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G = C20.31D10order 400 = 24·52

5th non-split extension by C20 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C20.31D10
 Chief series C1 — C5 — C52 — C5×C10 — C5×C20 — C5×C5⋊2C8 — C20.31D10
 Lower central C52 — C5×C10 — C20.31D10
 Upper central C1 — C4

Generators and relations for C20.31D10
G = < a,b,c | a20=c2=1, b10=a5, bab-1=cac=a9, cbc=a10b9 >

Subgroups: 364 in 56 conjugacy classes, 20 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C8, C2×C4, D5, C10, C10, M4(2), Dic5, C20, C20, D10, C52, C52C8, C40, C4×D5, C5⋊D5, C5×C10, C8⋊D5, C526C4, C5×C20, C2×C5⋊D5, C5×C52C8, C4×C5⋊D5, C20.31D10
Quotients: C1, C2, C4, C22, C2×C4, D5, M4(2), D10, C4×D5, C8⋊D5, D52, Dic52D5, C20.31D10

Smallest permutation representation of C20.31D10
On 40 points
Generators in S40
```(1 35 29 23 17 11 5 39 33 27 21 15 9 3 37 31 25 19 13 7)(2 28 14 40 26 12 38 24 10 36 22 8 34 20 6 32 18 4 30 16)
(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 5)(2 34)(3 23)(4 12)(6 30)(7 19)(9 37)(10 26)(11 15)(13 33)(14 22)(16 40)(17 29)(20 36)(21 25)(24 32)(27 39)(31 35)```

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

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

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

58 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A 5B 5C 5D 5E 5F 5G 5H 8A 8B 8C 8D 10A 10B 10C 10D 10E 10F 10G 10H 20A ··· 20H 20I ··· 20P 40A ··· 40P order 1 2 2 4 4 4 5 5 5 5 5 5 5 5 8 8 8 8 10 10 10 10 10 10 10 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 50 1 1 50 2 2 2 2 4 4 4 4 10 10 10 10 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 10 ··· 10

58 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + image C1 C2 C2 C4 C4 D5 M4(2) D10 C4×D5 C8⋊D5 D52 Dic5⋊2D5 C20.31D10 kernel C20.31D10 C5×C5⋊2C8 C4×C5⋊D5 C52⋊6C4 C2×C5⋊D5 C5⋊2C8 C52 C20 C10 C5 C4 C2 C1 # reps 1 2 1 2 2 4 2 4 8 16 4 4 8

Matrix representation of C20.31D10 in GL4(𝔽41) generated by

 9 0 0 0 0 9 0 0 0 0 0 1 0 0 40 34
,
 31 9 0 0 32 9 0 0 0 0 1 0 0 0 34 40
,
 1 7 0 0 0 40 0 0 0 0 1 0 0 0 34 40
`G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,0,40,0,0,1,34],[31,32,0,0,9,9,0,0,0,0,1,34,0,0,0,40],[1,0,0,0,7,40,0,0,0,0,1,34,0,0,0,40] >;`

C20.31D10 in GAP, Magma, Sage, TeX

`C_{20}._{31}D_{10}`
`% in TeX`

`G:=Group("C20.31D10");`
`// GroupNames label`

`G:=SmallGroup(400,63);`
`// by ID`

`G=gap.SmallGroup(400,63);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,31,50,970,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=c^2=1,b^10=a^5,b*a*b^-1=c*a*c=a^9,c*b*c=a^10*b^9>;`
`// generators/relations`

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