16),(19,26,39),(20,27,40),(21,28,41),(22,29,42),(23,30,37),(24,25,38)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(19,32),(20,31),(21,36),(22,35),(23,34),(24,33),(25,45),(26,44),(27,43),(28,48),(29,47),(30,46)]])
Matrix representation of C2×C6.6S4 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 69 | 65 | 0 | 0 |
| 29 | 3 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 53 | 40 |
| 0 | 0 | 21 | 20 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 53 | 52 |
| 0 | 0 | 33 | 20 |
| 69 | 65 | 0 | 0 |
| 29 | 3 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 |
| 10 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[69,29,0,0,65,3,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,53,21,0,0,40,20],[1,0,0,0,0,1,0,0,0,0,53,33,0,0,52,20],[69,29,0,0,65,3,0,0,0,0,72,72,0,0,1,0],[72,10,0,0,0,1,0,0,0,0,72,0,0,0,1,1] >;
C2×C6.6S4 in GAP, Magma, Sage, TeX
C_2\times C_6._6S_4
% in TeX
G:=Group("C2xC6.6S4"); // GroupNames label
G:=SmallGroup(288,911);
// by ID
G=gap.SmallGroup(288,911);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,170,675,2524,1908,172,1517,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^6=e^3=f^2=1,c^2=d^2=b^3,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,b*e=e*b,f*b*f=b^-1,d*c*d^-1=b^3*c,e*c*e^-1=b^3*c*d,f*c*f=c*d,e*d*e^-1=c,f*d*f=b^3*d,f*e*f=e^-1>;
// generators/relations
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