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## G = C2.F9order 144 = 24·32

### The central extension by C2 of F9

Aliases: C2.F9, C32⋊C16, (C3×C6).C8, C3⋊Dic3.C4, C322C8.1C2, SmallGroup(144,114)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2.F9
G = < a,b,c,d | a2=b3=c3=1, d8=a, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Character table of C2.F9

 class 1 2 3 4A 4B 6 8A 8B 8C 8D 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 8 9 9 8 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 trivial ρ2 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 -i -i -i -i i i i i linear of order 4 ρ4 1 1 1 1 1 1 -1 -1 -1 -1 i i i i -i -i -i -i linear of order 4 ρ5 1 1 1 -1 -1 1 i -i i -i ζ87 ζ87 ζ83 ζ83 ζ85 ζ85 ζ8 ζ8 linear of order 8 ρ6 1 1 1 -1 -1 1 i -i i -i ζ83 ζ83 ζ87 ζ87 ζ8 ζ8 ζ85 ζ85 linear of order 8 ρ7 1 1 1 -1 -1 1 -i i -i i ζ85 ζ85 ζ8 ζ8 ζ87 ζ87 ζ83 ζ83 linear of order 8 ρ8 1 1 1 -1 -1 1 -i i -i i ζ8 ζ8 ζ85 ζ85 ζ83 ζ83 ζ87 ζ87 linear of order 8 ρ9 1 -1 1 -i i -1 ζ166 ζ162 ζ1614 ζ1610 ζ1613 ζ165 ζ16 ζ169 ζ167 ζ1615 ζ1611 ζ163 linear of order 16 ρ10 1 -1 1 -i i -1 ζ166 ζ162 ζ1614 ζ1610 ζ165 ζ1613 ζ169 ζ16 ζ1615 ζ167 ζ163 ζ1611 linear of order 16 ρ11 1 -1 1 i -i -1 ζ162 ζ166 ζ1610 ζ1614 ζ167 ζ1615 ζ163 ζ1611 ζ165 ζ1613 ζ16 ζ169 linear of order 16 ρ12 1 -1 1 i -i -1 ζ162 ζ166 ζ1610 ζ1614 ζ1615 ζ167 ζ1611 ζ163 ζ1613 ζ165 ζ169 ζ16 linear of order 16 ρ13 1 -1 1 i -i -1 ζ1610 ζ1614 ζ162 ζ166 ζ163 ζ1611 ζ1615 ζ167 ζ169 ζ16 ζ165 ζ1613 linear of order 16 ρ14 1 -1 1 -i i -1 ζ1614 ζ1610 ζ166 ζ162 ζ169 ζ16 ζ1613 ζ165 ζ1611 ζ163 ζ1615 ζ167 linear of order 16 ρ15 1 -1 1 i -i -1 ζ1610 ζ1614 ζ162 ζ166 ζ1611 ζ163 ζ167 ζ1615 ζ16 ζ169 ζ1613 ζ165 linear of order 16 ρ16 1 -1 1 -i i -1 ζ1614 ζ1610 ζ166 ζ162 ζ16 ζ169 ζ165 ζ1613 ζ163 ζ1611 ζ167 ζ1615 linear of order 16 ρ17 8 8 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from F9 ρ18 8 -8 -1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C2.F9
On 48 points
Generators in S48
```(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(2 46 17)(3 47 18)(4 19 48)(6 21 34)(7 22 35)(8 36 23)(10 38 25)(11 39 26)(12 27 40)(14 29 42)(15 30 43)(16 44 31)
(1 45 32)(3 47 18)(4 48 19)(5 20 33)(7 22 35)(8 23 36)(9 37 24)(11 39 26)(12 40 27)(13 28 41)(15 30 43)(16 31 44)
(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)| (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (2,46,17)(3,47,18)(4,19,48)(6,21,34)(7,22,35)(8,36,23)(10,38,25)(11,39,26)(12,27,40)(14,29,42)(15,30,43)(16,44,31), (1,45,32)(3,47,18)(4,48,19)(5,20,33)(7,22,35)(8,23,36)(9,37,24)(11,39,26)(12,40,27)(13,28,41)(15,30,43)(16,31,44), (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( (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (2,46,17)(3,47,18)(4,19,48)(6,21,34)(7,22,35)(8,36,23)(10,38,25)(11,39,26)(12,27,40)(14,29,42)(15,30,43)(16,44,31), (1,45,32)(3,47,18)(4,48,19)(5,20,33)(7,22,35)(8,23,36)(9,37,24)(11,39,26)(12,40,27)(13,28,41)(15,30,43)(16,31,44), (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([[(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(2,46,17),(3,47,18),(4,19,48),(6,21,34),(7,22,35),(8,36,23),(10,38,25),(11,39,26),(12,27,40),(14,29,42),(15,30,43),(16,44,31)], [(1,45,32),(3,47,18),(4,48,19),(5,20,33),(7,22,35),(8,23,36),(9,37,24),(11,39,26),(12,40,27),(13,28,41),(15,30,43),(16,31,44)], [(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)]])`

C2.F9 is a maximal subgroup of   C4.3F9  C4.F9  C22.F9  C6.F9
C2.F9 is a maximal quotient of   C32⋊C32  He3⋊C16  C6.F9

Matrix representation of C2.F9 in GL9(𝔽97)

 96 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 96 1 0 0 0 0 0 0 0 96 0 0 0 0 0 0 42 10 0 37 96 96 0 0 0 0 0 60 0 1 0 0 0 0 0 0 38 0 0 0 0 1 0 95 33 0 59 0 0 96 96
,
 1 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 1 96 0 0 0 0 0 0 0 0 0 96 1 0 0 0 0 0 0 0 96 0 0 0 0 0 0 54 44 60 0 0 1 0 0 0 96 54 0 37 96 96 0 0 0 44 11 38 0 0 0 1 0 0 44 11 38 0 0 0 0 1
,
 8 0 0 0 0 0 0 0 0 0 0 0 0 0 96 1 0 0 0 42 10 37 37 95 96 0 0 0 0 0 0 0 0 0 96 1 0 95 33 59 59 0 0 95 96 0 83 20 10 10 44 43 60 0 0 83 20 11 10 44 43 60 0 0 82 63 80 80 11 53 38 0 0 50 82 80 80 11 53 38 0

`G:=sub<GL(9,GF(97))| [96,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,42,0,0,95,0,0,1,0,0,10,0,0,33,0,0,0,96,96,0,60,38,0,0,0,0,1,0,37,0,0,59,0,0,0,0,0,96,1,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,96],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,54,96,44,44,0,96,96,0,0,44,54,11,11,0,0,0,96,96,60,0,38,38,0,0,0,1,0,0,37,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,0,0,0,0,42,0,95,83,83,82,50,0,0,10,0,33,20,20,63,82,0,0,37,0,59,10,11,80,80,0,0,37,0,59,10,10,80,80,0,96,95,0,0,44,44,11,11,0,1,96,0,0,43,43,53,53,0,0,0,96,95,60,60,38,38,0,0,0,1,96,0,0,0,0] >;`

C2.F9 in GAP, Magma, Sage, TeX

`C_2.F_9`
`% in TeX`

`G:=Group("C2.F9");`
`// GroupNames label`

`G:=SmallGroup(144,114);`
`// by ID`

`G=gap.SmallGroup(144,114);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,3,12,31,50,1444,1690,256,4037,2315,881]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^2=b^3=c^3=1,d^8=a,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`

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