Copied to
clipboard

## G = C6.F9order 432 = 24·33

### 3rd non-split extension by C6 of F9 acting via F9/C32⋊C4=C2

Aliases: C6.3F9, C332C16, C2.(C3⋊F9), C3⋊(C2.F9), C32⋊(C3⋊C16), (C32×C6).2C8, C322C8.1S3, C3⋊Dic3.3Dic3, (C3×C6).(C3⋊C8), (C3×C3⋊Dic3).2C4, (C3×C322C8).4C2, SmallGroup(432,566)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C6.F9
 Chief series C1 — C3 — C33 — C32×C6 — C3×C3⋊Dic3 — C3×C32⋊2C8 — C6.F9
 Lower central C33 — C6.F9
 Upper central C1 — C2

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

Smallest permutation representation of C6.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

׿
×
𝔽