Copied to
clipboard

## G = C52⋊M4(2)  order 400 = 24·52

### The semidirect product of C52 and M4(2) acting faithfully

Aliases: C52⋊M4(2), D52.C4, C52⋊C8⋊C2, C52⋊C4.C4, D5⋊F5.2C2, C5⋊F5.1C22, C5⋊D5.2(C2×C4), SmallGroup(400,206)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C5⋊D5 — C52⋊M4(2)
 Chief series C1 — C52 — C5⋊D5 — C5⋊F5 — D5⋊F5 — C52⋊M4(2)
 Lower central C52 — C5⋊D5 — C52⋊M4(2)
 Upper central C1

Generators and relations for C52⋊M4(2)
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=b2, dad=a-1, cbc-1=a, bd=db, dcd=c5 >

10C2
25C2
2C5
4C5
25C22
25C4
25C4
2D5
10C10
10D5
20D5
25C8
25C2×C4
25C8
10F5
10F5
10D10
20F5
25M4(2)
10C2×F5

Character table of C52⋊M4(2)

 class 1 2A 2B 4A 4B 4C 5A 5B 8A 8B 8C 8D 10 size 1 10 25 25 25 50 8 16 50 50 50 50 40 ρ1 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 linear of order 2 ρ3 1 -1 1 1 1 -1 1 1 -1 1 -1 1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 linear of order 2 ρ5 1 -1 1 -1 -1 1 1 1 i i -i -i -1 linear of order 4 ρ6 1 1 1 -1 -1 -1 1 1 i -i -i i 1 linear of order 4 ρ7 1 1 1 -1 -1 -1 1 1 -i i i -i 1 linear of order 4 ρ8 1 -1 1 -1 -1 1 1 1 -i -i i i -1 linear of order 4 ρ9 2 0 -2 2i -2i 0 2 2 0 0 0 0 0 complex lifted from M4(2) ρ10 2 0 -2 -2i 2i 0 2 2 0 0 0 0 0 complex lifted from M4(2) ρ11 8 4 0 0 0 0 3 -2 0 0 0 0 -1 orthogonal faithful ρ12 8 -4 0 0 0 0 3 -2 0 0 0 0 1 orthogonal faithful ρ13 16 0 0 0 0 0 -4 1 0 0 0 0 0 orthogonal faithful

Permutation representations of C52⋊M4(2)
On 10 points - transitive group 10T28
Generators in S10
```(1 8 6 10 4)
(2 9 7 3 5)
(1 2)(3 4 5 6 7 8 9 10)
(4 8)(6 10)```

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

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

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

`G:=TransitiveGroup(10,28);`

On 20 points - transitive group 20T104
Generators in S20
```(2 5 18 14 9)(4 20 11 7 16)
(1 13 12 8 17)(3 6 19 15 10)
(1 2 3 4)(5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20)
(1 3)(2 4)(5 16)(6 13)(7 18)(8 15)(9 20)(10 17)(11 14)(12 19)```

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

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

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

`G:=TransitiveGroup(20,104);`

On 20 points - transitive group 20T107
Generators in S20
```(1 6 12 8 10)(2 11 9 5 7)(3 13 19 15 17)(4 20 18 14 16)
(1 12 10 6 8)(2 7 5 9 11)(3 13 19 15 17)(4 14 20 16 18)
(1 2)(3 4)(5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20)
(1 4)(2 3)(5 19)(6 16)(7 13)(8 18)(9 15)(10 20)(11 17)(12 14)```

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

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

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

`G:=TransitiveGroup(20,107);`

On 20 points - transitive group 20T109
Generators in S20
```(1 19 10 6 15)(3 12 17 13 8)
(2 20 11 7 16)(4 5 18 14 9)
(1 2 3 4)(5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20)
(6 10)(8 12)(13 17)(15 19)```

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

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

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

`G:=TransitiveGroup(20,109);`

On 20 points - transitive group 20T115
Generators in S20
```(1 16 14 18 20)(3 8 6 10 12)
(2 17 15 19 13)(4 9 7 11 5)
(1 2)(3 4)(5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20)
(1 3)(2 4)(5 13)(6 18)(7 15)(8 20)(9 17)(10 14)(11 19)(12 16)```

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

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

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

`G:=TransitiveGroup(20,115);`

On 25 points: primitive - transitive group 25T31
Generators in S25
```(1 7 5 9 3)(2 15 10 25 18)(4 12 20 17 19)(6 22 21 14 11)(8 23 13 24 16)
(1 8 6 2 4)(3 16 11 18 19)(5 13 21 10 20)(7 23 22 15 12)(9 24 14 25 17)
(2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17)(18 19 20 21 22 23 24 25)
(3 7)(5 9)(10 25)(11 22)(12 19)(13 24)(14 21)(15 18)(16 23)(17 20)```

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

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

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

`G:=TransitiveGroup(25,31);`

Polynomial with Galois group C52⋊M4(2) over ℚ
actionf(x)Disc(f)
10T28x10-6x9-45x8+136x7+1043x6+390x5-8071x4-22200x3-25799x2-14082x-2871222·32·58·232·432·794

Matrix representation of C52⋊M4(2) in GL8(ℤ)

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

`G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1],[0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,0,1,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,1,0] >;`

C52⋊M4(2) in GAP, Magma, Sage, TeX

`C_5^2\rtimes M_4(2)`
`% in TeX`

`G:=Group("C5^2:M4(2)");`
`// GroupNames label`

`G:=SmallGroup(400,206);`
`// by ID`

`G=gap.SmallGroup(400,206);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,5,48,121,31,50,964,1210,256,262,8645,587,1457,1463]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=b^2,d*a*d=a^-1,c*b*c^-1=a,b*d=d*b,d*c*d=c^5>;`
`// generators/relations`

Export

׿
×
𝔽