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## G = C24.167C23order 128 = 27

### 7th 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 — C24.167C23
 Chief series C1 — C2 — C22 — C23 — C24 — C22×D4 — C2×C4×D4 — C24.167C23
 Lower central C1 — C2 — C23 — C24.167C23
 Upper central C1 — C22 — C23×C4 — C24.167C23
 Jennings C1 — C2 — C24 — C24.167C23

Generators and relations for C24.167C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=c, f2=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=cd=dc, ce=ec, cf=fc, de=ed, df=fd, dg=gd, geg=bde, fg=gf >

Subgroups: 436 in 190 conjugacy classes, 62 normal (28 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×11], C22 [×3], C22 [×4], C22 [×18], C2×C4 [×2], C2×C4 [×4], C2×C4 [×31], D4 [×8], C23 [×3], C23 [×6], C23 [×6], C42 [×2], C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×5], C22×C4 [×4], C22×C4 [×10], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2.C42 [×2], C23⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×2], C4×D4 [×4], C23×C4 [×2], C22×D4, C23.9D4 [×2], C23.7Q8 [×2], C2×C23⋊C4 [×2], C2×C4×D4, C24.167C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C23.7Q8, C2×C23⋊C4, C23.C23, C24.167C23

Smallest permutation representation of C24.167C23
On 32 points
Generators in S32
```(1 22)(2 23)(3 24)(4 21)(5 14)(6 15)(7 16)(8 13)(9 32)(10 29)(11 30)(12 31)(17 28)(18 25)(19 26)(20 27)
(1 3)(2 5)(4 7)(6 8)(9 27)(10 12)(11 25)(13 15)(14 23)(16 21)(17 19)(18 30)(20 32)(22 24)(26 28)(29 31)
(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 6)(2 7)(3 8)(4 5)(9 25)(10 26)(11 27)(12 28)(13 24)(14 21)(15 22)(16 23)(17 31)(18 32)(19 29)(20 30)
(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 29 22 10)(2 11 23 30)(3 31 24 12)(4 9 21 32)(5 25 14 18)(6 19 15 26)(7 27 16 20)(8 17 13 28)
(1 2)(3 5)(4 8)(6 7)(9 17)(10 30)(11 29)(12 18)(13 21)(14 24)(15 16)(19 27)(20 26)(22 23)(25 31)(28 32)```

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 4A 4B 4C 4D 4E ··· 4L 4M ··· 4T order 1 2 2 2 2 ··· 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 size 1 1 1 1 2 ··· 2 4 4 2 2 2 2 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + - + image C1 C2 C2 C2 C2 C4 C4 C4 C4 D4 D4 Q8 C4○D4 C23⋊C4 C23.C23 kernel C24.167C23 C23.9D4 C23.7Q8 C2×C23⋊C4 C2×C4×D4 C2×C42 C2×C22⋊C4 C2×C4⋊C4 C23×C4 C22×C4 C2×D4 C2×D4 C23 C4 C2 # reps 1 2 2 2 1 2 2 2 2 4 2 2 4 2 2

Matrix representation of C24.167C23 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
,
 1 0 0 0 0 0 0 1 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
,
 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 4 0 0 0 0 4 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 4 0 0 0 0 0 0 4
,
 2 0 0 0 0 0 4 3 0 0 0 0 0 0 0 0 0 4 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 4 0 0
,
 4 1 0 0 0 0 3 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
,
 1 0 0 0 0 0 0 1 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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[2,4,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,1,0,0,0,0,4,0,0,0],[4,3,0,0,0,0,1,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C24.167C23 in GAP, Magma, Sage, TeX

`C_2^4._{167}C_2^3`
`% in TeX`

`G:=Group("C2^4.167C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,531);`
`// by ID`

`G=gap.SmallGroup(128,531);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,2804,1027]);`
`// Polycyclic`

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

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