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## G = C23.1D10order 160 = 25·5

### 1st non-split extension by C23 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C23.1D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C2×C5⋊D4 — C23.1D10
 Lower central C5 — C10 — C2×C10 — C23.1D10
 Upper central C1 — C2 — C23 — C22⋊C4

Generators and relations for C23.1D10
G = < a,b,c,d | a2=b2=c20=1, d2=a, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=ac-1 >

Smallest permutation representation of C23.1D10
On 40 points
Generators in S40
```(1 30)(3 32)(5 34)(7 36)(9 38)(11 40)(13 22)(15 24)(17 26)(19 28)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)
(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 24 30 15)(2 23)(3 13 32 22)(4 12)(5 40 34 11)(6 39)(7 9 36 38)(10 35)(14 31)(16 20)(17 28 26 19)(18 27)(21 33)(25 29)```

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

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

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

31 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H order 1 2 2 2 2 2 4 4 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 size 1 1 2 2 2 20 4 4 20 20 20 2 2 2 ··· 2 4 4 4 4 4 ··· 4

31 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 D5 D10 C4×D5 D20 C5⋊D4 C23⋊C4 C23.1D10 kernel C23.1D10 C23.D5 C5×C22⋊C4 C2×C5⋊D4 C2×Dic5 C22×D5 C2×C10 C22⋊C4 C23 C22 C22 C22 C5 C1 # reps 1 1 1 1 2 2 2 2 2 4 4 4 1 4

Matrix representation of C23.1D10 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 0 0 0 1 0 0 40 6 35 18 0 0 23 20 0 0
,
 35 18 0 0 23 6 0 0 0 0 0 1 0 0 1 0
`G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,0,35,23,0,0,18,20,0,40,0,0,1,6,0,0],[35,23,0,0,18,6,0,0,0,0,0,1,0,0,1,0] >;`

C23.1D10 in GAP, Magma, Sage, TeX

`C_2^3._1D_{10}`
`% in TeX`

`G:=Group("C2^3.1D10");`
`// GroupNames label`

`G:=SmallGroup(160,13);`
`// by ID`

`G=gap.SmallGroup(160,13);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,121,31,362,297,4613]);`
`// Polycyclic`

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

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