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## G = C23.20D4order 64 = 26

### 13rd non-split extension by C23 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C23.20D4
 Chief series C1 — C2 — C4 — C2×C4 — C4⋊C4 — C42⋊C2 — C23.20D4
 Lower central C1 — C2 — C2×C4 — C23.20D4
 Upper central C1 — C22 — C22×C4 — C23.20D4
 Jennings C1 — C2 — C2 — C2×C4 — C23.20D4

Generators and relations for C23.20D4
G = < a,b,c,d,e | a2=b2=c2=1, d4=e2=c, dad-1=ab=ba, ac=ca, eae-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd3 >

Character table of C23.20D4

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D size 1 1 1 1 4 2 2 2 2 4 4 4 4 8 8 4 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 2 2 2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 -2 2 0 0 2i -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ12 2 -2 2 -2 0 2 -2 0 0 0 0 2i -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ13 2 -2 2 -2 0 2 -2 0 0 0 0 -2i 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ14 2 -2 2 -2 0 -2 2 0 0 -2i 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ15 2 -2 -2 2 0 0 0 2i -2i 0 0 0 0 0 0 √-2 -√2 √2 -√-2 complex lifted from C4○D8 ρ16 2 -2 -2 2 0 0 0 -2i 2i 0 0 0 0 0 0 -√-2 -√2 √2 √-2 complex lifted from C4○D8 ρ17 2 -2 -2 2 0 0 0 2i -2i 0 0 0 0 0 0 -√-2 √2 -√2 √-2 complex lifted from C4○D8 ρ18 2 -2 -2 2 0 0 0 -2i 2i 0 0 0 0 0 0 √-2 √2 -√2 -√-2 complex lifted from C4○D8 ρ19 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C23.20D4
On 32 points
Generators in S32
```(2 28)(4 30)(6 32)(8 26)(9 18)(10 14)(11 20)(12 16)(13 22)(15 24)(17 21)(19 23)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)
(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)
(1 22 5 18)(2 12 6 16)(3 20 7 24)(4 10 8 14)(9 31 13 27)(11 29 15 25)(17 32 21 28)(19 30 23 26)```

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

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

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

Matrix representation of C23.20D4 in GL4(𝔽17) generated by

 1 0 0 0 0 16 0 0 0 0 1 0 0 0 0 16
,
 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16
,
 0 4 0 0 4 0 0 0 0 0 2 0 0 0 0 8
,
 0 1 0 0 1 0 0 0 0 0 0 8 0 0 2 0
`G:=sub<GL(4,GF(17))| [1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[0,4,0,0,4,0,0,0,0,0,2,0,0,0,0,8],[0,1,0,0,1,0,0,0,0,0,0,2,0,0,8,0] >;`

C23.20D4 in GAP, Magma, Sage, TeX

`C_2^3._{20}D_4`
`% in TeX`

`G:=Group("C2^3.20D4");`
`// GroupNames label`

`G:=SmallGroup(64,166);`
`// by ID`

`G=gap.SmallGroup(64,166);`
`# by ID`

`G:=PCGroup([6,-2,2,2,-2,2,-2,192,121,199,362,50,1444,376,88]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^4=e^2=c,d*a*d^-1=a*b=b*a,a*c=c*a,e*a*e^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^3>;`
`// generators/relations`

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