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## G = C23.32C23order 64 = 26

### 5th non-split extension by C23 of C23 acting via C23/C22=C2

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C23.32C23
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C22×Q8 — C23.32C23
 Lower central C1 — C2 — C23.32C23
 Upper central C1 — C22 — C23.32C23
 Jennings C1 — C22 — C23.32C23

Generators and relations for C23.32C23
G = < a,b,c,d,e,f | a2=b2=c2=1, d2=c, e2=f2=b, dad-1=ab=ba, ac=ca, ae=ea, af=fa, bc=cb, ede-1=bd=db, fef-1=be=eb, bf=fb, cd=dc, ce=ec, cf=fc, df=fd >

Subgroups: 145 in 133 conjugacy classes, 121 normal (6 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C42⋊C2, C4×Q8, C22×Q8, C23.32C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, 2- 1+4, C23.32C23

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

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4AB order 1 2 2 2 2 2 4 ··· 4 size 1 1 1 1 2 2 2 ··· 2

34 irreducible representations

 dim 1 1 1 1 1 4 type + + + + - image C1 C2 C2 C2 C4 2- 1+4 kernel C23.32C23 C42⋊C2 C4×Q8 C22×Q8 C2×Q8 C2 # reps 1 6 8 1 16 2

Matrix representation of C23.32C23 in GL5(𝔽5)

 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 2 2 0 4
,
 1 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4
,
 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4
,
 3 0 0 0 0 0 0 0 1 0 0 2 2 4 3 0 4 0 0 0 0 0 0 0 3
,
 1 0 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 3 0 0 0 1 2 2
,
 4 0 0 0 0 0 0 1 0 0 0 4 0 0 0 0 2 2 4 3 0 3 0 1 1

`G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,2,0,0,1,0,2,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,4,0,0,0,2,0,0,0,1,4,0,0,0,0,3,0,3],[1,0,0,0,0,0,2,0,0,0,0,0,3,0,1,0,0,0,3,2,0,0,0,0,2],[4,0,0,0,0,0,0,4,2,3,0,1,0,2,0,0,0,0,4,1,0,0,0,3,1] >;`

C23.32C23 in GAP, Magma, Sage, TeX

`C_2^3._{32}C_2^3`
`% in TeX`

`G:=Group("C2^3.32C2^3");`
`// GroupNames label`

`G:=SmallGroup(64,200);`
`// by ID`

`G=gap.SmallGroup(64,200);`
`# by ID`

`G:=PCGroup([6,-2,2,2,2,-2,2,192,217,103,188,86,579]);`
`// Polycyclic`

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

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