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## G = C34⋊C5order 405 = 34·5

### The semidirect product of C34 and C5 acting faithfully

Aliases: C34⋊C5, SmallGroup(405,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C34 — C34⋊C5
 Chief series C1 — C34 — C34⋊C5
 Lower central C34 — C34⋊C5
 Upper central C1

Generators and relations for C34⋊C5
G = < a,b,c,d,e | a3=b3=c3=d3=e5=1, ab=ba, ac=ca, ad=da, eae-1=a-1b-1c, bc=cb, bd=db, ebe-1=a, cd=dc, ece-1=a-1bd-1, ede-1=b-1c-1 >

Subgroups: 294 in 46 conjugacy classes, 3 normal (all characteristic)
C1, C3, C5, C32, C33, C34, C34⋊C5
Quotients: C1, C5, C34⋊C5

Character table of C34⋊C5

 class 1 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 3N 3O 3P 5A 5B 5C 5D size 1 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 81 81 81 81 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 ζ5 ζ52 ζ53 ζ54 linear of order 5 ρ3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ53 ζ5 ζ54 ζ52 linear of order 5 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ52 ζ54 ζ5 ζ53 linear of order 5 ρ5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ζ54 ζ53 ζ52 ζ5 linear of order 5 ρ6 5 -5-3√-3/2 1-3√-3/2 2 -1 1-3√-3/2 2 1+3√-3/2 -1 -5+3√-3/2 1+3√-3/2 -1 2 -1 2 -1 -1 0 0 0 0 complex faithful ρ7 5 1-3√-3/2 -5+3√-3/2 -1 2 -1 -1 -1 2 1+3√-3/2 -5-3√-3/2 1-3√-3/2 2 1+3√-3/2 2 -1 -1 0 0 0 0 complex faithful ρ8 5 1-3√-3/2 -1 2 -1 -5+3√-3/2 2 -5-3√-3/2 -1 1+3√-3/2 -1 1+3√-3/2 -1 1-3√-3/2 -1 2 2 0 0 0 0 complex faithful ρ9 5 1+3√-3/2 -5-3√-3/2 -1 2 -1 -1 -1 2 1-3√-3/2 -5+3√-3/2 1+3√-3/2 2 1-3√-3/2 2 -1 -1 0 0 0 0 complex faithful ρ10 5 -1 2 -5-3√-3/2 -1 -1 -5+3√-3/2 -1 -1 -1 2 2 1+3√-3/2 2 1-3√-3/2 1+3√-3/2 1-3√-3/2 0 0 0 0 complex faithful ρ11 5 -1 1+3√-3/2 -1 2 1-3√-3/2 -1 1+3√-3/2 2 -1 1-3√-3/2 -5+3√-3/2 -1 -5-3√-3/2 -1 2 2 0 0 0 0 complex faithful ρ12 5 2 2 1-3√-3/2 1-3√-3/2 -1 1+3√-3/2 -1 1+3√-3/2 2 2 -1 -1 -1 -1 -5-3√-3/2 -5+3√-3/2 0 0 0 0 complex faithful ρ13 5 -1 -1 1-3√-3/2 1+3√-3/2 2 1+3√-3/2 2 1-3√-3/2 -1 -1 2 -5-3√-3/2 2 -5+3√-3/2 -1 -1 0 0 0 0 complex faithful ρ14 5 -5+3√-3/2 1+3√-3/2 2 -1 1+3√-3/2 2 1-3√-3/2 -1 -5-3√-3/2 1-3√-3/2 -1 2 -1 2 -1 -1 0 0 0 0 complex faithful ρ15 5 1+3√-3/2 -1 2 -1 -5-3√-3/2 2 -5+3√-3/2 -1 1-3√-3/2 -1 1-3√-3/2 -1 1+3√-3/2 -1 2 2 0 0 0 0 complex faithful ρ16 5 -1 2 -5+3√-3/2 -1 -1 -5-3√-3/2 -1 -1 -1 2 2 1-3√-3/2 2 1+3√-3/2 1-3√-3/2 1+3√-3/2 0 0 0 0 complex faithful ρ17 5 2 -1 -1 -5+3√-3/2 2 -1 2 -5-3√-3/2 2 -1 -1 1+3√-3/2 -1 1-3√-3/2 1-3√-3/2 1+3√-3/2 0 0 0 0 complex faithful ρ18 5 2 -1 -1 -5-3√-3/2 2 -1 2 -5+3√-3/2 2 -1 -1 1-3√-3/2 -1 1+3√-3/2 1+3√-3/2 1-3√-3/2 0 0 0 0 complex faithful ρ19 5 -1 1-3√-3/2 -1 2 1+3√-3/2 -1 1-3√-3/2 2 -1 1+3√-3/2 -5-3√-3/2 -1 -5+3√-3/2 -1 2 2 0 0 0 0 complex faithful ρ20 5 -1 -1 1+3√-3/2 1-3√-3/2 2 1-3√-3/2 2 1+3√-3/2 -1 -1 2 -5+3√-3/2 2 -5-3√-3/2 -1 -1 0 0 0 0 complex faithful ρ21 5 2 2 1+3√-3/2 1+3√-3/2 -1 1-3√-3/2 -1 1-3√-3/2 2 2 -1 -1 -1 -1 -5+3√-3/2 -5-3√-3/2 0 0 0 0 complex faithful

Permutation representations of C34⋊C5
On 15 points - transitive group 15T26
Generators in S15
```(1 9 12)(2 13 10)(3 6 14)(4 7 15)(5 8 11)
(1 9 12)(2 10 13)(3 14 6)(4 7 15)(5 8 11)
(1 9 12)(2 10 13)(3 6 14)
(1 12 9)(2 10 13)(3 6 14)(5 11 8)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)```

`G:=sub<Sym(15)| (1,9,12)(2,13,10)(3,6,14)(4,7,15)(5,8,11), (1,9,12)(2,10,13)(3,14,6)(4,7,15)(5,8,11), (1,9,12)(2,10,13)(3,6,14), (1,12,9)(2,10,13)(3,6,14)(5,11,8), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)>;`

`G:=Group( (1,9,12)(2,13,10)(3,6,14)(4,7,15)(5,8,11), (1,9,12)(2,10,13)(3,14,6)(4,7,15)(5,8,11), (1,9,12)(2,10,13)(3,6,14), (1,12,9)(2,10,13)(3,6,14)(5,11,8), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15) );`

`G=PermutationGroup([[(1,9,12),(2,13,10),(3,6,14),(4,7,15),(5,8,11)], [(1,9,12),(2,10,13),(3,14,6),(4,7,15),(5,8,11)], [(1,9,12),(2,10,13),(3,6,14)], [(1,12,9),(2,10,13),(3,6,14),(5,11,8)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15)]])`

`G:=TransitiveGroup(15,26);`

Polynomial with Galois group C34⋊C5 over ℚ
actionf(x)Disc(f)
15T26x15-150x13-520x12+2400x11+12366x10-1700x9-73410x8-60675x7+161150x6+214578x5-119280x4-247825x3-3750x2+93525x+23255320·524·716·432·1512·34572·54072·159438609894012

Matrix representation of C34⋊C5 in GL5(𝔽31)

 5 0 0 0 0 0 5 0 0 0 0 0 5 0 0 0 0 0 5 0 4 30 3 0 25
,
 5 0 0 0 0 0 5 0 0 0 0 0 5 0 0 12 23 2 25 0 0 0 0 0 5
,
 5 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 17 19 5 0 0 6 13 0 5
,
 25 0 0 0 0 0 25 0 0 0 0 0 1 0 0 19 8 19 5 0 27 1 13 0 5
,
 0 1 0 0 0 0 0 1 0 0 2 9 21 24 0 0 0 0 10 1 0 0 0 15 0

`G:=sub<GL(5,GF(31))| [5,0,0,0,4,0,5,0,0,30,0,0,5,0,3,0,0,0,5,0,0,0,0,0,25],[5,0,0,12,0,0,5,0,23,0,0,0,5,2,0,0,0,0,25,0,0,0,0,0,5],[5,0,0,0,0,0,1,0,17,6,0,0,1,19,13,0,0,0,5,0,0,0,0,0,5],[25,0,0,19,27,0,25,0,8,1,0,0,1,19,13,0,0,0,5,0,0,0,0,0,5],[0,0,2,0,0,1,0,9,0,0,0,1,21,0,0,0,0,24,10,15,0,0,0,1,0] >;`

C34⋊C5 in GAP, Magma, Sage, TeX

`C_3^4\rtimes C_5`
`% in TeX`

`G:=Group("C3^4:C5");`
`// GroupNames label`

`G:=SmallGroup(405,15);`
`// by ID`

`G=gap.SmallGroup(405,15);`
`# by ID`

`G:=PCGroup([5,-5,-3,3,3,3,3751,827,3303,9129]);`
`// Polycyclic`

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

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