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## G = C23.D5order 80 = 24·5

### The non-split extension by C23 of D5 acting via D5/C5=C2

Aliases: C23.D5, C22⋊Dic5, C10.11D4, C22.7D10, (C2×C10)⋊4C4, C53(C22⋊C4), C10.16(C2×C4), (C2×Dic5)⋊2C2, C2.3(C5⋊D4), C2.5(C2×Dic5), (C22×C10).2C2, (C2×C10).7C22, SmallGroup(80,19)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C23.D5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C23.D5
 Lower central C5 — C10 — C23.D5
 Upper central C1 — C22 — C23

Generators and relations for C23.D5
G = < a,b,c,d,e | a2=b2=c2=d5=1, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Character table of C23.D5

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 5A 5B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 10M 10N size 1 1 1 1 2 2 10 10 10 10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 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 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 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 linear of order 2 ρ5 1 -1 -1 1 1 -1 i i -i -i 1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 linear of order 4 ρ6 1 -1 -1 1 -1 1 i -i -i i 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 linear of order 4 ρ7 1 -1 -1 1 1 -1 -i -i i i 1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 linear of order 4 ρ8 1 -1 -1 1 -1 1 -i i i -i 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 linear of order 4 ρ9 2 -2 2 -2 0 0 0 0 0 0 2 2 0 0 0 0 0 -2 -2 0 0 0 -2 2 2 -2 orthogonal lifted from D4 ρ10 2 2 -2 -2 0 0 0 0 0 0 2 2 0 0 0 0 0 -2 2 0 0 0 2 -2 -2 -2 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 0 0 0 0 -1-√5/2 -1+√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D10 ρ12 2 2 2 2 2 2 0 0 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ13 2 2 2 2 2 2 0 0 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ14 2 2 2 2 -2 -2 0 0 0 0 -1+√5/2 -1-√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D10 ρ15 2 -2 -2 2 -2 2 0 0 0 0 -1+√5/2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 -1-√5/2 1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1+√5/2 -1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ16 2 -2 -2 2 2 -2 0 0 0 0 -1+√5/2 -1-√5/2 1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 -1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ17 2 -2 -2 2 -2 2 0 0 0 0 -1-√5/2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 -1+√5/2 1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 1-√5/2 -1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ18 2 -2 -2 2 2 -2 0 0 0 0 -1-√5/2 -1+√5/2 1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1-√5/2 1+√5/2 1-√5/2 -1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ19 2 -2 2 -2 0 0 0 0 0 0 -1-√5/2 -1+√5/2 -ζ53+ζ52 -ζ53+ζ52 ζ54-ζ5 -ζ54+ζ5 ζ53-ζ52 1-√5/2 1+√5/2 ζ54-ζ5 -ζ54+ζ5 ζ53-ζ52 1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 complex lifted from C5⋊D4 ρ20 2 2 -2 -2 0 0 0 0 0 0 -1+√5/2 -1-√5/2 ζ54-ζ5 -ζ54+ζ5 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 1+√5/2 -1+√5/2 ζ53-ζ52 -ζ53+ζ52 -ζ54+ζ5 -1-√5/2 1-√5/2 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ21 2 2 -2 -2 0 0 0 0 0 0 -1-√5/2 -1+√5/2 ζ53-ζ52 -ζ53+ζ52 ζ54-ζ5 -ζ54+ζ5 ζ53-ζ52 1-√5/2 -1-√5/2 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 -1+√5/2 1+√5/2 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ22 2 2 -2 -2 0 0 0 0 0 0 -1+√5/2 -1-√5/2 -ζ54+ζ5 ζ54-ζ5 ζ53-ζ52 -ζ53+ζ52 -ζ54+ζ5 1+√5/2 -1+√5/2 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 -1-√5/2 1-√5/2 1+√5/2 1-√5/2 complex lifted from C5⋊D4 ρ23 2 2 -2 -2 0 0 0 0 0 0 -1-√5/2 -1+√5/2 -ζ53+ζ52 ζ53-ζ52 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 1-√5/2 -1-√5/2 ζ54-ζ5 -ζ54+ζ5 ζ53-ζ52 -1+√5/2 1+√5/2 1-√5/2 1+√5/2 complex lifted from C5⋊D4 ρ24 2 -2 2 -2 0 0 0 0 0 0 -1-√5/2 -1+√5/2 ζ53-ζ52 ζ53-ζ52 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 1-√5/2 1+√5/2 -ζ54+ζ5 ζ54-ζ5 -ζ53+ζ52 1-√5/2 -1-√5/2 -1+√5/2 1+√5/2 complex lifted from C5⋊D4 ρ25 2 -2 2 -2 0 0 0 0 0 0 -1+√5/2 -1-√5/2 -ζ54+ζ5 -ζ54+ζ5 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 1+√5/2 1-√5/2 -ζ53+ζ52 ζ53-ζ52 ζ54-ζ5 1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 complex lifted from C5⋊D4 ρ26 2 -2 2 -2 0 0 0 0 0 0 -1+√5/2 -1-√5/2 ζ54-ζ5 ζ54-ζ5 ζ53-ζ52 -ζ53+ζ52 -ζ54+ζ5 1+√5/2 1-√5/2 ζ53-ζ52 -ζ53+ζ52 -ζ54+ζ5 1+√5/2 -1+√5/2 -1-√5/2 1-√5/2 complex lifted from C5⋊D4

Smallest permutation representation of C23.D5
On 40 points
Generators in S40
```(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 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 35 14 25)(2 34 15 24)(3 33 11 23)(4 32 12 22)(5 31 13 21)(6 38 16 28)(7 37 17 27)(8 36 18 26)(9 40 19 30)(10 39 20 29)```

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

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

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

Matrix representation of C23.D5 in GL3(𝔽41) generated by

 40 0 0 0 1 0 0 0 40
,
 40 0 0 0 1 0 0 0 1
,
 1 0 0 0 40 0 0 0 40
,
 1 0 0 0 37 0 0 0 10
,
 32 0 0 0 0 10 0 37 0
`G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,40],[40,0,0,0,1,0,0,0,1],[1,0,0,0,40,0,0,0,40],[1,0,0,0,37,0,0,0,10],[32,0,0,0,0,37,0,10,0] >;`

C23.D5 in GAP, Magma, Sage, TeX

`C_2^3.D_5`
`% in TeX`

`G:=Group("C2^3.D5");`
`// GroupNames label`

`G:=SmallGroup(80,19);`
`// by ID`

`G=gap.SmallGroup(80,19);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-5,20,101,1604]);`
`// Polycyclic`

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

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