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## G = C42.677C23order 128 = 27

### 92nd 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 — C22 — C42.677C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C22×C42 — C42.677C23
 Lower central C1 — C22 — C42.677C23
 Upper central C1 — C42 — C42.677C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.677C23

Generators and relations for C42.677C23
G = < a,b,c,d,e | a4=b4=1, c2=b, d2=e2=a2, ab=ba, cac-1=a-1b2, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2c, de=ed >

Subgroups: 332 in 242 conjugacy classes, 152 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×12], C4 [×6], C22, C22 [×6], C22 [×14], C8 [×8], C2×C4 [×2], C2×C4 [×22], C2×C4 [×30], C23, C23 [×2], C23 [×6], C42 [×4], C42 [×12], C2×C8 [×8], M4(2) [×8], C22×C4 [×2], C22×C4 [×16], C22×C4 [×12], C24, C4×C8 [×4], C8⋊C4 [×4], C22⋊C8 [×8], C4⋊C8 [×8], C2×C42 [×2], C2×C42 [×10], C2×M4(2) [×4], C23×C4, C23×C4 [×2], C4×M4(2) [×2], C24.4C4 [×2], C4⋊M4(2) [×2], C42.12C4 [×4], C42.6C4 [×4], C22×C42, C42.677C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×8], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C2×M4(2) [×12], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, C22×M4(2) [×2], C42.677C23

Smallest permutation representation of C42.677C23
On 32 points
Generators in S32
```(1 10 31 23)(2 20 32 15)(3 12 25 17)(4 22 26 9)(5 14 27 19)(6 24 28 11)(7 16 29 21)(8 18 30 13)
(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 19 31 14)(2 20 32 15)(3 21 25 16)(4 22 26 9)(5 23 27 10)(6 24 28 11)(7 17 29 12)(8 18 30 13)
(1 19 31 14)(2 15 32 20)(3 21 25 16)(4 9 26 22)(5 23 27 10)(6 11 28 24)(7 17 29 12)(8 13 30 18)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A ··· 4L 4M ··· 4AD 8A ··· 8P order 1 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 M4(2) C4○D4 kernel C42.677C23 C4×M4(2) C24.4C4 C4⋊M4(2) C42.12C4 C42.6C4 C22×C42 C2×C42 C23×C4 C2×C4 C2×C4 # reps 1 2 2 2 4 4 1 12 4 16 8

Matrix representation of C42.677C23 in GL4(𝔽17) generated by

 13 0 0 0 0 13 0 0 0 0 13 0 0 0 0 4
,
 13 0 0 0 0 13 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 13 0 0 0 0 0 0 1 0 0 1 0
,
 13 0 0 0 0 13 0 0 0 0 4 0 0 0 0 4
,
 13 0 0 0 0 4 0 0 0 0 4 0 0 0 0 13
`G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,4],[13,0,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[0,13,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[13,0,0,0,0,4,0,0,0,0,4,0,0,0,0,13] >;`

C42.677C23 in GAP, Magma, Sage, TeX

`C_4^2._{677}C_2^3`
`% in TeX`

`G:=Group("C4^2.677C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1652);`
`// by ID`

`G=gap.SmallGroup(128,1652);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,2019,124]);`
`// Polycyclic`

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

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