Copied to
clipboard

## G = F9⋊C4order 288 = 25·32

### The semidirect product of F9 and C4 acting via C4/C2=C2

Aliases: F9⋊C4, C2.3AΓL1(𝔽9), C32⋊C4.Q8, C3⋊S3.SD16, C32⋊(C4.Q8), (C2×F9).3C2, (C3×C6).3SD16, C2.PSU3(𝔽2).1C2, C3⋊S3.(C4⋊C4), (C2×C3⋊S3).3D4, C32⋊C4.3(C2×C4), C3⋊S3.Q8.3C2, (C2×C32⋊C4).3C22, SmallGroup(288,843)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C32⋊C4 — F9⋊C4
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — C2×C32⋊C4 — C2×F9 — F9⋊C4
 Lower central C32 — C3⋊S3 — C32⋊C4 — F9⋊C4
 Upper central C1 — C2

Generators and relations for F9⋊C4
G = < a,b,c,d | a3=b3=c8=d4=1, cac-1=ab=ba, dad-1=a-1b, cbc-1=a, bd=db, dcd-1=c3 >

Character table of F9⋊C4

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6 8A 8B 8C 8D 12A 12B size 1 1 9 9 8 12 12 18 18 36 36 8 18 18 18 18 24 24 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 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 linear of order 2 ρ5 1 -1 -1 1 1 i -i 1 -1 -i i -1 -1 1 -1 1 i -i linear of order 4 ρ6 1 -1 -1 1 1 -i i 1 -1 i -i -1 -1 1 -1 1 -i i linear of order 4 ρ7 1 -1 -1 1 1 -i i 1 -1 -i i -1 1 -1 1 -1 -i i linear of order 4 ρ8 1 -1 -1 1 1 i -i 1 -1 i -i -1 1 -1 1 -1 i -i linear of order 4 ρ9 2 2 2 2 2 0 0 -2 -2 0 0 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 2 0 0 -2 2 0 0 -2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ11 2 -2 2 -2 2 0 0 0 0 0 0 -2 √-2 √-2 -√-2 -√-2 0 0 complex lifted from SD16 ρ12 2 2 -2 -2 2 0 0 0 0 0 0 2 -√-2 √-2 √-2 -√-2 0 0 complex lifted from SD16 ρ13 2 2 -2 -2 2 0 0 0 0 0 0 2 √-2 -√-2 -√-2 √-2 0 0 complex lifted from SD16 ρ14 2 -2 2 -2 2 0 0 0 0 0 0 -2 -√-2 -√-2 √-2 √-2 0 0 complex lifted from SD16 ρ15 8 8 0 0 -1 -2 -2 0 0 0 0 -1 0 0 0 0 1 1 orthogonal lifted from AΓL1(𝔽9) ρ16 8 8 0 0 -1 2 2 0 0 0 0 -1 0 0 0 0 -1 -1 orthogonal lifted from AΓL1(𝔽9) ρ17 8 -8 0 0 -1 2i -2i 0 0 0 0 1 0 0 0 0 -i i complex faithful ρ18 8 -8 0 0 -1 -2i 2i 0 0 0 0 1 0 0 0 0 i -i complex faithful

Smallest permutation representation of F9⋊C4
On 36 points
Generators in S36
```(1 15 19)(2 31 35)(3 22 26)(4 12 8)(5 11 10)(6 7 9)(13 16 14)(17 18 20)(21 28 23)(24 25 27)(29 32 30)(33 34 36)
(1 16 20)(2 32 36)(3 23 27)(4 5 9)(6 12 11)(7 8 10)(13 18 19)(14 17 15)(21 24 22)(25 26 28)(29 34 35)(30 33 31)
(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)
(1 3 4 2)(5 32 16 23)(6 35 17 26)(7 30 18 21)(8 33 19 24)(9 36 20 27)(10 31 13 22)(11 34 14 25)(12 29 15 28)```

`G:=sub<Sym(36)| (1,15,19)(2,31,35)(3,22,26)(4,12,8)(5,11,10)(6,7,9)(13,16,14)(17,18,20)(21,28,23)(24,25,27)(29,32,30)(33,34,36), (1,16,20)(2,32,36)(3,23,27)(4,5,9)(6,12,11)(7,8,10)(13,18,19)(14,17,15)(21,24,22)(25,26,28)(29,34,35)(30,33,31), (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), (1,3,4,2)(5,32,16,23)(6,35,17,26)(7,30,18,21)(8,33,19,24)(9,36,20,27)(10,31,13,22)(11,34,14,25)(12,29,15,28)>;`

`G:=Group( (1,15,19)(2,31,35)(3,22,26)(4,12,8)(5,11,10)(6,7,9)(13,16,14)(17,18,20)(21,28,23)(24,25,27)(29,32,30)(33,34,36), (1,16,20)(2,32,36)(3,23,27)(4,5,9)(6,12,11)(7,8,10)(13,18,19)(14,17,15)(21,24,22)(25,26,28)(29,34,35)(30,33,31), (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), (1,3,4,2)(5,32,16,23)(6,35,17,26)(7,30,18,21)(8,33,19,24)(9,36,20,27)(10,31,13,22)(11,34,14,25)(12,29,15,28) );`

`G=PermutationGroup([[(1,15,19),(2,31,35),(3,22,26),(4,12,8),(5,11,10),(6,7,9),(13,16,14),(17,18,20),(21,28,23),(24,25,27),(29,32,30),(33,34,36)], [(1,16,20),(2,32,36),(3,23,27),(4,5,9),(6,12,11),(7,8,10),(13,18,19),(14,17,15),(21,24,22),(25,26,28),(29,34,35),(30,33,31)], [(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)], [(1,3,4,2),(5,32,16,23),(6,35,17,26),(7,30,18,21),(8,33,19,24),(9,36,20,27),(10,31,13,22),(11,34,14,25),(12,29,15,28)]])`

Matrix representation of F9⋊C4 in GL8(𝔽73)

 0 0 1 0 0 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 72 72 72 72 72 72 72 72 1 0 0 0 0 0 0 0
,
 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 72 72 72 72 72 72 72 72 1 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 0 0
,
 0 0 1 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 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0
,
 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 27

`G:=sub<GL(8,GF(73))| [0,0,0,0,0,0,72,1,0,0,0,0,0,1,72,0,1,0,0,0,0,0,72,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,72,0,0,0,0,0,1,0,72,0,0,0,1,0,0,0,72,0],[0,0,0,72,1,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,1,0,0,0,0,72,0,0,0,1,0,0,0,72,0,1,0,0,1,0,0,72,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,72,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0],[0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,27] >;`

F9⋊C4 in GAP, Magma, Sage, TeX

`F_9\rtimes C_4`
`% in TeX`

`G:=Group("F9:C4");`
`// GroupNames label`

`G:=SmallGroup(288,843);`
`// by ID`

`G=gap.SmallGroup(288,843);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,309,64,219,100,4037,4716,2371,201,10982,4717,3156,622]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^8=d^4=1,c*a*c^-1=a*b=b*a,d*a*d^-1=a^-1*b,c*b*c^-1=a,b*d=d*b,d*c*d^-1=c^3>;`
`// generators/relations`

Export

׿
×
𝔽