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## G = C32⋊3F5order 180 = 22·32·5

### 2nd semidirect product of C32 and F5 acting via F5/D5=C2

Aliases: C323F5, C151Dic3, C3⋊(C3⋊F5), (C3×C15)⋊2C4, C5⋊(C3⋊Dic3), D5.(C3⋊S3), (C3×D5).3S3, (C32×D5).1C2, SmallGroup(180,22)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C32⋊3F5
 Chief series C1 — C5 — C15 — C3×C15 — C32×D5 — C32⋊3F5
 Lower central C3×C15 — C32⋊3F5
 Upper central C1

Generators and relations for C323F5
G = < a,b,c,d | a3=b3=c5=d4=1, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c3 >

Character table of C323F5

 class 1 2 3A 3B 3C 3D 4A 4B 5 6A 6B 6C 6D 15A 15B 15C 15D 15E 15F 15G 15H size 1 5 2 2 2 2 45 45 4 10 10 10 10 4 4 4 4 4 4 4 4 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 -i i 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 4 ρ4 1 -1 1 1 1 1 i -i 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 4 ρ5 2 2 -1 -1 2 -1 0 0 2 2 -1 -1 -1 -1 -1 -1 -1 2 2 -1 -1 orthogonal lifted from S3 ρ6 2 2 -1 -1 -1 2 0 0 2 -1 2 -1 -1 2 -1 -1 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 -1 2 -1 -1 0 0 2 -1 -1 -1 2 -1 2 2 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 2 2 -1 -1 -1 0 0 2 -1 -1 2 -1 -1 -1 -1 -1 -1 -1 2 2 orthogonal lifted from S3 ρ9 2 -2 -1 2 -1 -1 0 0 2 1 1 1 -2 -1 2 2 -1 -1 -1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ10 2 -2 2 -1 -1 -1 0 0 2 1 1 -2 1 -1 -1 -1 -1 -1 -1 2 2 symplectic lifted from Dic3, Schur index 2 ρ11 2 -2 -1 -1 2 -1 0 0 2 -2 1 1 1 -1 -1 -1 -1 2 2 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ12 2 -2 -1 -1 -1 2 0 0 2 1 -2 1 1 2 -1 -1 2 -1 -1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ13 4 0 4 4 4 4 0 0 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ14 4 0 -2 -2 4 -2 0 0 -1 0 0 0 0 1+√-15/2 1-√-15/2 1+√-15/2 1-√-15/2 -1 -1 1+√-15/2 1-√-15/2 complex lifted from C3⋊F5 ρ15 4 0 -2 -2 -2 4 0 0 -1 0 0 0 0 -1 1-√-15/2 1+√-15/2 -1 1-√-15/2 1+√-15/2 1-√-15/2 1+√-15/2 complex lifted from C3⋊F5 ρ16 4 0 -2 4 -2 -2 0 0 -1 0 0 0 0 1+√-15/2 -1 -1 1-√-15/2 1+√-15/2 1-√-15/2 1-√-15/2 1+√-15/2 complex lifted from C3⋊F5 ρ17 4 0 -2 -2 4 -2 0 0 -1 0 0 0 0 1-√-15/2 1+√-15/2 1-√-15/2 1+√-15/2 -1 -1 1-√-15/2 1+√-15/2 complex lifted from C3⋊F5 ρ18 4 0 -2 4 -2 -2 0 0 -1 0 0 0 0 1-√-15/2 -1 -1 1+√-15/2 1-√-15/2 1+√-15/2 1+√-15/2 1-√-15/2 complex lifted from C3⋊F5 ρ19 4 0 4 -2 -2 -2 0 0 -1 0 0 0 0 1+√-15/2 1+√-15/2 1-√-15/2 1-√-15/2 1-√-15/2 1+√-15/2 -1 -1 complex lifted from C3⋊F5 ρ20 4 0 -2 -2 -2 4 0 0 -1 0 0 0 0 -1 1+√-15/2 1-√-15/2 -1 1+√-15/2 1-√-15/2 1+√-15/2 1-√-15/2 complex lifted from C3⋊F5 ρ21 4 0 4 -2 -2 -2 0 0 -1 0 0 0 0 1-√-15/2 1-√-15/2 1+√-15/2 1+√-15/2 1+√-15/2 1-√-15/2 -1 -1 complex lifted from C3⋊F5

Smallest permutation representation of C323F5
On 45 points
Generators in S45
```(1 31 16)(2 32 17)(3 33 18)(4 34 19)(5 35 20)(6 36 21)(7 37 22)(8 38 23)(9 39 24)(10 40 25)(11 41 26)(12 42 27)(13 43 28)(14 44 29)(15 45 30)
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)
(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)
(2 3 5 4)(6 11)(7 13 10 14)(8 15 9 12)(16 31)(17 33 20 34)(18 35 19 32)(21 41)(22 43 25 44)(23 45 24 42)(26 36)(27 38 30 39)(28 40 29 37)```

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

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

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

C323F5 is a maximal subgroup of   C3⋊S3×F5  S3×C3⋊F5
C323F5 is a maximal quotient of   C30.Dic3

Matrix representation of C323F5 in GL6(𝔽61)

 0 60 0 0 0 0 1 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 1 0 0 0 0 60 0 0 0 0 0 0 0 33 0 6 6 0 0 55 27 55 0 0 0 0 55 27 55 0 0 6 6 0 33
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 60 60 60 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 11 0 0 0 0 0 11 50 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 60 60 60 60

`G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,60,0,0,0,0,1,0,0,0,0,0,0,0,33,55,0,6,0,0,0,27,55,6,0,0,6,55,27,0,0,0,6,0,55,33],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,1,0,0,0,60,0,0,1,0,0,60,0,0,0],[11,11,0,0,0,0,0,50,0,0,0,0,0,0,1,0,0,60,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,1,0,60] >;`

C323F5 in GAP, Magma, Sage, TeX

`C_3^2\rtimes_3F_5`
`% in TeX`

`G:=Group("C3^2:3F5");`
`// GroupNames label`

`G:=SmallGroup(180,22);`
`// by ID`

`G=gap.SmallGroup(180,22);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-3,-5,10,122,483,2704,1809]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^3=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;`
`// generators/relations`

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