direct product, metabelian, soluble, monomial, A-group
Aliases: C2×C3⋊F9, C6⋊F9, C3⋊2(C2×F9), C33⋊3(C2×C8), (C32×C6)⋊2C8, C32⋊C4.6D6, C32⋊C4.Dic3, (C3×C6)⋊(C3⋊C8), C3⋊S3⋊2(C3⋊C8), (C3×C3⋊S3)⋊2C8, C32⋊2(C2×C3⋊C8), (C6×C3⋊S3).2C4, (C3×C32⋊C4).2C4, (C6×C32⋊C4).9C2, (C2×C32⋊C4).5S3, C3⋊S3.1(C2×Dic3), (C2×C3⋊S3).3Dic3, (C3×C32⋊C4).9C22, (C3×C3⋊S3).3(C2×C4), SmallGroup(432,752)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C2×C3⋊F9 |
Generators and relations for C2×C3⋊F9
G = < a,b,c,d,e | a2=b3=c3=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, ece-1=cd=dc, ede-1=c >
Subgroups: 368 in 58 conjugacy classes, 23 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, C2×C4, C32, C32, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C2×C12, C33, C32⋊C4, S3×C6, C2×C3⋊S3, C2×C3⋊C8, C3×C3⋊S3, C32×C6, F9, C2×C32⋊C4, C3×C32⋊C4, C6×C3⋊S3, C2×F9, C3⋊F9, C6×C32⋊C4, C2×C3⋊F9
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, C3⋊C8, C2×Dic3, C2×C3⋊C8, F9, C2×F9, C3⋊F9, C2×C3⋊F9
Character table of C2×C3⋊F9
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 9 | 9 | 2 | 8 | 8 | 8 | 9 | 9 | 9 | 9 | 2 | 8 | 8 | 8 | 18 | 18 | 27 | 27 | 27 | 27 | 27 | 27 | 27 | 27 | 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 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -i | i | i | -i | -i | i | i | -i | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | i | i | i | -i | -i | -i | -i | i | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | i | -i | -i | i | i | -i | -i | i | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -i | -i | -i | i | i | i | i | -i | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ9 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | -i | i | i | 1 | 1 | 1 | 1 | -1 | -1 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | i | -i | i | -i | linear of order 8 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | i | i | -i | -i | 1 | 1 | 1 | 1 | -1 | -1 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | -i | i | -i | i | linear of order 8 |
| ρ11 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | -1 | -1 | -1 | 1 | ζ85 | ζ87 | ζ83 | ζ85 | ζ8 | ζ83 | ζ87 | ζ8 | -i | -i | i | i | linear of order 8 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | -1 | -1 | 1 | ζ83 | ζ8 | ζ85 | ζ83 | ζ87 | ζ85 | ζ8 | ζ87 | i | i | -i | -i | linear of order 8 |
| ρ13 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | i | i | -i | -i | 1 | 1 | 1 | 1 | -1 | -1 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | -i | i | -i | i | linear of order 8 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | -1 | -1 | 1 | ζ87 | ζ85 | ζ8 | ζ87 | ζ83 | ζ8 | ζ85 | ζ83 | i | i | -i | -i | linear of order 8 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | -1 | -1 | -1 | 1 | ζ8 | ζ83 | ζ87 | ζ8 | ζ85 | ζ87 | ζ83 | ζ85 | -i | -i | i | i | linear of order 8 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | -i | i | i | 1 | 1 | 1 | 1 | -1 | -1 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | i | -i | i | -i | linear of order 8 |
| ρ17 | 2 | -2 | -2 | 2 | -1 | -1 | 2 | -1 | -2 | 2 | 2 | -2 | 1 | 1 | -2 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ19 | 2 | -2 | -2 | 2 | -1 | -1 | 2 | -1 | 2 | -2 | -2 | 2 | 1 | 1 | -2 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ20 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -2 | -2 | -2 | -2 | -1 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ21 | 2 | 2 | -2 | -2 | -1 | -1 | 2 | -1 | 2i | 2i | -2i | -2i | -1 | -1 | 2 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | i | -i | complex lifted from C3⋊C8 |
| ρ22 | 2 | 2 | -2 | -2 | -1 | -1 | 2 | -1 | -2i | -2i | 2i | 2i | -1 | -1 | 2 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | -i | i | complex lifted from C3⋊C8 |
| ρ23 | 2 | -2 | 2 | -2 | -1 | -1 | 2 | -1 | 2i | -2i | 2i | -2i | 1 | 1 | -2 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | -i | i | i | complex lifted from C3⋊C8 |
| ρ24 | 2 | -2 | 2 | -2 | -1 | -1 | 2 | -1 | -2i | 2i | -2i | 2i | 1 | 1 | -2 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | i | -i | -i | complex lifted from C3⋊C8 |
| ρ25 | 8 | -8 | 0 | 0 | 8 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -8 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×F9 |
| ρ26 | 8 | 8 | 0 | 0 | 8 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 8 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from F9 |
| ρ27 | 8 | 8 | 0 | 0 | -4 | 1+3√-3/2 | -1 | 1-3√-3/2 | 0 | 0 | 0 | 0 | -4 | 1+3√-3/2 | -1 | 1-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3⋊F9 |
| ρ28 | 8 | -8 | 0 | 0 | -4 | 1-3√-3/2 | -1 | 1+3√-3/2 | 0 | 0 | 0 | 0 | 4 | -1+3√-3/2 | 1 | -1-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ29 | 8 | 8 | 0 | 0 | -4 | 1-3√-3/2 | -1 | 1+3√-3/2 | 0 | 0 | 0 | 0 | -4 | 1-3√-3/2 | -1 | 1+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3⋊F9 |
| ρ30 | 8 | -8 | 0 | 0 | -4 | 1+3√-3/2 | -1 | 1-3√-3/2 | 0 | 0 | 0 | 0 | 4 | -1-3√-3/2 | 1 | -1+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 48 points
Generators in S48
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 40)(16 33)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(1 46 11)(2 12 47)(3 48 13)(4 14 41)(5 42 15)(6 16 43)(7 44 9)(8 10 45)(17 28 34)(18 35 29)(19 30 36)(20 37 31)(21 32 38)(22 39 25)(23 26 40)(24 33 27)
(1 46 11)(2 12 47)(4 14 41)(5 15 42)(6 43 16)(8 45 10)(18 29 35)(19 30 36)(20 37 31)(22 39 25)(23 40 26)(24 27 33)
(1 46 11)(2 47 12)(3 13 48)(5 15 42)(6 16 43)(7 44 9)(17 28 34)(19 30 36)(20 31 37)(21 38 32)(23 40 26)(24 33 27)
(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)
G:=sub<Sym(48)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (1,46,11)(2,12,47)(3,48,13)(4,14,41)(5,42,15)(6,16,43)(7,44,9)(8,10,45)(17,28,34)(18,35,29)(19,30,36)(20,37,31)(21,32,38)(22,39,25)(23,26,40)(24,33,27), (1,46,11)(2,12,47)(4,14,41)(5,15,42)(6,43,16)(8,45,10)(18,29,35)(19,30,36)(20,37,31)(22,39,25)(23,40,26)(24,27,33), (1,46,11)(2,47,12)(3,13,48)(5,15,42)(6,16,43)(7,44,9)(17,28,34)(19,30,36)(20,31,37)(21,38,32)(23,40,26)(24,33,27), (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)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,40)(16,33)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48), (1,46,11)(2,12,47)(3,48,13)(4,14,41)(5,42,15)(6,16,43)(7,44,9)(8,10,45)(17,28,34)(18,35,29)(19,30,36)(20,37,31)(21,32,38)(22,39,25)(23,26,40)(24,33,27), (1,46,11)(2,12,47)(4,14,41)(5,15,42)(6,43,16)(8,45,10)(18,29,35)(19,30,36)(20,37,31)(22,39,25)(23,40,26)(24,27,33), (1,46,11)(2,47,12)(3,13,48)(5,15,42)(6,16,43)(7,44,9)(17,28,34)(19,30,36)(20,31,37)(21,38,32)(23,40,26)(24,33,27), (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) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,40),(16,33),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48)], [(1,46,11),(2,12,47),(3,48,13),(4,14,41),(5,42,15),(6,16,43),(7,44,9),(8,10,45),(17,28,34),(18,35,29),(19,30,36),(20,37,31),(21,32,38),(22,39,25),(23,26,40),(24,33,27)], [(1,46,11),(2,12,47),(4,14,41),(5,15,42),(6,43,16),(8,45,10),(18,29,35),(19,30,36),(20,37,31),(22,39,25),(23,40,26),(24,27,33)], [(1,46,11),(2,47,12),(3,13,48),(5,15,42),(6,16,43),(7,44,9),(17,28,34),(19,30,36),(20,31,37),(21,38,32),(23,40,26),(24,33,27)], [(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)]])
Matrix representation of C2×C3⋊F9 ►in GL8(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 | 0 | 0 |
| 3 | 3 | 0 | 0 | 0 | 64 | 0 | 0 |
| 39 | 39 | 0 | 0 | 0 | 0 | 64 | 0 |
| 31 | 31 | 0 | 0 | 0 | 0 | 0 | 64 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 29 | 1 | 0 | 0 | 0 | 0 | 0 |
| 60 | 53 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 70 | 0 | 0 | 8 | 8 | 0 | 0 |
| 39 | 0 | 0 | 0 | 0 | 0 | 64 | 0 |
| 0 | 42 | 0 | 0 | 3 | 0 | 0 | 8 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 42 | 8 | 0 | 0 | 0 | 0 | 0 |
| 31 | 0 | 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 27 | 24 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 34 | 0 | 0 | 46 | 0 | 8 | 0 |
| 31 | 0 | 0 | 0 | 24 | 0 | 0 | 64 |
| 27 | 27 | 0 | 0 | 1 | 66 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 59 | 0 | 72 |
| 0 | 0 | 0 | 0 | 0 | 13 | 72 | 0 |
| 60 | 14 | 0 | 66 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 14 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 13 | 0 | 0 |
G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[8,0,0,0,0,3,39,31,0,8,0,0,0,3,39,31,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64],[8,0,14,60,0,0,39,0,0,64,29,53,0,70,0,42,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,64,8,0,3,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8],[8,0,0,31,0,27,0,31,0,64,42,0,0,24,34,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,46,24,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64],[27,0,0,0,60,0,0,0,27,0,0,0,14,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,66,72,46,27,1,72,0,0,0,0,0,0,66,0,59,13,0,46,14,13,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0] >;
C2×C3⋊F9 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes F_9
% in TeX
G:=Group("C2xC3:F9"); // GroupNames label
G:=SmallGroup(432,752);
// by ID
G=gap.SmallGroup(432,752);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,58,2244,718,165,677,691,530,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
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