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## G = C102⋊4C4order 400 = 24·52

### 4th semidirect product of C102 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C102⋊4C4
 Chief series C1 — C5 — C52 — C5⋊D5 — C2×C5⋊D5 — C2×C52⋊C4 — C102⋊4C4
 Lower central C52 — C5×C10 — C102⋊4C4
 Upper central C1 — C2 — C22

Generators and relations for C1024C4
G = < a,b,c | a10=b10=c4=1, ab=ba, cac-1=a7b5, cbc-1=b3 >

Subgroups: 908 in 100 conjugacy classes, 20 normal (10 characteristic)
C1, C2, C2 [×4], C4 [×2], C22, C22 [×4], C5 [×2], C5 [×2], C2×C4 [×2], C23, D5 [×14], C10 [×2], C10 [×8], C22⋊C4, F5 [×4], D10 [×20], C2×C10 [×2], C2×C10 [×2], C52, C2×F5 [×4], C22×D5 [×4], C5⋊D5 [×2], C5⋊D5, C5×C10, C5×C10, C22⋊F5 [×2], C52⋊C4 [×2], C2×C5⋊D5 [×2], C2×C5⋊D5 [×2], C102, C2×C52⋊C4 [×2], C22×C5⋊D5, C1024C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, F5 [×2], C2×F5 [×2], C22⋊F5 [×2], C52⋊C4, C2×C52⋊C4, C1024C4

Permutation representations of C1024C4
On 20 points - transitive group 20T103
Generators in S20
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 6 4 9 2 7 5 10 3 8)(11 12 13 14 15 16 17 18 19 20)
(1 12)(2 20 5 14)(3 18 4 16)(6 19 8 15)(7 17)(9 13 10 11)```

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

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

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

`G:=TransitiveGroup(20,103);`

34 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A ··· 5F 10A ··· 10R order 1 2 2 2 2 2 4 4 4 4 5 ··· 5 10 ··· 10 size 1 1 2 25 25 50 50 50 50 50 4 ··· 4 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 2 4 4 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C4 C4 D4 F5 C2×F5 C22⋊F5 C52⋊C4 C2×C52⋊C4 C102⋊4C4 kernel C102⋊4C4 C2×C52⋊C4 C22×C5⋊D5 C2×C5⋊D5 C102 C5⋊D5 C2×C10 C10 C5 C22 C2 C1 # reps 1 2 1 2 2 2 2 2 4 4 4 8

Matrix representation of C1024C4 in GL4(𝔽41) generated by

 7 34 0 0 7 40 0 0 29 22 6 40 38 30 36 1
,
 1 34 0 0 7 34 0 0 1 35 6 40 6 39 36 1
,
 0 0 34 1 1 1 39 40 1 35 40 0 0 6 34 0
`G:=sub<GL(4,GF(41))| [7,7,29,38,34,40,22,30,0,0,6,36,0,0,40,1],[1,7,1,6,34,34,35,39,0,0,6,36,0,0,40,1],[0,1,1,0,0,1,35,6,34,39,40,34,1,40,0,0] >;`

C1024C4 in GAP, Magma, Sage, TeX

`C_{10}^2\rtimes_4C_4`
`% in TeX`

`G:=Group("C10^2:4C4");`
`// GroupNames label`

`G:=SmallGroup(400,162);`
`// by ID`

`G=gap.SmallGroup(400,162);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,1444,496,5765,2897]);`
`// Polycyclic`

`G:=Group<a,b,c|a^10=b^10=c^4=1,a*b=b*a,c*a*c^-1=a^7*b^5,c*b*c^-1=b^3>;`
`// generators/relations`

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