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## G = C102.C4order 400 = 24·52

### 4th non-split extension by C102 of C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C102.C4
 Chief series C1 — C5 — C52 — C5×C10 — C5×Dic5 — C52⋊3C8 — C102.C4
 Lower central C52 — C5×C10 — C102.C4
 Upper central C1 — C2 — C22

Generators and relations for C102.C4
G = < a,b,c | a10=b10=1, c4=b5, ab=ba, cac-1=a-1b5, cbc-1=b7 >

Smallest permutation representation of C102.C4
On 40 points
Generators in S40
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)
(1 6 4 9 2 7 5 10 3 8)(11 20 13 17 15 19 12 16 14 18)(21 24 27 30 23 26 29 22 25 28)(31 38 35 32 39 36 33 40 37 34)
(1 38 20 29 7 33 12 24)(2 32 16 23 8 37 13 28)(3 36 17 27 9 31 14 22)(4 40 18 21 10 35 15 26)(5 34 19 25 6 39 11 30)```

`G:=sub<Sym(40)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40), (1,6,4,9,2,7,5,10,3,8)(11,20,13,17,15,19,12,16,14,18)(21,24,27,30,23,26,29,22,25,28)(31,38,35,32,39,36,33,40,37,34), (1,38,20,29,7,33,12,24)(2,32,16,23,8,37,13,28)(3,36,17,27,9,31,14,22)(4,40,18,21,10,35,15,26)(5,34,19,25,6,39,11,30)>;`

`G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40), (1,6,4,9,2,7,5,10,3,8)(11,20,13,17,15,19,12,16,14,18)(21,24,27,30,23,26,29,22,25,28)(31,38,35,32,39,36,33,40,37,34), (1,38,20,29,7,33,12,24)(2,32,16,23,8,37,13,28)(3,36,17,27,9,31,14,22)(4,40,18,21,10,35,15,26)(5,34,19,25,6,39,11,30) );`

`G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40)], [(1,6,4,9,2,7,5,10,3,8),(11,20,13,17,15,19,12,16,14,18),(21,24,27,30,23,26,29,22,25,28),(31,38,35,32,39,36,33,40,37,34)], [(1,38,20,29,7,33,12,24),(2,32,16,23,8,37,13,28),(3,36,17,27,9,31,14,22),(4,40,18,21,10,35,15,26),(5,34,19,25,6,39,11,30)]])`

46 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A 5B 5C ··· 5G 8A 8B 8C 8D 10A ··· 10F 10G ··· 10U 20A ··· 20H order 1 2 2 4 4 4 5 5 5 ··· 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 2 5 5 10 2 2 4 ··· 4 50 50 50 50 2 ··· 2 4 ··· 4 10 ··· 10

46 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + - + - + + - image C1 C2 C2 C4 C4 D5 M4(2) Dic5 D10 Dic5 C4.Dic5 F5 C2×F5 C22.F5 D5.D5 C2×D5.D5 C102.C4 kernel C102.C4 C52⋊3C8 C10×Dic5 C5×Dic5 C102 C2×Dic5 C52 Dic5 Dic5 C2×C10 C5 C2×C10 C10 C5 C22 C2 C1 # reps 1 2 1 2 2 2 2 2 2 2 8 1 1 2 4 4 8

Matrix representation of C102.C4 in GL6(𝔽41)

 16 0 0 0 0 0 0 23 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 16 0 0 0 0 0 0 18 0 0 0 0 0 0 37 0 0 0 0 0 0 10
,
 0 4 0 0 0 0 8 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0

`G:=sub<GL(6,GF(41))| [16,0,0,0,0,0,0,23,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,16,0,0,0,0,0,0,18,0,0,0,0,0,0,37,0,0,0,0,0,0,10],[0,8,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C102.C4 in GAP, Magma, Sage, TeX

`C_{10}^2.C_4`
`% in TeX`

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

`G:=SmallGroup(400,147);`
`// by ID`

`G=gap.SmallGroup(400,147);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,50,1924,8645,2897]);`
`// Polycyclic`

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

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