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## G = C32⋊F5order 180 = 22·32·5

### The semidirect product of C32 and F5 acting via F5/C5=C4

Aliases: C32⋊F5, C5⋊(C32⋊C4), (C3×C15)⋊1C4, C3⋊D15.C2, SmallGroup(180,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C32⋊F5
 Chief series C1 — C5 — C3×C15 — C3⋊D15 — C32⋊F5
 Lower central C3×C15 — C32⋊F5
 Upper central C1

Generators and relations for C32⋊F5
G = < a,b,c,d | a3=b3=c5=d4=1, dbd-1=ab=ba, ac=ca, dad-1=a-1b, bc=cb, dcd-1=c3 >

45C2
2C3
2C3
45C4
30S3
30S3
9D5
2C15
2C15
9F5
6D15
6D15

Character table of C32⋊F5

 class 1 2 3A 3B 4A 4B 5 15A 15B 15C 15D 15E 15F 15G 15H size 1 45 4 4 45 45 4 4 4 4 4 4 4 4 4 ρ1 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 linear of order 2 ρ3 1 -1 1 1 -i i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ4 1 -1 1 1 i -i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ5 4 0 4 4 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ6 4 0 1 -2 0 0 4 1 1 -2 1 1 -2 -2 -2 orthogonal lifted from C32⋊C4 ρ7 4 0 -2 1 0 0 4 -2 -2 1 -2 -2 1 1 1 orthogonal lifted from C32⋊C4 ρ8 4 0 -2 1 0 0 -1 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 orthogonal faithful ρ9 4 0 -2 1 0 0 -1 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 orthogonal faithful ρ10 4 0 1 -2 0 0 -1 ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 orthogonal faithful ρ11 4 0 -2 1 0 0 -1 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 orthogonal faithful ρ12 4 0 1 -2 0 0 -1 -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 orthogonal faithful ρ13 4 0 -2 1 0 0 -1 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 orthogonal faithful ρ14 4 0 1 -2 0 0 -1 -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 orthogonal faithful ρ15 4 0 1 -2 0 0 -1 -ζ32ζ54+ζ32ζ5-2ζ54-ζ5-1 -ζ3ζ53+ζ3ζ52-2ζ53-ζ52-1 2ζ32ζ54+2ζ32ζ52+ζ32+ζ54+ζ52+1 -ζ3ζ54+ζ3ζ5-2ζ54-ζ5-1 ζ3ζ53-ζ3ζ52-ζ53-2ζ52-1 2ζ3ζ54+2ζ3ζ53+ζ3+ζ54+ζ53+1 2ζ3ζ52+2ζ3ζ5+ζ3+ζ52+ζ5+1 2ζ3ζ54+2ζ3ζ52+ζ3+ζ54+ζ52+1 orthogonal faithful

Permutation representations of C32⋊F5
On 30 points - transitive group 30T46
Generators in S30
```(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)
(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 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)
(1 18)(2 20 5 16)(3 17 4 19)(6 28 11 23)(7 30 15 21)(8 27 14 24)(9 29 13 22)(10 26 12 25)```

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

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

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

`G:=TransitiveGroup(30,46);`

C32⋊F5 is a maximal subgroup of   C32⋊F5⋊C2
C32⋊F5 is a maximal quotient of   (C3×C6).F5

Matrix representation of C32⋊F5 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 8 45 0 0 16 52
,
 52 16 0 0 45 8 0 0 0 0 8 45 0 0 16 52
,
 43 60 0 0 1 0 0 0 0 0 18 18 0 0 43 60
,
 0 0 1 0 0 0 0 1 1 0 0 0 43 60 0 0
`G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,8,16,0,0,45,52],[52,45,0,0,16,8,0,0,0,0,8,16,0,0,45,52],[43,1,0,0,60,0,0,0,0,0,18,43,0,0,18,60],[0,0,1,43,0,0,0,60,1,0,0,0,0,1,0,0] >;`

C32⋊F5 in GAP, Magma, Sage, TeX

`C_3^2\rtimes F_5`
`% in TeX`

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

`G:=SmallGroup(180,25);`
`// by ID`

`G=gap.SmallGroup(180,25);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,3,-5,10,422,67,643,248,1804,1809]);`
`// Polycyclic`

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

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