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## G = C10.D20order 400 = 24·52

### 6th non-split extension by C10 of D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C10.D20
 Chief series C1 — C5 — C52 — C5×C10 — C102 — C10×Dic5 — C10.D20
 Lower central C52 — C5×C10 — C10.D20
 Upper central C1 — C22

Generators and relations for C10.D20
G = < a,b,c | a10=b20=c2=1, bab-1=cac=a-1, cbc=a5b-1 >

Subgroups: 764 in 100 conjugacy classes, 28 normal (8 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C5, C2×C4, C23, D5, C10, C10, C22⋊C4, Dic5, C20, D10, C2×C10, C2×C10, C52, C2×Dic5, C2×C20, C22×D5, C5⋊D5, C5×C10, C5×C10, D10⋊C4, C5×Dic5, C2×C5⋊D5, C2×C5⋊D5, C102, C10×Dic5, C22×C5⋊D5, C10.D20
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D10, C4×D5, D20, C5⋊D4, D10⋊C4, D52, Dic52D5, C5⋊D20, C10.D20

Smallest permutation representation of C10.D20
On 40 points
Generators in S40
```(1 30 13 22 5 34 17 26 9 38)(2 39 10 27 18 35 6 23 14 31)(3 32 15 24 7 36 19 28 11 40)(4 21 12 29 20 37 8 25 16 33)
(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 15)(2 27)(3 13)(4 25)(5 11)(6 23)(7 9)(8 21)(10 39)(12 37)(14 35)(16 33)(17 19)(18 31)(20 29)(22 40)(24 38)(26 36)(28 34)(30 32)```

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

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

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

58 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 5C 5D 5E 5F 5G 5H 10A ··· 10L 10M ··· 10X 20A ··· 20P order 1 2 2 2 2 2 4 4 4 4 5 5 5 5 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 50 50 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 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + image C1 C2 C2 C4 D4 D5 D10 C4×D5 D20 C5⋊D4 D52 Dic5⋊2D5 C5⋊D20 kernel C10.D20 C10×Dic5 C22×C5⋊D5 C2×C5⋊D5 C5×C10 C2×Dic5 C2×C10 C10 C10 C10 C22 C2 C2 # reps 1 2 1 4 2 4 4 8 8 8 4 4 8

Matrix representation of C10.D20 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 6 0 0 0 0 35 35
,
 1 39 0 0 0 0 0 40 0 0 0 0 0 0 25 39 0 0 0 0 2 13 0 0 0 0 0 0 1 0 0 0 0 0 6 40
,
 1 0 0 0 0 0 1 40 0 0 0 0 0 0 35 1 0 0 0 0 6 6 0 0 0 0 0 0 1 0 0 0 0 0 6 40

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,35,0,0,0,0,6,35],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,25,2,0,0,0,0,39,13,0,0,0,0,0,0,1,6,0,0,0,0,0,40],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,1,6,0,0,0,0,0,0,1,6,0,0,0,0,0,40] >;`

C10.D20 in GAP, Magma, Sage, TeX

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

`G:=Group("C10.D20");`
`// GroupNames label`

`G:=SmallGroup(400,73);`
`// by ID`

`G=gap.SmallGroup(400,73);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,79,970,11525]);`
`// Polycyclic`

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

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