Copied to
clipboard

## G = C5⋊2F9order 360 = 23·32·5

### The semidirect product of C5 and F9 acting via F9/C32⋊C4=C2

Aliases: C52F9, (C3×C15)⋊2C8, C32⋊(C52C8), C3⋊S3.Dic5, C32⋊C4.1D5, (C5×C3⋊S3).2C4, (C5×C32⋊C4).3C2, SmallGroup(360,124)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C5⋊2F9
 Chief series C1 — C5 — C3×C15 — C5×C3⋊S3 — C5×C32⋊C4 — C5⋊2F9
 Lower central C3×C15 — C5⋊2F9
 Upper central C1

Generators and relations for C52F9
G = < a,b,c,d | a5=b3=c3=d8=1, ab=ba, ac=ca, dad-1=a-1, dbd-1=bc=cb, dcd-1=b >

Character table of C52F9

 class 1 2 3 4A 4B 5A 5B 8A 8B 8C 8D 10A 10B 15A 15B 15C 15D 20A 20B 20C 20D size 1 9 8 9 9 2 2 45 45 45 45 18 18 8 8 8 8 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 trivial ρ2 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 i -i i -i 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ4 1 1 1 -1 -1 1 1 -i i -i i 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ5 1 -1 1 i -i 1 1 ζ87 ζ85 ζ83 ζ8 -1 -1 1 1 1 1 i i -i -i linear of order 8 ρ6 1 -1 1 -i i 1 1 ζ85 ζ87 ζ8 ζ83 -1 -1 1 1 1 1 -i -i i i linear of order 8 ρ7 1 -1 1 i -i 1 1 ζ83 ζ8 ζ87 ζ85 -1 -1 1 1 1 1 i i -i -i linear of order 8 ρ8 1 -1 1 -i i 1 1 ζ8 ζ83 ζ85 ζ87 -1 -1 1 1 1 1 -i -i i i linear of order 8 ρ9 2 2 2 2 2 -1+√5/2 -1-√5/2 0 0 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ10 2 2 2 2 2 -1-√5/2 -1+√5/2 0 0 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ11 2 2 2 -2 -2 -1+√5/2 -1-√5/2 0 0 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ12 2 2 2 -2 -2 -1-√5/2 -1+√5/2 0 0 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ13 2 -2 2 2i -2i -1-√5/2 -1+√5/2 0 0 0 0 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 complex lifted from C5⋊2C8, Schur index 2 ρ14 2 -2 2 2i -2i -1+√5/2 -1-√5/2 0 0 0 0 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 complex lifted from C5⋊2C8, Schur index 2 ρ15 2 -2 2 -2i 2i -1-√5/2 -1+√5/2 0 0 0 0 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 ζ43ζ54+ζ43ζ5 ζ43ζ53+ζ43ζ52 ζ4ζ53+ζ4ζ52 ζ4ζ54+ζ4ζ5 complex lifted from C5⋊2C8, Schur index 2 ρ16 2 -2 2 -2i 2i -1+√5/2 -1-√5/2 0 0 0 0 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 ζ43ζ53+ζ43ζ52 ζ43ζ54+ζ43ζ5 ζ4ζ54+ζ4ζ5 ζ4ζ53+ζ4ζ52 complex lifted from C5⋊2C8, Schur index 2 ρ17 8 0 -1 0 0 8 8 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 orthogonal lifted from F9 ρ18 8 0 -1 0 0 -2+2√5 -2-2√5 0 0 0 0 0 0 ζ53-2ζ52 ζ54-2ζ5 -2ζ53+ζ52 -2ζ54+ζ5 0 0 0 0 complex faithful ρ19 8 0 -1 0 0 -2-2√5 -2+2√5 0 0 0 0 0 0 -2ζ54+ζ5 ζ53-2ζ52 ζ54-2ζ5 -2ζ53+ζ52 0 0 0 0 complex faithful ρ20 8 0 -1 0 0 -2+2√5 -2-2√5 0 0 0 0 0 0 -2ζ53+ζ52 -2ζ54+ζ5 ζ53-2ζ52 ζ54-2ζ5 0 0 0 0 complex faithful ρ21 8 0 -1 0 0 -2-2√5 -2+2√5 0 0 0 0 0 0 ζ54-2ζ5 -2ζ53+ζ52 -2ζ54+ζ5 ζ53-2ζ52 0 0 0 0 complex faithful

Smallest permutation representation of C52F9
On 45 points
Generators in S45
```(1 3 5 4 2)(6 19 36 28 44)(7 45 29 37 20)(8 21 30 22 38)(9 39 23 31 14)(10 15 32 24 40)(11 41 25 33 16)(12 17 34 26 42)(13 43 27 35 18)
(1 20 16)(2 37 33)(3 7 11)(4 29 25)(5 45 41)(6 35 8)(9 32 34)(10 12 31)(13 30 36)(14 15 17)(18 21 19)(22 28 43)(23 40 42)(24 26 39)(27 38 44)
(1 21 17)(2 8 12)(3 30 34)(4 38 42)(5 22 26)(6 31 37)(7 36 9)(10 33 35)(11 13 32)(14 20 19)(15 16 18)(23 29 44)(24 41 43)(25 27 40)(28 39 45)
(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)```

`G:=sub<Sym(45)| (1,3,5,4,2)(6,19,36,28,44)(7,45,29,37,20)(8,21,30,22,38)(9,39,23,31,14)(10,15,32,24,40)(11,41,25,33,16)(12,17,34,26,42)(13,43,27,35,18), (1,20,16)(2,37,33)(3,7,11)(4,29,25)(5,45,41)(6,35,8)(9,32,34)(10,12,31)(13,30,36)(14,15,17)(18,21,19)(22,28,43)(23,40,42)(24,26,39)(27,38,44), (1,21,17)(2,8,12)(3,30,34)(4,38,42)(5,22,26)(6,31,37)(7,36,9)(10,33,35)(11,13,32)(14,20,19)(15,16,18)(23,29,44)(24,41,43)(25,27,40)(28,39,45), (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)>;`

`G:=Group( (1,3,5,4,2)(6,19,36,28,44)(7,45,29,37,20)(8,21,30,22,38)(9,39,23,31,14)(10,15,32,24,40)(11,41,25,33,16)(12,17,34,26,42)(13,43,27,35,18), (1,20,16)(2,37,33)(3,7,11)(4,29,25)(5,45,41)(6,35,8)(9,32,34)(10,12,31)(13,30,36)(14,15,17)(18,21,19)(22,28,43)(23,40,42)(24,26,39)(27,38,44), (1,21,17)(2,8,12)(3,30,34)(4,38,42)(5,22,26)(6,31,37)(7,36,9)(10,33,35)(11,13,32)(14,20,19)(15,16,18)(23,29,44)(24,41,43)(25,27,40)(28,39,45), (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) );`

`G=PermutationGroup([[(1,3,5,4,2),(6,19,36,28,44),(7,45,29,37,20),(8,21,30,22,38),(9,39,23,31,14),(10,15,32,24,40),(11,41,25,33,16),(12,17,34,26,42),(13,43,27,35,18)], [(1,20,16),(2,37,33),(3,7,11),(4,29,25),(5,45,41),(6,35,8),(9,32,34),(10,12,31),(13,30,36),(14,15,17),(18,21,19),(22,28,43),(23,40,42),(24,26,39),(27,38,44)], [(1,21,17),(2,8,12),(3,30,34),(4,38,42),(5,22,26),(6,31,37),(7,36,9),(10,33,35),(11,13,32),(14,20,19),(15,16,18),(23,29,44),(24,41,43),(25,27,40),(28,39,45)], [(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)]])`

Matrix representation of C52F9 in GL10(𝔽241)

 98 0 0 0 0 0 0 0 0 0 74 91 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 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 1
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 192 0 0 0 0 0 0 0 0 0 240 0 0 0 0 0 0 0 0 1 240 0 0 0 0 0 0 0 0 0 240 0 0 1 0 0 0 0 0 0 240 1 0 0 0 0 0 0 0 0 240 0 1 0 0 0 0 0 192 240 239 240 240 240 240 240 0 0 0 0 240 0 0 0 1 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 0 192 0 0 0 0 0 0 0 0 0 240 1 0 0 0 0 0 0 0 0 240 0 1 0 0 0 0 0 0 0 240 0 0 1 0 0 0 0 0 0 240 0 0 0 1 0 0 192 240 240 239 240 240 240 240 0 0 0 0 0 240 0 0 0 0 0 0 0 1 0 240 0 0 0 0
,
 27 144 0 0 0 0 0 0 0 0 64 214 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 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 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 192 240 240 240 240 240 240 240

`G:=sub<GL(10,GF(241))| [98,74,0,0,0,0,0,0,0,0,0,91,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,192,0,0,0,0,0,1,0,0,0,240,0,0,0,192,240,240,240,240,240,239,240,0,0,0,0,0,0,1,0,240,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,0,1,0,0,240,0,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,0,0,240,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,192,0,0,0,0,0,0,0,0,0,240,0,1,0,0,0,0,0,0,0,240,0,0,0,0,192,240,240,240,240,239,240,240,0,0,0,1,0,0,0,240,0,0,0,0,0,0,1,0,0,240,0,0,0,0,0,0,0,1,0,240,0,0,0,0,0,0,0,0,1,240,0,0],[27,64,0,0,0,0,0,0,0,0,144,214,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,192,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,1,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,240,0,0,0,0,1,0,0,0,0,240,0,0,0,0,0,0,1,0,0,240,0,0,0,1,0,0,0,0,0,240] >;`

C52F9 in GAP, Magma, Sage, TeX

`C_5\rtimes_2F_9`
`% in TeX`

`G:=Group("C5:2F9");`
`// GroupNames label`

`G:=SmallGroup(360,124);`
`// by ID`

`G=gap.SmallGroup(360,124);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,3,-5,12,31,771,489,111,244,490,376,10373]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^3=c^3=d^8=1,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