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## G = C24⋊C5order 80 = 24·5

### The semidirect product of C24 and C5 acting faithfully

Aliases: C24⋊C5, SmallGroup(80,49)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C24⋊C5
 Chief series C1 — C24 — C24⋊C5
 Lower central C24 — C24⋊C5
 Upper central C1

Generators and relations for C24⋊C5
G = < a,b,c,d,e | a2=b2=c2=d2=e5=1, ab=ba, ac=ca, ad=da, eae-1=abc, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, ede-1=a >

5C2
5C2
5C2
16C5
5C22
5C22
5C22
5C22
5C22
5C22
5C22
5C23
5C23
5C23

Character table of C24⋊C5

 class 1 2A 2B 2C 5A 5B 5C 5D size 1 5 5 5 16 16 16 16 ρ1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 ζ52 ζ5 ζ54 ζ53 linear of order 5 ρ3 1 1 1 1 ζ53 ζ54 ζ5 ζ52 linear of order 5 ρ4 1 1 1 1 ζ54 ζ52 ζ53 ζ5 linear of order 5 ρ5 1 1 1 1 ζ5 ζ53 ζ52 ζ54 linear of order 5 ρ6 5 -3 1 1 0 0 0 0 orthogonal faithful ρ7 5 1 -3 1 0 0 0 0 orthogonal faithful ρ8 5 1 1 -3 0 0 0 0 orthogonal faithful

Permutation representations of C24⋊C5
On 10 points - transitive group 10T8
Generators in S10
```(2 9)(3 10)(4 6)(5 7)
(4 6)(5 7)
(1 8)(4 6)
(1 8)(3 10)(4 6)(5 7)
(1 2 3 4 5)(6 7 8 9 10)```

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

`G:=Group( (2,9)(3,10)(4,6)(5,7), (4,6)(5,7), (1,8)(4,6), (1,8)(3,10)(4,6)(5,7), (1,2,3,4,5)(6,7,8,9,10) );`

`G=PermutationGroup([[(2,9),(3,10),(4,6),(5,7)], [(4,6),(5,7)], [(1,8),(4,6)], [(1,8),(3,10),(4,6),(5,7)], [(1,2,3,4,5),(6,7,8,9,10)]])`

`G:=TransitiveGroup(10,8);`

On 16 points: primitive - transitive group 16T178
Generators in S16
```(1 9)(2 12)(3 11)(4 5)(6 7)(8 14)(10 15)(13 16)
(1 13)(2 4)(3 10)(5 12)(6 14)(7 8)(9 16)(11 15)
(1 6)(2 11)(3 12)(4 15)(5 10)(7 9)(8 16)(13 14)
(1 10)(2 8)(3 13)(4 7)(5 6)(9 15)(11 16)(12 14)
(2 3 4 5 6)(7 8 9 10 11)(12 13 14 15 16)```

`G:=sub<Sym(16)| (1,9)(2,12)(3,11)(4,5)(6,7)(8,14)(10,15)(13,16), (1,13)(2,4)(3,10)(5,12)(6,14)(7,8)(9,16)(11,15), (1,6)(2,11)(3,12)(4,15)(5,10)(7,9)(8,16)(13,14), (1,10)(2,8)(3,13)(4,7)(5,6)(9,15)(11,16)(12,14), (2,3,4,5,6)(7,8,9,10,11)(12,13,14,15,16)>;`

`G:=Group( (1,9)(2,12)(3,11)(4,5)(6,7)(8,14)(10,15)(13,16), (1,13)(2,4)(3,10)(5,12)(6,14)(7,8)(9,16)(11,15), (1,6)(2,11)(3,12)(4,15)(5,10)(7,9)(8,16)(13,14), (1,10)(2,8)(3,13)(4,7)(5,6)(9,15)(11,16)(12,14), (2,3,4,5,6)(7,8,9,10,11)(12,13,14,15,16) );`

`G=PermutationGroup([[(1,9),(2,12),(3,11),(4,5),(6,7),(8,14),(10,15),(13,16)], [(1,13),(2,4),(3,10),(5,12),(6,14),(7,8),(9,16),(11,15)], [(1,6),(2,11),(3,12),(4,15),(5,10),(7,9),(8,16),(13,14)], [(1,10),(2,8),(3,13),(4,7),(5,6),(9,15),(11,16),(12,14)], [(2,3,4,5,6),(7,8,9,10,11),(12,13,14,15,16)]])`

`G:=TransitiveGroup(16,178);`

On 20 points - transitive group 20T17
Generators in S20
```(1 7)(2 8)(3 18)(4 10)(5 14)(6 20)(9 12)(11 17)(13 19)(15 16)
(1 15)(2 11)(3 18)(4 19)(7 16)(8 17)(9 12)(10 13)
(1 15)(3 9)(5 20)(6 14)(7 16)(12 18)
(1 15)(2 8)(3 9)(4 19)(5 6)(7 16)(10 13)(11 17)(12 18)(14 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)```

`G:=sub<Sym(20)| (1,7)(2,8)(3,18)(4,10)(5,14)(6,20)(9,12)(11,17)(13,19)(15,16), (1,15)(2,11)(3,18)(4,19)(7,16)(8,17)(9,12)(10,13), (1,15)(3,9)(5,20)(6,14)(7,16)(12,18), (1,15)(2,8)(3,9)(4,19)(5,6)(7,16)(10,13)(11,17)(12,18)(14,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)>;`

`G:=Group( (1,7)(2,8)(3,18)(4,10)(5,14)(6,20)(9,12)(11,17)(13,19)(15,16), (1,15)(2,11)(3,18)(4,19)(7,16)(8,17)(9,12)(10,13), (1,15)(3,9)(5,20)(6,14)(7,16)(12,18), (1,15)(2,8)(3,9)(4,19)(5,6)(7,16)(10,13)(11,17)(12,18)(14,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20) );`

`G=PermutationGroup([[(1,7),(2,8),(3,18),(4,10),(5,14),(6,20),(9,12),(11,17),(13,19),(15,16)], [(1,15),(2,11),(3,18),(4,19),(7,16),(8,17),(9,12),(10,13)], [(1,15),(3,9),(5,20),(6,14),(7,16),(12,18)], [(1,15),(2,8),(3,9),(4,19),(5,6),(7,16),(10,13),(11,17),(12,18),(14,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)]])`

`G:=TransitiveGroup(20,17);`

On 20 points - transitive group 20T23
Generators in S20
```(2 7)(3 13)(4 14)(5 10)(8 18)(9 19)(12 17)(15 20)
(1 16)(3 18)(4 9)(5 10)(6 11)(8 13)(14 19)(15 20)
(1 11)(2 17)(3 18)(4 14)(6 16)(7 12)(8 13)(9 19)
(1 6)(3 8)(4 14)(5 15)(9 19)(10 20)(11 16)(13 18)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)```

`G:=sub<Sym(20)| (2,7)(3,13)(4,14)(5,10)(8,18)(9,19)(12,17)(15,20), (1,16)(3,18)(4,9)(5,10)(6,11)(8,13)(14,19)(15,20), (1,11)(2,17)(3,18)(4,14)(6,16)(7,12)(8,13)(9,19), (1,6)(3,8)(4,14)(5,15)(9,19)(10,20)(11,16)(13,18), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)>;`

`G:=Group( (2,7)(3,13)(4,14)(5,10)(8,18)(9,19)(12,17)(15,20), (1,16)(3,18)(4,9)(5,10)(6,11)(8,13)(14,19)(15,20), (1,11)(2,17)(3,18)(4,14)(6,16)(7,12)(8,13)(9,19), (1,6)(3,8)(4,14)(5,15)(9,19)(10,20)(11,16)(13,18), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20) );`

`G=PermutationGroup([[(2,7),(3,13),(4,14),(5,10),(8,18),(9,19),(12,17),(15,20)], [(1,16),(3,18),(4,9),(5,10),(6,11),(8,13),(14,19),(15,20)], [(1,11),(2,17),(3,18),(4,14),(6,16),(7,12),(8,13),(9,19)], [(1,6),(3,8),(4,14),(5,15),(9,19),(10,20),(11,16),(13,18)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)]])`

`G:=TransitiveGroup(20,23);`

C24⋊C5 is a maximal subgroup of   C24⋊D5  F16
C24⋊C5 is a maximal quotient of   2- 1+4⋊C5  C24⋊C25

Polynomial with Galois group C24⋊C5 over ℚ
actionf(x)Disc(f)
10T8x10-20x8+149x6-519x4+851x2-529210·118·236

Matrix representation of C24⋊C5 in GL5(ℤ)

 1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 1
,
 -1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 1
,
 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0

`G:=sub<GL(5,Integers())| [1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1],[-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1],[-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1],[0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0] >;`

C24⋊C5 in GAP, Magma, Sage, TeX

`C_2^4\rtimes C_5`
`% in TeX`

`G:=Group("C2^4:C5");`
`// GroupNames label`

`G:=SmallGroup(80,49);`
`// by ID`

`G=gap.SmallGroup(80,49);`
`# by ID`

`G:=PCGroup([5,-5,-2,2,2,2,401,677,1103,1879]);`
`// Polycyclic`

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

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