direct product, metabelian, soluble, monomial, A-group
Aliases: C3×F9, C32⋊C24, C33⋊1C8, C3⋊S3.C12, C32⋊C4.1C6, (C3×C3⋊S3).1C4, (C3×C32⋊C4).1C2, SmallGroup(216,154)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3×F9 |
Generators and relations for C3×F9
G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Character table of C3×F9
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | |
| size | 1 | 9 | 1 | 1 | 8 | 8 | 8 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | |
| 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 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | -1 | -1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ65 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ6 | linear of order 6 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | -1 | -1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ6 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ65 | linear of order 6 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | -1 | -1 | -1 | -1 | -i | -i | i | i | -i | -i | i | i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | -1 | -1 | -1 | -1 | i | i | -i | -i | i | i | -i | -i | linear of order 4 |
| ρ9 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | ζ83 | ζ85 | ζ8 | ζ87 | i | -i | -i | i | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | linear of order 8 |
| ρ10 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | ζ85 | ζ83 | ζ87 | ζ8 | -i | i | i | -i | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | linear of order 8 |
| ρ11 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | ζ87 | ζ8 | ζ85 | ζ83 | i | -i | -i | i | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | linear of order 8 |
| ρ12 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | ζ8 | ζ87 | ζ83 | ζ85 | -i | i | i | -i | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | linear of order 8 |
| ρ13 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | -1 | -1 | ζ32 | ζ3 | -i | i | i | -i | ζ6 | ζ65 | ζ6 | ζ65 | ζ4ζ3 | ζ4ζ3 | ζ43ζ3 | ζ43ζ3 | ζ4ζ32 | ζ4ζ32 | ζ43ζ32 | ζ43ζ32 | linear of order 12 |
| ρ14 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | -1 | -1 | ζ3 | ζ32 | i | -i | -i | i | ζ65 | ζ6 | ζ65 | ζ6 | ζ43ζ32 | ζ43ζ32 | ζ4ζ32 | ζ4ζ32 | ζ43ζ3 | ζ43ζ3 | ζ4ζ3 | ζ4ζ3 | linear of order 12 |
| ρ15 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | -1 | -1 | ζ32 | ζ3 | i | -i | -i | i | ζ6 | ζ65 | ζ6 | ζ65 | ζ43ζ3 | ζ43ζ3 | ζ4ζ3 | ζ4ζ3 | ζ43ζ32 | ζ43ζ32 | ζ4ζ32 | ζ4ζ32 | linear of order 12 |
| ρ16 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | -1 | -1 | ζ3 | ζ32 | -i | i | i | -i | ζ65 | ζ6 | ζ65 | ζ6 | ζ4ζ32 | ζ4ζ32 | ζ43ζ32 | ζ43ζ32 | ζ4ζ3 | ζ4ζ3 | ζ43ζ3 | ζ43ζ3 | linear of order 12 |
| ρ17 | 1 | -1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | -i | i | ζ65 | ζ6 | ζ85 | ζ83 | ζ87 | ζ8 | ζ86ζ3 | ζ82ζ32 | ζ82ζ3 | ζ86ζ32 | ζ83ζ32 | ζ87ζ32 | ζ8ζ32 | ζ85ζ32 | ζ83ζ3 | ζ87ζ3 | ζ8ζ3 | ζ85ζ3 | linear of order 24 |
| ρ18 | 1 | -1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | -i | i | ζ6 | ζ65 | ζ85 | ζ83 | ζ87 | ζ8 | ζ86ζ32 | ζ82ζ3 | ζ82ζ32 | ζ86ζ3 | ζ83ζ3 | ζ87ζ3 | ζ8ζ3 | ζ85ζ3 | ζ83ζ32 | ζ87ζ32 | ζ8ζ32 | ζ85ζ32 | linear of order 24 |
| ρ19 | 1 | -1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | -i | i | ζ6 | ζ65 | ζ8 | ζ87 | ζ83 | ζ85 | ζ86ζ32 | ζ82ζ3 | ζ82ζ32 | ζ86ζ3 | ζ87ζ3 | ζ83ζ3 | ζ85ζ3 | ζ8ζ3 | ζ87ζ32 | ζ83ζ32 | ζ85ζ32 | ζ8ζ32 | linear of order 24 |
| ρ20 | 1 | -1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | -i | i | ζ65 | ζ6 | ζ8 | ζ87 | ζ83 | ζ85 | ζ86ζ3 | ζ82ζ32 | ζ82ζ3 | ζ86ζ32 | ζ87ζ32 | ζ83ζ32 | ζ85ζ32 | ζ8ζ32 | ζ87ζ3 | ζ83ζ3 | ζ85ζ3 | ζ8ζ3 | linear of order 24 |
| ρ21 | 1 | -1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | i | -i | ζ65 | ζ6 | ζ87 | ζ8 | ζ85 | ζ83 | ζ82ζ3 | ζ86ζ32 | ζ86ζ3 | ζ82ζ32 | ζ8ζ32 | ζ85ζ32 | ζ83ζ32 | ζ87ζ32 | ζ8ζ3 | ζ85ζ3 | ζ83ζ3 | ζ87ζ3 | linear of order 24 |
| ρ22 | 1 | -1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | i | -i | ζ6 | ζ65 | ζ83 | ζ85 | ζ8 | ζ87 | ζ82ζ32 | ζ86ζ3 | ζ86ζ32 | ζ82ζ3 | ζ85ζ3 | ζ8ζ3 | ζ87ζ3 | ζ83ζ3 | ζ85ζ32 | ζ8ζ32 | ζ87ζ32 | ζ83ζ32 | linear of order 24 |
| ρ23 | 1 | -1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | i | -i | ζ6 | ζ65 | ζ87 | ζ8 | ζ85 | ζ83 | ζ82ζ32 | ζ86ζ3 | ζ86ζ32 | ζ82ζ3 | ζ8ζ3 | ζ85ζ3 | ζ83ζ3 | ζ87ζ3 | ζ8ζ32 | ζ85ζ32 | ζ83ζ32 | ζ87ζ32 | linear of order 24 |
| ρ24 | 1 | -1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | i | -i | ζ65 | ζ6 | ζ83 | ζ85 | ζ8 | ζ87 | ζ82ζ3 | ζ86ζ32 | ζ86ζ3 | ζ82ζ32 | ζ85ζ32 | ζ8ζ32 | ζ87ζ32 | ζ83ζ32 | ζ85ζ3 | ζ8ζ3 | ζ87ζ3 | ζ83ζ3 | linear of order 24 |
| ρ25 | 8 | 0 | 8 | 8 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from F9 |
| ρ26 | 8 | 0 | -4-4√-3 | -4+4√-3 | -1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ27 | 8 | 0 | -4+4√-3 | -4-4√-3 | -1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 24 points - transitive group 24T567
Generators in S24
(1 12 21)(2 13 22)(3 14 23)(4 15 24)(5 16 17)(6 9 18)(7 10 19)(8 11 20)
(2 13 22)(3 14 23)(4 24 15)(6 18 9)(7 19 10)(8 11 20)
(1 12 21)(3 14 23)(4 15 24)(5 17 16)(7 19 10)(8 20 11)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (1,12,21)(2,13,22)(3,14,23)(4,15,24)(5,16,17)(6,9,18)(7,10,19)(8,11,20), (2,13,22)(3,14,23)(4,24,15)(6,18,9)(7,19,10)(8,11,20), (1,12,21)(3,14,23)(4,15,24)(5,17,16)(7,19,10)(8,20,11), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;
G:=Group( (1,12,21)(2,13,22)(3,14,23)(4,15,24)(5,16,17)(6,9,18)(7,10,19)(8,11,20), (2,13,22)(3,14,23)(4,24,15)(6,18,9)(7,19,10)(8,11,20), (1,12,21)(3,14,23)(4,15,24)(5,17,16)(7,19,10)(8,20,11), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(1,12,21),(2,13,22),(3,14,23),(4,15,24),(5,16,17),(6,9,18),(7,10,19),(8,11,20)], [(2,13,22),(3,14,23),(4,24,15),(6,18,9),(7,19,10),(8,11,20)], [(1,12,21),(3,14,23),(4,15,24),(5,17,16),(7,19,10),(8,20,11)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,567);
►On 27 points - transitive group 27T78
Generators in S27
(1 3 2)(4 19 23)(5 12 24)(6 13 25)(7 14 26)(8 15 27)(9 16 20)(10 17 21)(11 18 22)
(1 15 19)(2 8 4)(3 27 23)(5 10 11)(6 9 7)(12 17 18)(13 16 14)(20 26 25)(21 22 24)
(1 16 12)(2 9 5)(3 20 24)(4 6 11)(7 10 8)(13 18 19)(14 17 15)(21 27 26)(22 23 25)
(4 5 6 7 8 9 10 11)(12 13 14 15 16 17 18 19)(20 21 22 23 24 25 26 27)
G:=sub<Sym(27)| (1,3,2)(4,19,23)(5,12,24)(6,13,25)(7,14,26)(8,15,27)(9,16,20)(10,17,21)(11,18,22), (1,15,19)(2,8,4)(3,27,23)(5,10,11)(6,9,7)(12,17,18)(13,16,14)(20,26,25)(21,22,24), (1,16,12)(2,9,5)(3,20,24)(4,6,11)(7,10,8)(13,18,19)(14,17,15)(21,27,26)(22,23,25), (4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27)>;
G:=Group( (1,3,2)(4,19,23)(5,12,24)(6,13,25)(7,14,26)(8,15,27)(9,16,20)(10,17,21)(11,18,22), (1,15,19)(2,8,4)(3,27,23)(5,10,11)(6,9,7)(12,17,18)(13,16,14)(20,26,25)(21,22,24), (1,16,12)(2,9,5)(3,20,24)(4,6,11)(7,10,8)(13,18,19)(14,17,15)(21,27,26)(22,23,25), (4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27) );
G=PermutationGroup([[(1,3,2),(4,19,23),(5,12,24),(6,13,25),(7,14,26),(8,15,27),(9,16,20),(10,17,21),(11,18,22)], [(1,15,19),(2,8,4),(3,27,23),(5,10,11),(6,9,7),(12,17,18),(13,16,14),(20,26,25),(21,22,24)], [(1,16,12),(2,9,5),(3,20,24),(4,6,11),(7,10,8),(13,18,19),(14,17,15),(21,27,26),(22,23,25)], [(4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19),(20,21,22,23,24,25,26,27)]])
G:=TransitiveGroup(27,78);
C3×F9 is a maximal subgroup of
F9⋊S3
Matrix representation of C3×F9 ►in GL8(𝔽73)
| 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 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 64 | 0 | 0 | 0 | 0 |
| 64 | 0 | 0 | 0 | 64 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 8 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 8 | 0 |
| 64 | 0 | 0 | 0 | 0 | 0 | 0 | 64 |
| 64 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 65 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 65 | 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 64 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(8,GF(73))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64],[1,0,1,9,64,72,72,64,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64],[64,65,65,0,8,0,9,9,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,7,72,72,72,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
C3×F9 in GAP, Magma, Sage, TeX
C_3\times F_9
% in TeX
G:=Group("C3xF9"); // GroupNames label
G:=SmallGroup(216,154);
// by ID
G=gap.SmallGroup(216,154);
# by ID
G:=PCGroup([6,-2,-3,-2,-2,-3,3,36,50,2164,856,142,6053,1169,455]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
Export
Subgroup lattice of C3×F9 in TeX
Character table of C3×F9 in TeX