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G = C52⋊3C42order 400 = 24·52

2nd semidirect product of C52 and C42 acting via C42/C2=C2×C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C52⋊3C42
 Chief series C1 — C5 — C52 — C5⋊D5 — C2×C5⋊D5 — C2×C5⋊F5 — C52⋊3C42
 Lower central C52 — C52⋊3C42
 Upper central C1 — C2

Generators and relations for C523C42
G = < a,b,c,d | a5=b5=c4=d4=1, ab=ba, cac-1=a2, dad-1=a-1, cbc-1=b2, bd=db, cd=dc >

Subgroups: 556 in 76 conjugacy classes, 23 normal (11 characteristic)
C1, C2, C2, C4, C22, C5, C5, C2×C4, D5, C10, C10, C42, Dic5, C20, F5, D10, C52, C4×D5, C2×F5, C5⋊D5, C5×C10, C4×F5, C5×Dic5, C5⋊F5, C52⋊C4, C2×C5⋊D5, Dic52D5, C2×C5⋊F5, C2×C52⋊C4, C523C42
Quotients: C1, C2, C4, C22, C2×C4, C42, F5, C2×F5, C4×F5, D5⋊F5, C523C42

Character table of C523C42

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 5C 5D 10A 10B 10C 10D 20A 20B 20C 20D size 1 1 25 25 5 5 5 5 25 25 25 25 25 25 25 25 4 4 8 8 4 4 8 8 20 20 20 20 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 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 1 1 1 1 linear of order 2 ρ3 1 1 1 1 -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 ρ4 1 1 1 1 -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 ρ5 1 -1 1 -1 i -i -i i -i -1 1 -1 1 i -i i 1 1 1 1 -1 -1 -1 -1 -i i -i i linear of order 4 ρ6 1 1 -1 -1 -1 -1 1 1 -i -i -i i i i i -i 1 1 1 1 1 1 1 1 -1 1 1 -1 linear of order 4 ρ7 1 -1 -1 1 i -i i -i -1 -i i i -i -1 1 1 1 1 1 1 -1 -1 -1 -1 -i -i i i linear of order 4 ρ8 1 1 -1 -1 -1 -1 1 1 i i i -i -i -i -i i 1 1 1 1 1 1 1 1 -1 1 1 -1 linear of order 4 ρ9 1 -1 1 -1 i -i -i i i 1 -1 1 -1 -i i -i 1 1 1 1 -1 -1 -1 -1 -i i -i i linear of order 4 ρ10 1 -1 -1 1 i -i i -i 1 i -i -i i 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -i -i i i linear of order 4 ρ11 1 1 -1 -1 1 1 -1 -1 -i i i -i -i i i -i 1 1 1 1 1 1 1 1 1 -1 -1 1 linear of order 4 ρ12 1 -1 1 -1 -i i i -i -i 1 -1 1 -1 i -i i 1 1 1 1 -1 -1 -1 -1 i -i i -i linear of order 4 ρ13 1 -1 -1 1 -i i -i i -1 i -i -i i -1 1 1 1 1 1 1 -1 -1 -1 -1 i i -i -i linear of order 4 ρ14 1 -1 1 -1 -i i i -i i -1 1 -1 1 -i i -i 1 1 1 1 -1 -1 -1 -1 i -i i -i linear of order 4 ρ15 1 1 -1 -1 1 1 -1 -1 i -i -i i i -i -i i 1 1 1 1 1 1 1 1 1 -1 -1 1 linear of order 4 ρ16 1 -1 -1 1 -i i -i i 1 -i i i -i 1 -1 -1 1 1 1 1 -1 -1 -1 -1 i i -i -i linear of order 4 ρ17 4 4 0 0 0 0 4 4 0 0 0 0 0 0 0 0 4 -1 -1 -1 -1 4 -1 -1 0 -1 -1 0 orthogonal lifted from F5 ρ18 4 4 0 0 0 0 -4 -4 0 0 0 0 0 0 0 0 4 -1 -1 -1 -1 4 -1 -1 0 1 1 0 orthogonal lifted from C2×F5 ρ19 4 4 0 0 4 4 0 0 0 0 0 0 0 0 0 0 -1 4 -1 -1 4 -1 -1 -1 -1 0 0 -1 orthogonal lifted from F5 ρ20 4 4 0 0 -4 -4 0 0 0 0 0 0 0 0 0 0 -1 4 -1 -1 4 -1 -1 -1 1 0 0 1 orthogonal lifted from C2×F5 ρ21 4 -4 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 4 -1 -1 -1 1 -4 1 1 0 i -i 0 complex lifted from C4×F5 ρ22 4 -4 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 -1 4 -1 -1 -4 1 1 1 -i 0 0 i complex lifted from C4×F5 ρ23 4 -4 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 -1 4 -1 -1 -4 1 1 1 i 0 0 -i complex lifted from C4×F5 ρ24 4 -4 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 4 -1 -1 -1 1 -4 1 1 0 -i i 0 complex lifted from C4×F5 ρ25 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 3 -2 2 2 2 -3 0 0 0 0 orthogonal faithful ρ26 8 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 3 -2 -2 -2 -2 3 0 0 0 0 orthogonal lifted from D5⋊F5 ρ27 8 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 -2 3 -2 -2 3 -2 0 0 0 0 orthogonal lifted from D5⋊F5 ρ28 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 -2 3 2 2 -3 2 0 0 0 0 orthogonal faithful

Permutation representations of C523C42
On 20 points - transitive group 20T91
Generators in S20
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 5 4 3 2)(6 10 9 8 7)(11 12 13 14 15)(16 17 18 19 20)
(1 6)(2 9 5 8)(3 7 4 10)(11 20 12 18)(13 16 15 17)(14 19)
(1 19 6 14)(2 18 7 13)(3 17 8 12)(4 16 9 11)(5 20 10 15)```

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

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

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

`G:=TransitiveGroup(20,91);`

Matrix representation of C523C42 in GL8(ℤ)

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

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

C523C42 in GAP, Magma, Sage, TeX

`C_5^2\rtimes_3C_4^2`
`% in TeX`

`G:=Group("C5^2:3C4^2");`
`// GroupNames label`

`G:=SmallGroup(400,124);`
`// by ID`

`G=gap.SmallGroup(400,124);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,1444,970,496,8645,2897]);`
`// Polycyclic`

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

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