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## G = C42.264C23order 128 = 27

### 125th 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 — C42.264C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C23×C8 — C42.264C23
 Lower central C1 — C22 — C42.264C23
 Upper central C1 — C2×C8 — C42.264C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.264C23

Generators and relations for C42.264C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2b-1, ab=ba, cac-1=a-1b2, ad=da, eae=ab2, bc=cb, bd=db, be=eb, dcd-1=a2b2c, ce=ec, ede=b2d >

Subgroups: 380 in 260 conjugacy classes, 144 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×20], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×12], C2×C4 [×14], D4 [×14], Q8 [×2], C23, C23 [×4], C23 [×6], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×10], C2×C8 [×14], M4(2) [×8], 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 [×6], C22×C8 [×4], C2×M4(2) [×2], C2×M4(2) [×2], C8○D4 [×4], C23×C4, C2×C4○D4, C82M4(2), C24.4C4, (C22×C8)⋊C2, C42.6C22, C8×D4 [×4], C89D4 [×4], C22.19C24, C23×C8, C2×C8○D4, C42.264C23
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], C8○D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C8○D4 [×2], C42.264C23

Smallest permutation representation of C42.264C23
On 32 points
Generators in S32
```(1 13 27 18)(2 14 28 19)(3 15 29 20)(4 16 30 21)(5 9 31 22)(6 10 32 23)(7 11 25 24)(8 12 26 17)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 20 13 24)(10 21 14 17)(11 22 15 18)(12 23 16 19)
(1 3 5 7)(2 26 6 30)(4 28 8 32)(9 20 13 24)(10 12 14 16)(11 22 15 18)(17 19 21 23)(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 27)(2 32)(3 29)(4 26)(5 31)(6 28)(7 25)(8 30)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)```

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 4A 4B 4C 4D 4E ··· 4J 4K ··· 4P 8A ··· 8H 8I ··· 8T 8U ··· 8AB order 1 2 2 2 2 ··· 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 8 ··· 8 size 1 1 1 1 2 ··· 2 4 4 1 1 1 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 C8○D4 kernel C42.264C23 C8○2M4(2) C24.4C4 (C22×C8)⋊C2 C42.6C22 C8×D4 C8⋊9D4 C22.19C24 C23×C8 C2×C8○D4 C22≀C2 C4⋊D4 C22⋊Q8 C22.D4 C2×C8 C2×C4 C22 # reps 1 1 1 1 1 4 4 1 1 1 4 4 4 4 4 4 16

Matrix representation of C42.264C23 in GL4(𝔽17) generated by

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

C42.264C23 in GAP, Magma, Sage, TeX

`C_4^2._{264}C_2^3`
`% in TeX`

`G:=Group("C4^2.264C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1661);`
`// by ID`

`G=gap.SmallGroup(128,1661);`
`# by ID`

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

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

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