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## G = C4.C25order 128 = 27

### 13rd non-split extension by C4 of C25 acting via C25/C24=C2

p-group, metabelian, nilpotent (class 3), monomial, rational

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C4.C25
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2- 1+4 — C4.C25
 Lower central C1 — C2 — C4 — C4.C25
 Upper central C1 — C2 — C2×C4○D4 — C4.C25
 Jennings C1 — C2 — C2 — C4 — C4.C25

Generators and relations for C4.C25
G = < a,b,c,d,e,f | a4=b2=d2=e2=f2=1, c2=a2, bab=cac-1=a-1, ad=da, ae=ea, af=fa, cbc-1=ab, bd=db, be=eb, fbf=a2b, cd=dc, ce=ec, cf=fc, ede=a2d, df=fd, ef=fe >

Subgroups: 1044 in 706 conjugacy classes, 426 normal (13 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C2×M4(2), C8○D4, C2×SD16, C2×Q16, C4○D8, C8⋊C22, C8.C22, C22×Q8, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, 2- 1+4, 2- 1+4, Q8○M4(2), C2×C8.C22, D8⋊C22, D4○SD16, Q8○D8, C2×2- 1+4, C2.C25, C4.C25
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, C25, D4×C23, C4.C25

Smallest permutation representation of C4.C25
On 32 points
Generators in S32
```(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 11)(2 10)(3 9)(4 12)(5 15)(6 14)(7 13)(8 16)(17 26)(18 25)(19 28)(20 27)(21 32)(22 31)(23 30)(24 29)
(1 19 3 17)(2 18 4 20)(5 21 7 23)(6 24 8 22)(9 25 11 27)(10 28 12 26)(13 29 15 31)(14 32 16 30)
(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)
(1 7)(2 8)(3 5)(4 6)(9 15)(10 16)(11 13)(12 14)(17 21)(18 22)(19 23)(20 24)(25 31)(26 32)(27 29)(28 30)
(1 7)(2 8)(3 5)(4 6)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)```

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

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

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

41 conjugacy classes

 class 1 2A 2B ··· 2H 2I ··· 2N 4A ··· 4H 4I ··· 4R 8A ··· 8H order 1 2 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

41 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 8 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 C4.C25 kernel C4.C25 Q8○M4(2) C2×C8.C22 D8⋊C22 D4○SD16 Q8○D8 C2×2- 1+4 C2.C25 C2×D4 C2×Q8 C4○D4 C1 # reps 1 1 6 6 8 8 1 1 3 1 4 1

Matrix representation of C4.C25 in GL8(𝔽17)

 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0
,
 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 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 1 0 0 0
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0
,
 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0

`G:=sub<GL(8,GF(17))| [0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0],[0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0] >;`

C4.C25 in GAP, Magma, Sage, TeX

`C_4.C_2^5`
`% in TeX`

`G:=Group("C4.C2^5");`
`// GroupNames label`

`G:=SmallGroup(128,2318);`
`// by ID`

`G=gap.SmallGroup(128,2318);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,456,521,2804,4037,2028,124]);`
`// Polycyclic`

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

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