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## G = C24.176C23order 128 = 27

### 16th 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 — C24.176C23
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C4×C22⋊C4 — C24.176C23
 Lower central C1 — C2 — C23 — C24.176C23
 Upper central C1 — C22 — C23×C4 — C24.176C23
 Jennings C1 — C2 — C24 — C24.176C23

Generators and relations for C24.176C23
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=c, f2=abc, g2=b, ab=ba, ac=ca, eae-1=faf-1=ad=da, ag=ga, bc=cb, bd=db, geg-1=be=eb, bf=fb, bg=gb, fcf-1=cd=dc, ce=ec, cg=gc, de=ed, df=fd, dg=gd, fef-1=ae, fg=gf >

Subgroups: 364 in 162 conjugacy classes, 58 normal (28 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×13], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×4], C2×C4 [×31], Q8 [×4], C23 [×3], C23 [×4], C23 [×2], C42 [×2], C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×8], C22×C4 [×6], C22×C4 [×4], C22×C4 [×6], C2×Q8 [×4], C24, C2.C42 [×2], C2×C42, C2×C22⋊C4 [×6], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4, C22⋊Q8 [×4], C23×C4, C22×Q8, C23.9D4 [×4], C4×C22⋊C4, C23.7Q8, C2×C22⋊Q8, C24.176C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, C23.67C23, C2×C23⋊C4, C23.C23, C24.176C23

Smallest permutation representation of C24.176C23
On 32 points
Generators in S32
```(1 7)(2 15)(3 9)(4 13)(5 8)(6 16)(10 11)(12 14)(17 28)(18 27)(19 26)(20 25)(21 31)(22 30)(23 29)(24 32)
(1 4)(2 3)(5 12)(6 11)(7 13)(8 14)(9 15)(10 16)(17 22)(18 23)(19 24)(20 21)(25 31)(26 32)(27 29)(28 30)
(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)
(1 6)(2 5)(3 12)(4 11)(7 16)(8 15)(9 14)(10 13)(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 25 13 23)(2 19 9 30)(3 24 15 28)(4 31 7 18)(5 17 14 32)(6 27 10 21)(8 26 12 22)(11 29 16 20)
(1 3 4 2)(5 6 12 11)(7 9 13 15)(8 16 14 10)(17 27 22 29)(18 30 23 28)(19 25 24 31)(20 32 21 26)```

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

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

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

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 1 2 2 2 4 4 type + + + + + - + + image C1 C2 C2 C2 C2 C4 C4 Q8 D4 C4○D4 C23⋊C4 C23.C23 kernel C24.176C23 C23.9D4 C4×C22⋊C4 C23.7Q8 C2×C22⋊Q8 C2×C4⋊C4 C22×Q8 C22⋊C4 C22×C4 C23 C4 C2 # reps 1 4 1 1 1 6 2 4 4 4 2 2

Matrix representation of C24.176C23 in GL6(𝔽5)

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

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

C24.176C23 in GAP, Magma, Sage, TeX

`C_2^4._{176}C_2^3`
`% in TeX`

`G:=Group("C2^4.176C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,728);`
`// by ID`

`G=gap.SmallGroup(128,728);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,400,422,100,2019,1018,2028]);`
`// Polycyclic`

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

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