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## G = C24.169C23order 128 = 27

### 9th non-split extension by C24 of C23 acting via C23/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C24.169C23
 Chief series C1 — C2 — C22 — C23 — C24 — C2×C22⋊C4 — C2×C42⋊C2 — C24.169C23
 Lower central C1 — C2 — C23 — C24.169C23
 Upper central C1 — C22 — C23×C4 — C24.169C23
 Jennings C1 — C2 — C24 — C24.169C23

Generators and relations for C24.169C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=c, f2=d, g2=a, ab=ba, ac=ca, ad=da, fef-1=ae=ea, af=fa, ag=ga, bc=cb, ebe-1=bd=db, bf=fb, bg=gb, gcg-1=cd=dc, ce=ec, cf=fc, de=ed, df=fd, dg=gd, geg-1=bde, fg=gf >

Subgroups: 324 in 150 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×12], C22, C22 [×6], C22 [×10], C2×C4 [×4], C2×C4 [×32], C23, C23 [×6], C23 [×2], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×8], C22×C4 [×8], C24, C2.C42 [×4], C2×C42 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4 [×2], C42⋊C2 [×4], C23×C4, C23.9D4 [×4], C23.34D4 [×2], C2×C42⋊C2, C24.169C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C23.34D4, C23.C23 [×2], C24.169C23

Smallest permutation representation of C24.169C23
On 32 points
Generators in S32
```(1 27)(2 28)(3 25)(4 26)(5 10)(6 11)(7 12)(8 9)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(29 31)(30 32)
(1 10)(2 8)(3 12)(4 6)(5 27)(7 25)(9 28)(11 26)(13 21)(14 20)(15 23)(16 18)(17 32)(19 30)(22 31)(24 29)
(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 25)(2 26)(3 27)(4 28)(5 12)(6 9)(7 10)(8 11)(13 30)(14 31)(15 32)(16 29)(17 23)(18 24)(19 21)(20 22)
(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 26 25 2)(3 28 27 4)(5 6 12 9)(7 8 10 11)(13 24 30 18)(14 23 31 17)(15 22 32 20)(16 21 29 19)
(1 20 27 18)(2 32 28 30)(3 24 25 22)(4 13 26 15)(5 16 10 14)(6 21 11 23)(7 31 12 29)(8 17 9 19)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4N 4O ··· 4V order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 4 ··· 4 size 1 1 1 1 2 ··· 2 2 2 2 2 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 type + + + + + image C1 C2 C2 C2 C4 C4 D4 C4○D4 C23.C23 kernel C24.169C23 C23.9D4 C23.34D4 C2×C42⋊C2 C2×C42 C2×C4⋊C4 C22×C4 C23 C2 # reps 1 4 2 1 4 4 4 8 4

Matrix representation of C24.169C23 in GL6(𝔽5)

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

`G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,4,0,0,0,0,3,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,2,0,0,0,0,3,0],[1,0,0,0,0,0,4,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[2,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C24.169C23 in GAP, Magma, Sage, TeX

`C_2^4._{169}C_2^3`
`% in TeX`

`G:=Group("C2^4.169C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,552);`
`// by ID`

`G=gap.SmallGroup(128,552);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,58,2804,1027]);`
`// Polycyclic`

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

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