metabelian, soluble, monomial, A-group
Aliases: C6.3F9, C33⋊2C16, C2.(C3⋊F9), C3⋊(C2.F9), C32⋊(C3⋊C16), (C32×C6).2C8, C32⋊2C8.1S3, C3⋊Dic3.3Dic3, (C3×C6).(C3⋊C8), (C3×C3⋊Dic3).2C4, (C3×C32⋊2C8).4C2, SmallGroup(432,566)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C3×C32⋊2C8 — C6.F9 |
| C33 — C6.F9 |
Generators and relations for C6.F9
G = < a,b,c,d | a6=b3=c3=1, d8=a3, ab=ba, ac=ca, dad-1=a-1, dbd-1=bc=cb, dcd-1=b >
Character table of C6.F9
| class | 1 | 2 | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 12A | 12B | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 8 | 8 | 8 | 9 | 9 | 2 | 8 | 8 | 8 | 9 | 9 | 9 | 9 | 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 | -i | i | i | i | -i | -i | -i | i | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ4 | 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 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | i | -i | -i | i | -1 | -1 | ζ8 | ζ87 | ζ83 | ζ83 | ζ85 | ζ85 | ζ8 | ζ87 | -i | i | -i | i | linear of order 8 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | i | i | -i | -1 | -1 | ζ87 | ζ8 | ζ85 | ζ85 | ζ83 | ζ83 | ζ87 | ζ8 | i | -i | i | -i | linear of order 8 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | i | -i | -i | i | -1 | -1 | ζ85 | ζ83 | ζ87 | ζ87 | ζ8 | ζ8 | ζ85 | ζ83 | -i | i | -i | i | linear of order 8 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | i | i | -i | -1 | -1 | ζ83 | ζ85 | ζ8 | ζ8 | ζ87 | ζ87 | ζ83 | ζ85 | i | -i | i | -i | linear of order 8 |
| ρ9 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | -1 | ζ162 | ζ1614 | ζ166 | ζ1610 | i | -i | ζ169 | ζ1615 | ζ163 | ζ1611 | ζ165 | ζ1613 | ζ16 | ζ167 | ζ1614 | ζ1610 | ζ166 | ζ162 | linear of order 16 |
| ρ10 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | ζ166 | ζ1610 | ζ162 | ζ1614 | -i | i | ζ1611 | ζ1613 | ζ169 | ζ16 | ζ1615 | ζ167 | ζ163 | ζ165 | ζ1610 | ζ1614 | ζ162 | ζ166 | linear of order 16 |
| ρ11 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | -1 | ζ1610 | ζ166 | ζ1614 | ζ162 | i | -i | ζ1613 | ζ1611 | ζ1615 | ζ167 | ζ169 | ζ16 | ζ165 | ζ163 | ζ166 | ζ162 | ζ1614 | ζ1610 | linear of order 16 |
| ρ12 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | ζ1614 | ζ162 | ζ1610 | ζ166 | -i | i | ζ1615 | ζ169 | ζ165 | ζ1613 | ζ163 | ζ1611 | ζ167 | ζ16 | ζ162 | ζ166 | ζ1610 | ζ1614 | linear of order 16 |
| ρ13 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | ζ1614 | ζ162 | ζ1610 | ζ166 | -i | i | ζ167 | ζ16 | ζ1613 | ζ165 | ζ1611 | ζ163 | ζ1615 | ζ169 | ζ162 | ζ166 | ζ1610 | ζ1614 | linear of order 16 |
| ρ14 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | -1 | ζ1610 | ζ166 | ζ1614 | ζ162 | i | -i | ζ165 | ζ163 | ζ167 | ζ1615 | ζ16 | ζ169 | ζ1613 | ζ1611 | ζ166 | ζ162 | ζ1614 | ζ1610 | linear of order 16 |
| ρ15 | 1 | -1 | 1 | 1 | 1 | 1 | -i | i | -1 | -1 | -1 | -1 | ζ166 | ζ1610 | ζ162 | ζ1614 | -i | i | ζ163 | ζ165 | ζ16 | ζ169 | ζ167 | ζ1615 | ζ1611 | ζ1613 | ζ1610 | ζ1614 | ζ162 | ζ166 | linear of order 16 |
| ρ16 | 1 | -1 | 1 | 1 | 1 | 1 | i | -i | -1 | -1 | -1 | -1 | ζ162 | ζ1614 | ζ166 | ζ1610 | i | -i | ζ16 | ζ167 | ζ1611 | ζ163 | ζ1613 | ζ165 | ζ169 | ζ1615 | ζ1614 | ζ1610 | ζ166 | ζ162 | linear of order 16 |
| ρ17 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ18 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | -2 | -2 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ19 | 2 | 2 | -1 | -1 | -1 | 2 | -2 | -2 | -1 | -1 | -1 | 2 | 2i | -2i | -2i | 2i | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | i | -i | i | -i | complex lifted from C3⋊C8 |
| ρ20 | 2 | 2 | -1 | -1 | -1 | 2 | -2 | -2 | -1 | -1 | -1 | 2 | -2i | 2i | 2i | -2i | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -i | i | -i | i | complex lifted from C3⋊C8 |
| ρ21 | 2 | -2 | -1 | -1 | -1 | 2 | 2i | -2i | 1 | 1 | 1 | -2 | 2ζ85 | 2ζ83 | 2ζ87 | 2ζ8 | -i | i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ87 | ζ85 | ζ83 | ζ8 | complex lifted from C3⋊C16, Schur index 2 |
| ρ22 | 2 | -2 | -1 | -1 | -1 | 2 | 2i | -2i | 1 | 1 | 1 | -2 | 2ζ8 | 2ζ87 | 2ζ83 | 2ζ85 | -i | i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ83 | ζ8 | ζ87 | ζ85 | complex lifted from C3⋊C16, Schur index 2 |
| ρ23 | 2 | -2 | -1 | -1 | -1 | 2 | -2i | 2i | 1 | 1 | 1 | -2 | 2ζ87 | 2ζ8 | 2ζ85 | 2ζ83 | i | -i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ85 | ζ87 | ζ8 | ζ83 | complex lifted from C3⋊C16, Schur index 2 |
| ρ24 | 2 | -2 | -1 | -1 | -1 | 2 | -2i | 2i | 1 | 1 | 1 | -2 | 2ζ83 | 2ζ85 | 2ζ8 | 2ζ87 | i | -i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ8 | ζ83 | ζ85 | ζ87 | complex lifted from C3⋊C16, Schur index 2 |
| ρ25 | 8 | 8 | 8 | -1 | -1 | -1 | 0 | 0 | 8 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from F9 |
| ρ26 | 8 | -8 | 8 | -1 | -1 | -1 | 0 | 0 | -8 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C2.F9, Schur index 2 |
| ρ27 | 8 | 8 | -4 | 1+3√-3/2 | 1-3√-3/2 | -1 | 0 | 0 | -4 | 1-3√-3/2 | 1+3√-3/2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3⋊F9 |
| ρ28 | 8 | -8 | -4 | 1-3√-3/2 | 1+3√-3/2 | -1 | 0 | 0 | 4 | -1-3√-3/2 | -1+3√-3/2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ29 | 8 | -8 | -4 | 1+3√-3/2 | 1-3√-3/2 | -1 | 0 | 0 | 4 | -1+3√-3/2 | -1-3√-3/2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ30 | 8 | 8 | -4 | 1-3√-3/2 | 1+3√-3/2 | -1 | 0 | 0 | -4 | 1+3√-3/2 | 1-3√-3/2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3⋊F9 |
►On 48 points
Generators in S48
(1 23 35 9 31 43)(2 44 32 10 36 24)(3 25 37 11 17 45)(4 46 18 12 38 26)(5 27 39 13 19 47)(6 48 20 14 40 28)(7 29 41 15 21 33)(8 34 22 16 42 30)
(2 32 36)(3 17 37)(4 38 18)(6 40 20)(7 41 21)(8 22 42)(10 24 44)(11 25 45)(12 46 26)(14 48 28)(15 33 29)(16 30 34)
(1 31 35)(3 17 37)(4 18 38)(5 39 19)(7 41 21)(8 42 22)(9 23 43)(11 25 45)(12 26 46)(13 47 27)(15 33 29)(16 34 30)
(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)| (1,23,35,9,31,43)(2,44,32,10,36,24)(3,25,37,11,17,45)(4,46,18,12,38,26)(5,27,39,13,19,47)(6,48,20,14,40,28)(7,29,41,15,21,33)(8,34,22,16,42,30), (2,32,36)(3,17,37)(4,38,18)(6,40,20)(7,41,21)(8,22,42)(10,24,44)(11,25,45)(12,46,26)(14,48,28)(15,33,29)(16,30,34), (1,31,35)(3,17,37)(4,18,38)(5,39,19)(7,41,21)(8,42,22)(9,23,43)(11,25,45)(12,26,46)(13,47,27)(15,33,29)(16,34,30), (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( (1,23,35,9,31,43)(2,44,32,10,36,24)(3,25,37,11,17,45)(4,46,18,12,38,26)(5,27,39,13,19,47)(6,48,20,14,40,28)(7,29,41,15,21,33)(8,34,22,16,42,30), (2,32,36)(3,17,37)(4,38,18)(6,40,20)(7,41,21)(8,22,42)(10,24,44)(11,25,45)(12,46,26)(14,48,28)(15,33,29)(16,30,34), (1,31,35)(3,17,37)(4,18,38)(5,39,19)(7,41,21)(8,42,22)(9,23,43)(11,25,45)(12,26,46)(13,47,27)(15,33,29)(16,34,30), (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([[(1,23,35,9,31,43),(2,44,32,10,36,24),(3,25,37,11,17,45),(4,46,18,12,38,26),(5,27,39,13,19,47),(6,48,20,14,40,28),(7,29,41,15,21,33),(8,34,22,16,42,30)], [(2,32,36),(3,17,37),(4,38,18),(6,40,20),(7,41,21),(8,22,42),(10,24,44),(11,25,45),(12,46,26),(14,48,28),(15,33,29),(16,30,34)], [(1,31,35),(3,17,37),(4,18,38),(5,39,19),(7,41,21),(8,42,22),(9,23,43),(11,25,45),(12,26,46),(13,47,27),(15,33,29),(16,34,30)], [(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 C6.F9 ►in GL10(𝔽97)
| 1 | 96 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 61 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 61 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 61 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 61 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 35 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 35 | 0 |
| 0 | 0 | 46 | 11 | 65 | 72 | 0 | 0 | 0 | 35 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 61 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 35 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 61 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 61 | 0 |
| 0 | 0 | 7 | 8 | 0 | 72 | 0 | 4 | 6 | 35 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 61 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 61 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 61 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 39 | 89 | 85 | 2 | 23 | 50 | 0 | 1 |
| 0 | 70 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 70 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 8 | 12 | 70 | 29 | 47 | 22 | 34 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 40 | 60 | 80 | 58 | 26 | 13 | 85 |
G:=sub<GL(10,GF(97))| [1,1,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,0,61,0,0,0,0,0,0,46,0,0,0,61,0,0,0,0,0,11,0,0,0,0,61,0,0,0,0,65,0,0,0,0,0,61,0,0,0,72,0,0,0,0,0,0,35,0,0,0,0,0,0,0,0,0,0,35,0,0,0,0,0,0,0,0,0,0,35,0,0,0,0,0,0,0,0,0,0,35],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,1,0,0,0,0,0,8,0,0,0,0,35,0,0,0,0,0,0,0,0,0,0,61,0,0,0,72,0,0,0,0,0,0,35,0,0,0,0,0,0,0,0,0,0,61,0,4,0,0,0,0,0,0,0,0,61,6,0,0,0,0,0,0,0,0,0,35],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,61,0,0,0,0,0,0,39,0,0,0,35,0,0,0,0,0,89,0,0,0,0,35,0,0,0,0,85,0,0,0,0,0,61,0,0,0,2,0,0,0,0,0,0,61,0,0,23,0,0,0,0,0,0,0,35,0,50,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,70,0,0,0,0,0,0,0,0,70,0,0,0,0,0,0,0,0,0,0,0,0,0,7,0,0,0,1,14,0,0,0,0,8,0,0,0,0,40,0,0,0,0,12,0,0,1,0,60,0,0,0,0,70,0,1,0,0,80,0,0,0,1,29,0,0,0,0,58,0,0,1,0,47,0,0,0,0,26,0,0,0,0,22,1,0,0,0,13,0,0,0,0,34,0,0,0,0,85] >;
C6.F9 in GAP, Magma, Sage, TeX
C_6.F_9
% in TeX
G:=Group("C6.F9"); // GroupNames label
G:=SmallGroup(432,566);
// by ID
G=gap.SmallGroup(432,566);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,14,36,58,2244,1411,298,677,1356,1027,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^3=c^3=1,d^8=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
Export
Subgroup lattice of C6.F9 in TeX
Character table of C6.F9 in TeX