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## G = C15⋊2F5order 300 = 22·3·52

### 2nd semidirect product of C15 and F5 acting via F5/C5=C4

Aliases: C152F5, C526Dic3, (C5×C15)⋊4C4, C3⋊(C52⋊C4), C52(C3⋊F5), C5⋊D5.3S3, (C3×C5⋊D5).2C2, SmallGroup(300,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C15 — C15⋊2F5
 Chief series C1 — C5 — C52 — C5×C15 — C3×C5⋊D5 — C15⋊2F5
 Lower central C5×C15 — C15⋊2F5
 Upper central C1

Generators and relations for C152F5
G = < a,b,c | a15=b5=c4=1, ab=ba, cac-1=a2, cbc-1=b3 >

25C2
2C5
2C5
75C4
25C6
5D5
5D5
10D5
10D5
2C15
2C15
25Dic3
15F5
15F5
10C3×D5
10C3×D5

Character table of C152F5

 class 1 2 3 4A 4B 5A 5B 5C 5D 5E 5F 6 15A 15B 15C 15D 15E 15F 15G 15H 15I 15J 15K 15L size 1 25 2 75 75 4 4 4 4 4 4 50 4 4 4 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 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 1 1 1 linear of order 2 ρ3 1 -1 1 -i i 1 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ4 1 -1 1 i -i 1 1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ5 2 2 -1 0 0 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 -2 -1 0 0 2 2 2 2 2 2 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ7 4 0 4 0 0 -1 -1 -1 4 -1 -1 0 -1 -1 -1 4 -1 -1 -1 -1 -1 -1 -1 4 orthogonal lifted from F5 ρ8 4 0 4 0 0 -1 -1 4 -1 -1 -1 0 -1 -1 -1 -1 -1 -1 -1 -1 4 -1 4 -1 orthogonal lifted from F5 ρ9 4 0 4 0 0 3+√5/2 -1-√5 -1 -1 -1+√5 3-√5/2 0 -1+√5 3+√5/2 -1-√5 -1 3-√5/2 -1-√5 -1+√5 3+√5/2 -1 3-√5/2 -1 -1 orthogonal lifted from C52⋊C4 ρ10 4 0 4 0 0 -1-√5 3-√5/2 -1 -1 3+√5/2 -1+√5 0 3+√5/2 -1-√5 3-√5/2 -1 -1+√5 3-√5/2 3+√5/2 -1-√5 -1 -1+√5 -1 -1 orthogonal lifted from C52⋊C4 ρ11 4 0 4 0 0 3-√5/2 -1+√5 -1 -1 -1-√5 3+√5/2 0 -1-√5 3-√5/2 -1+√5 -1 3+√5/2 -1+√5 -1-√5 3-√5/2 -1 3+√5/2 -1 -1 orthogonal lifted from C52⋊C4 ρ12 4 0 4 0 0 -1+√5 3+√5/2 -1 -1 3-√5/2 -1-√5 0 3-√5/2 -1+√5 3+√5/2 -1 -1-√5 3+√5/2 3-√5/2 -1+√5 -1 -1-√5 -1 -1 orthogonal lifted from C52⋊C4 ρ13 4 0 -2 0 0 3-√5/2 -1+√5 -1 -1 -1-√5 3+√5/2 0 1+√5/2 ζ32ζ53+ζ32ζ52-2ζ32-2 1-√5/2 1+√-15/2 ζ32ζ54+ζ32ζ5-2ζ32-2 1-√5/2 1+√5/2 ζ3ζ53+ζ3ζ52-2ζ3-2 1+√-15/2 ζ3ζ54+ζ3ζ5-2ζ3-2 1-√-15/2 1-√-15/2 complex faithful ρ14 4 0 -2 0 0 -1+√5 3+√5/2 -1 -1 3-√5/2 -1-√5 0 ζ3ζ53+ζ3ζ52-2ζ3-2 1-√5/2 ζ32ζ54+ζ32ζ5-2ζ32-2 1-√-15/2 1+√5/2 ζ3ζ54+ζ3ζ5-2ζ3-2 ζ32ζ53+ζ32ζ52-2ζ32-2 1-√5/2 1+√-15/2 1+√5/2 1-√-15/2 1+√-15/2 complex faithful ρ15 4 0 -2 0 0 3+√5/2 -1-√5 -1 -1 -1+√5 3-√5/2 0 1-√5/2 ζ3ζ54+ζ3ζ5-2ζ3-2 1+√5/2 1+√-15/2 ζ3ζ53+ζ3ζ52-2ζ3-2 1+√5/2 1-√5/2 ζ32ζ54+ζ32ζ5-2ζ32-2 1+√-15/2 ζ32ζ53+ζ32ζ52-2ζ32-2 1-√-15/2 1-√-15/2 complex faithful ρ16 4 0 -2 0 0 -1-√5 3-√5/2 -1 -1 3+√5/2 -1+√5 0 ζ3ζ54+ζ3ζ5-2ζ3-2 1+√5/2 ζ32ζ53+ζ32ζ52-2ζ32-2 1+√-15/2 1-√5/2 ζ3ζ53+ζ3ζ52-2ζ3-2 ζ32ζ54+ζ32ζ5-2ζ32-2 1+√5/2 1-√-15/2 1-√5/2 1+√-15/2 1-√-15/2 complex faithful ρ17 4 0 -2 0 0 -1 -1 -1 4 -1 -1 0 1+√-15/2 1-√-15/2 1+√-15/2 -2 1+√-15/2 1-√-15/2 1-√-15/2 1+√-15/2 1+√-15/2 1-√-15/2 1-√-15/2 -2 complex lifted from C3⋊F5 ρ18 4 0 -2 0 0 -1 -1 4 -1 -1 -1 0 1+√-15/2 1+√-15/2 1+√-15/2 1-√-15/2 1-√-15/2 1-√-15/2 1-√-15/2 1-√-15/2 -2 1+√-15/2 -2 1+√-15/2 complex lifted from C3⋊F5 ρ19 4 0 -2 0 0 3+√5/2 -1-√5 -1 -1 -1+√5 3-√5/2 0 1-√5/2 ζ32ζ54+ζ32ζ5-2ζ32-2 1+√5/2 1-√-15/2 ζ32ζ53+ζ32ζ52-2ζ32-2 1+√5/2 1-√5/2 ζ3ζ54+ζ3ζ5-2ζ3-2 1-√-15/2 ζ3ζ53+ζ3ζ52-2ζ3-2 1+√-15/2 1+√-15/2 complex faithful ρ20 4 0 -2 0 0 -1-√5 3-√5/2 -1 -1 3+√5/2 -1+√5 0 ζ32ζ54+ζ32ζ5-2ζ32-2 1+√5/2 ζ3ζ53+ζ3ζ52-2ζ3-2 1-√-15/2 1-√5/2 ζ32ζ53+ζ32ζ52-2ζ32-2 ζ3ζ54+ζ3ζ5-2ζ3-2 1+√5/2 1+√-15/2 1-√5/2 1-√-15/2 1+√-15/2 complex faithful ρ21 4 0 -2 0 0 -1 -1 4 -1 -1 -1 0 1-√-15/2 1-√-15/2 1-√-15/2 1+√-15/2 1+√-15/2 1+√-15/2 1+√-15/2 1+√-15/2 -2 1-√-15/2 -2 1-√-15/2 complex lifted from C3⋊F5 ρ22 4 0 -2 0 0 -1+√5 3+√5/2 -1 -1 3-√5/2 -1-√5 0 ζ32ζ53+ζ32ζ52-2ζ32-2 1-√5/2 ζ3ζ54+ζ3ζ5-2ζ3-2 1+√-15/2 1+√5/2 ζ32ζ54+ζ32ζ5-2ζ32-2 ζ3ζ53+ζ3ζ52-2ζ3-2 1-√5/2 1-√-15/2 1+√5/2 1+√-15/2 1-√-15/2 complex faithful ρ23 4 0 -2 0 0 3-√5/2 -1+√5 -1 -1 -1-√5 3+√5/2 0 1+√5/2 ζ3ζ53+ζ3ζ52-2ζ3-2 1-√5/2 1-√-15/2 ζ3ζ54+ζ3ζ5-2ζ3-2 1-√5/2 1+√5/2 ζ32ζ53+ζ32ζ52-2ζ32-2 1-√-15/2 ζ32ζ54+ζ32ζ5-2ζ32-2 1+√-15/2 1+√-15/2 complex faithful ρ24 4 0 -2 0 0 -1 -1 -1 4 -1 -1 0 1-√-15/2 1+√-15/2 1-√-15/2 -2 1-√-15/2 1+√-15/2 1+√-15/2 1-√-15/2 1-√-15/2 1+√-15/2 1+√-15/2 -2 complex lifted from C3⋊F5

Permutation representations of C152F5
On 30 points - transitive group 30T76
Generators in S30
```(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 4 7 10 13)(2 5 8 11 14)(3 6 9 12 15)(16 28 25 22 19)(17 29 26 23 20)(18 30 27 24 21)
(1 20)(2 28 5 22)(3 21 9 24)(4 29 13 26)(6 30)(7 23 10 17)(8 16 14 19)(11 25)(12 18 15 27)```

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

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

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

`G:=TransitiveGroup(30,76);`

Matrix representation of C152F5 in GL4(𝔽61) generated by

 0 53 40 2 55 8 26 39 0 0 38 13 0 0 48 0
,
 0 18 43 60 44 43 18 17 0 0 0 60 0 0 1 17
,
 17 1 0 1 60 43 0 0 16 43 1 1 60 44 0 0
`G:=sub<GL(4,GF(61))| [0,55,0,0,53,8,0,0,40,26,38,48,2,39,13,0],[0,44,0,0,18,43,0,0,43,18,0,1,60,17,60,17],[17,60,16,60,1,43,43,44,0,0,1,0,1,0,1,0] >;`

C152F5 in GAP, Magma, Sage, TeX

`C_{15}\rtimes_2F_5`
`% in TeX`

`G:=Group("C15:2F5");`
`// GroupNames label`

`G:=SmallGroup(300,35);`
`// by ID`

`G=gap.SmallGroup(300,35);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-5,-5,10,122,483,488,4504,3009]);`
`// Polycyclic`

`G:=Group<a,b,c|a^15=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^2,c*b*c^-1=b^3>;`
`// generators/relations`

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