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## G = D10.C23order 160 = 25·5

### 11st non-split extension by D10 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D10.C23
 Chief series C1 — C5 — D5 — D10 — C2×F5 — C4×F5 — D10.C23
 Lower central C5 — C10 — D10.C23
 Upper central C1 — C4 — C2×C4

Generators and relations for D10.C23
G = < a,b,c,d,e | a10=b2=e2=1, c2=a-1b, d2=a5, bab=a-1, cac-1=a3, ad=da, ae=ea, cbc-1=a2b, bd=db, be=eb, cd=dc, ece=a5c, de=ed >

Subgroups: 244 in 76 conjugacy classes, 36 normal (24 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×6], C22, C22 [×4], C5, C2×C4, C2×C4 [×9], C23, D5 [×2], D5, C10, C10, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×2], C2×C10, C42⋊C2, C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C22×D5, C4×F5 [×2], C4⋊F5 [×2], C22⋊F5 [×2], C2×C4×D5, D10.C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, C22×C4, C4○D4 [×2], F5, C42⋊C2, C2×F5 [×3], C22×F5, D10.C23

Character table of D10.C23

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 5 10A 10B 10C 20A 20B 20C 20D size 1 1 2 5 5 10 1 1 2 5 5 10 10 10 10 10 10 10 10 10 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 -1 1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 linear of order 2 ρ3 1 1 -1 1 1 -1 1 1 -1 1 1 1 -1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 -1 1 1 -1 -1 -1 1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 1 1 -1 linear of order 2 ρ6 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 linear of order 2 ρ8 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ9 1 1 -1 -1 -1 1 -1 -1 1 1 1 i -1 -i i i -i i -i -i 1 -1 1 -1 -1 1 1 -1 linear of order 4 ρ10 1 1 1 -1 -1 -1 1 1 1 -1 -1 -i -1 i -i i -i i -i i 1 1 1 1 1 1 1 1 linear of order 4 ρ11 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -i 1 -i i i -i -i i i 1 -1 1 -1 1 -1 -1 1 linear of order 4 ρ12 1 1 1 -1 -1 -1 -1 -1 -1 1 1 i 1 i -i i -i -i i -i 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ13 1 1 -1 -1 -1 1 1 1 -1 -1 -1 i 1 i -i -i i i -i -i 1 -1 1 -1 1 -1 -1 1 linear of order 4 ρ14 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -i 1 -i i -i i i -i i 1 1 1 1 -1 -1 -1 -1 linear of order 4 ρ15 1 1 -1 -1 -1 1 -1 -1 1 1 1 -i -1 i -i -i i -i i i 1 -1 1 -1 -1 1 1 -1 linear of order 4 ρ16 1 1 1 -1 -1 -1 1 1 1 -1 -1 i -1 -i i -i i -i i -i 1 1 1 1 1 1 1 1 linear of order 4 ρ17 2 -2 0 2 -2 0 -2i 2i 0 2i -2i 0 0 0 0 0 0 0 0 0 2 0 -2 0 2i 0 0 -2i complex lifted from C4○D4 ρ18 2 -2 0 2 -2 0 2i -2i 0 -2i 2i 0 0 0 0 0 0 0 0 0 2 0 -2 0 -2i 0 0 2i complex lifted from C4○D4 ρ19 2 -2 0 -2 2 0 -2i 2i 0 -2i 2i 0 0 0 0 0 0 0 0 0 2 0 -2 0 2i 0 0 -2i complex lifted from C4○D4 ρ20 2 -2 0 -2 2 0 2i -2i 0 2i -2i 0 0 0 0 0 0 0 0 0 2 0 -2 0 -2i 0 0 2i complex lifted from C4○D4 ρ21 4 4 -4 0 0 0 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 1 -1 -1 1 orthogonal lifted from C2×F5 ρ22 4 4 4 0 0 0 4 4 4 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ23 4 4 -4 0 0 0 4 4 -4 0 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 -1 1 1 -1 orthogonal lifted from C2×F5 ρ24 4 4 4 0 0 0 -4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from C2×F5 ρ25 4 -4 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 0 0 -1 -√5 1 √5 i -√-5 √-5 -i complex faithful ρ26 4 -4 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 -1 -√5 1 √5 -i √-5 -√-5 i complex faithful ρ27 4 -4 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 0 0 0 0 -1 √5 1 -√5 i √-5 -√-5 -i complex faithful ρ28 4 -4 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 0 0 0 0 -1 √5 1 -√5 -i -√-5 √-5 i complex faithful

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

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

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

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

Matrix representation of D10.C23 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 1 0 0 40 40 40 40 0 0 1 0 0 0 0 0 0 1 0 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0
,
 0 9 0 0 0 0 32 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 40 40 40 40
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,0,0,1,40,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,1,0,40],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

D10.C23 in GAP, Magma, Sage, TeX

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

`G:=Group("D10.C2^3");`
`// GroupNames label`

`G:=SmallGroup(160,205);`
`// by ID`

`G=gap.SmallGroup(160,205);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,103,362,2309,599]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^10=b^2=e^2=1,c^2=a^-1*b,d^2=a^5,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,a*e=e*a,c*b*c^-1=a^2*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^5*c,d*e=e*d>;`
`// generators/relations`

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