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## G = C42.691C23order 128 = 27

### 106th non-split extension by C42 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 — C42.691C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C2×C4×D4 — C42.691C23
 Lower central C1 — C2 — C42.691C23
 Upper central C1 — C2×C4 — C42.691C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.691C23

Generators and relations for C42.691C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=b, e2=a2, ab=ba, ac=ca, dad=a-1, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2c, ede-1=a2d >

Subgroups: 388 in 250 conjugacy classes, 174 normal (16 characteristic)
C1, C2 [×3], C2 [×10], C4 [×6], C4 [×7], C22, C22 [×10], C22 [×18], C8 [×8], C2×C4 [×2], C2×C4 [×12], C2×C4 [×17], D4 [×16], C23, C23 [×12], C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×8], C22×C4 [×3], C22×C4 [×10], C22×C4 [×4], C2×D4 [×12], C24 [×2], C4×C8 [×4], C22⋊C8 [×12], C4⋊C8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C22×C8 [×8], C23×C4 [×2], C22×D4, C2×C22⋊C8 [×4], C42.12C4 [×2], C8×D4 [×8], C2×C4×D4, C42.691C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, C22×C8 [×14], C23×C4, 2+ 1+4 [×2], C22.11C24, C23×C8, Q8○M4(2), C42.691C23

Smallest permutation representation of C42.691C23
On 32 points
Generators in S32
```(1 23 31 10)(2 24 32 11)(3 17 25 12)(4 18 26 13)(5 19 27 14)(6 20 28 15)(7 21 29 16)(8 22 30 9)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 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)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)
(1 14 31 19)(2 20 32 15)(3 16 25 21)(4 22 26 9)(5 10 27 23)(6 24 28 11)(7 12 29 17)(8 18 30 13)```

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2M 4A 4B 4C 4D 4E ··· 4V 8A ··· 8AF order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 4 4 type + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 C8 2+ 1+4 Q8○M4(2) kernel C42.691C23 C2×C22⋊C8 C42.12C4 C8×D4 C2×C4×D4 C2×C22⋊C4 C2×C4⋊C4 C4×D4 C22×D4 C2×D4 C4 C2 # reps 1 4 2 8 1 4 2 8 2 32 2 2

Matrix representation of C42.691C23 in GL5(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 16 0
,
 13 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 9 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0
,
 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 16 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 16 0

`G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0],[13,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[9,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,1,0] >;`

C42.691C23 in GAP, Magma, Sage, TeX

`C_4^2._{691}C_2^3`
`% in TeX`

`G:=Group("C4^2.691C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1704);`
`// by ID`

`G=gap.SmallGroup(128,1704);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,219,675,124]);`
`// Polycyclic`

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

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