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## G = C42.265C23order 128 = 27

### 126th non-split extension by C42 of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C42.265C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C22×M4(2) — C42.265C23
 Lower central C1 — C22 — C42.265C23
 Upper central C1 — C2×C4 — C42.265C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.265C23

Generators and relations for C42.265C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2b, ab=ba, cac-1=a-1b2, dad-1=eae=ab2, bc=cb, bd=db, be=eb, dcd-1=a2c, ce=ec, ede=b2d >

Subgroups: 380 in 246 conjugacy classes, 140 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×18], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×12], C2×C4 [×14], D4 [×10], Q8 [×2], C23, C23 [×4], C23 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×10], C2×C8 [×6], M4(2) [×16], C22×C4 [×2], C22×C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×6], C2×Q8, C4○D4 [×4], C24, C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×8], C4⋊C8 [×4], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C22×C8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×M4(2) [×6], C2×M4(2) [×4], C8○D4 [×4], C23×C4, C2×C4○D4, C82M4(2), C24.4C4, (C22×C8)⋊C2, C42.6C22, C89D4 [×4], C86D4 [×4], C22.19C24, C22×M4(2), C2×C8○D4, C42.265C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, Q8○M4(2) [×2], C42.265C23

Smallest permutation representation of C42.265C23
On 32 points
Generators in S32
```(1 16 31 22)(2 13 32 19)(3 10 25 24)(4 15 26 21)(5 12 27 18)(6 9 28 23)(7 14 29 20)(8 11 30 17)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 17 13 21)(10 18 14 22)(11 19 15 23)(12 20 16 24)
(1 3 5 7)(2 26 6 30)(4 28 8 32)(9 15 13 11)(10 22 14 18)(12 24 16 20)(17 23 21 19)(25 27 29 31)
(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 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 23)(10 20)(11 17)(12 22)(13 19)(14 24)(15 21)(16 18)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G ··· 4N 8A ··· 8H 8I ··· 8T order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 2 4 4 4 4 1 1 1 1 2 2 4 ··· 4 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 Q8○M4(2) kernel C42.265C23 C8○2M4(2) C24.4C4 (C22×C8)⋊C2 C42.6C22 C8⋊9D4 C8⋊6D4 C22.19C24 C22×M4(2) C2×C8○D4 C22≀C2 C4⋊D4 C22⋊Q8 C22.D4 C2×C8 C2×C4 C2 # reps 1 1 1 1 1 4 4 1 1 1 4 4 4 4 4 4 4

Matrix representation of C42.265C23 in GL6(𝔽17)

 0 1 0 0 0 0 16 0 0 0 0 0 0 0 13 0 16 2 0 0 0 0 2 0 0 0 0 9 0 0 0 0 1 13 15 4
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 1 0 13 4
,
 0 13 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 0 0 0 2 0 9 16 0 0 13 0 0 0 0 0 15 4 2 0
,
 1 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 16 0 0 0 4 0 0 16

`G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,1,0,0,0,0,9,13,0,0,16,2,0,15,0,0,2,0,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,1,0,0,0,4,0,0,0,0,0,0,13,13,0,0,0,0,0,4],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,2,13,15,0,0,0,0,0,4,0,0,1,9,0,2,0,0,0,16,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;`

C42.265C23 in GAP, Magma, Sage, TeX

`C_4^2._{265}C_2^3`
`% in TeX`

`G:=Group("C4^2.265C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1662);`
`// by ID`

`G=gap.SmallGroup(128,1662);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,184,2019,124]);`
`// Polycyclic`

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

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