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## G = C25.C4order 128 = 27

### 4th non-split extension by C25 of C4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C25.C4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — D4×C23 — C25.C4
 Lower central C1 — C2 — C23 — C25.C4
 Upper central C1 — C22 — C23×C4 — C25.C4
 Jennings C1 — C2 — C2 — C22×C4 — C25.C4

Generators and relations for C25.C4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=1, f4=e, ab=ba, ac=ca, ad=da, ae=ea, faf-1=ace, bc=cb, fbf-1=bd=db, be=eb, cd=dc, fcf-1=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 1012 in 410 conjugacy classes, 72 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×14], C4 [×4], C4 [×2], C22, C22 [×6], C22 [×78], C8 [×4], C2×C4 [×8], C2×C4 [×10], D4 [×32], C23, C23 [×10], C23 [×86], C2×C8 [×4], M4(2) [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×4], C2×D4 [×8], C2×D4 [×52], C24, C24 [×4], C24 [×20], C22⋊C8 [×4], C4.D4 [×8], C2×M4(2) [×4], C23×C4, C22×D4 [×4], C22×D4 [×12], C25 [×2], C24.4C4 [×2], C2×C4.D4 [×4], D4×C23, C25.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, C22⋊C4 [×12], C22×C4, C2×D4 [×6], C4.D4 [×4], C2×C22⋊C4 [×3], C22≀C2 [×4], C243C4, C2×C4.D4 [×2], C25.C4

Permutation representations of C25.C4
On 16 points - transitive group 16T253
Generators in S16
```(1 9)(2 8)(3 15)(4 6)(5 13)(7 11)(10 12)(14 16)
(1 5)(2 16)(3 7)(4 10)(6 12)(8 14)(9 13)(11 15)
(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)```

`G:=sub<Sym(16)| (1,9)(2,8)(3,15)(4,6)(5,13)(7,11)(10,12)(14,16), (1,5)(2,16)(3,7)(4,10)(6,12)(8,14)(9,13)(11,15), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)>;`

`G:=Group( (1,9)(2,8)(3,15)(4,6)(5,13)(7,11)(10,12)(14,16), (1,5)(2,16)(3,7)(4,10)(6,12)(8,14)(9,13)(11,15), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16) );`

`G=PermutationGroup([(1,9),(2,8),(3,15),(4,6),(5,13),(7,11),(10,12),(14,16)], [(1,5),(2,16),(3,7),(4,10),(6,12),(8,14),(9,13),(11,15)], [(1,15),(2,12),(3,9),(4,14),(5,11),(6,16),(7,13),(8,10)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)])`

`G:=TransitiveGroup(16,253);`

32 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J ··· 2Q 4A 4B 4C 4D 4E 4F 8A ··· 8H order 1 2 2 2 2 ··· 2 2 ··· 2 4 4 4 4 4 4 8 ··· 8 size 1 1 1 1 2 ··· 2 4 ··· 4 2 2 2 2 4 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 D4 D4 C4.D4 kernel C25.C4 C24.4C4 C2×C4.D4 D4×C23 C22×D4 C25 C22×C4 C2×D4 C22 # reps 1 2 4 1 4 4 4 8 4

Matrix representation of C25.C4 in GL6(𝔽17)

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

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

C25.C4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(128,515);`
`// by ID`

`G=gap.SmallGroup(128,515);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,2028,124]);`
`// Polycyclic`

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

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