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## G = C42.257C23order 128 = 27

### 118th 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 — C42.257C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C2×C42⋊C2 — C42.257C23
 Lower central C1 — C22 — C42.257C23
 Upper central C1 — C2×C4 — C42.257C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.257C23

Generators and relations for C42.257C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1, dad=ab2, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=b2c, de=ed >

Subgroups: 332 in 242 conjugacy classes, 172 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C24, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C4⋊M4(2), C42.6C22, C2×C42⋊C2, C22×M4(2), C42.257C23
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, C22×C4⋊C4, Q8○M4(2), C42.257C23

Smallest permutation representation of C42.257C23
On 32 points
Generators in S32
```(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)
(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)
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)
(1 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 18)(10 23)(11 20)(12 17)(13 22)(14 19)(15 24)(16 21)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 4K ··· 4R 8A ··· 8P order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C4 D4 Q8 Q8○M4(2) kernel C42.257C23 C4⋊M4(2) C42.6C22 C2×C42⋊C2 C22×M4(2) C2×C22⋊C4 C2×C4⋊C4 C42⋊C2 C22×C4 C22×C4 C2 # reps 1 4 8 1 2 4 4 8 4 4 4

Matrix representation of C42.257C23 in GL6(𝔽17)

 4 0 0 0 0 0 1 13 0 0 0 0 0 0 5 15 0 0 0 0 12 12 0 0 0 0 4 0 4 15 0 0 1 13 16 13
,
 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 8 0 0 0 0 4 1 0 0 0 0 0 0 0 0 1 0 0 0 15 0 9 1 0 0 13 0 0 0 0 0 2 13 2 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 5 16 0 0 0 0 0 0 1 0 0 0 4 0 4 16
,
 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))| [4,1,0,0,0,0,0,13,0,0,0,0,0,0,5,12,4,1,0,0,15,12,0,13,0,0,0,0,4,16,0,0,0,0,15,13],[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],[16,4,0,0,0,0,8,1,0,0,0,0,0,0,0,15,13,2,0,0,0,0,0,13,0,0,1,9,0,2,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,5,0,4,0,0,0,16,0,0,0,0,0,0,1,4,0,0,0,0,0,16],[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] >;`

C42.257C23 in GAP, Magma, Sage, TeX

`C_4^2._{257}C_2^3`
`% in TeX`

`G:=Group("C4^2.257C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1637);`
`// by ID`

`G=gap.SmallGroup(128,1637);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,723,2019,124]);`
`// Polycyclic`

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

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