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## G = C25.3C4order 128 = 27

### 3rd 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 — C25.3C4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C22⋊C4 — C25.3C4
 Lower central C1 — C2 — C23 — C25.3C4
 Upper central C1 — C22 — C23×C4 — C25.3C4
 Jennings C1 — C2 — C22 — C22×C4 — C25.3C4

Generators and relations for C25.3C4
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=acd, bc=cb, bd=db, fbf-1=be=eb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 564 in 220 conjugacy classes, 52 normal (16 characteristic)
C1, C2 [×3], C2 [×10], C4 [×6], C22, C22 [×6], C22 [×46], C8 [×4], C2×C4 [×4], C2×C4 [×16], C23 [×3], C23 [×4], C23 [×46], C22⋊C4 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C22×C4 [×6], C22×C4 [×6], C24, C24 [×2], C24 [×10], C22⋊C8 [×4], C22⋊C8 [×2], C2×C22⋊C4 [×4], C2×C22⋊C4 [×4], C2×M4(2) [×2], C23×C4 [×2], C25, C23⋊C8 [×4], C24.4C4 [×2], C22×C22⋊C4, C25.3C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C4.D4 [×2], C2×C22⋊C4, C2×M4(2) [×2], C24.4C4, C2×C23⋊C4, C2×C4.D4, C25.3C4

Permutation representations of C25.3C4
On 16 points - transitive group 16T252
Generators in S16
```(1 5)(2 14)(3 11)(6 10)(7 15)(9 13)
(1 9)(2 14)(3 11)(4 16)(5 13)(6 10)(7 15)(8 12)
(1 5)(2 10)(3 7)(4 12)(6 14)(8 16)(9 13)(11 15)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)
(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)| (1,5)(2,14)(3,11)(6,10)(7,15)(9,13), (1,9)(2,14)(3,11)(4,16)(5,13)(6,10)(7,15)(8,12), (1,5)(2,10)(3,7)(4,12)(6,14)(8,16)(9,13)(11,15), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (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( (1,5)(2,14)(3,11)(6,10)(7,15)(9,13), (1,9)(2,14)(3,11)(4,16)(5,13)(6,10)(7,15)(8,12), (1,5)(2,10)(3,7)(4,12)(6,14)(8,16)(9,13)(11,15), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (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([(1,5),(2,14),(3,11),(6,10),(7,15),(9,13)], [(1,9),(2,14),(3,11),(4,16),(5,13),(6,10),(7,15),(8,12)], [(1,5),(2,10),(3,7),(4,12),(6,14),(8,16),(9,13),(11,15)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12)], [(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(16,252);`

32 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E ··· 4J 8A ··· 8H order 1 2 2 2 2 ··· 2 2 2 2 2 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 4 4 4 4 2 2 2 2 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C4 C4 C4 D4 M4(2) C23⋊C4 C4.D4 kernel C25.3C4 C23⋊C8 C24.4C4 C22×C22⋊C4 C2×C22⋊C4 C23×C4 C25 C22×C4 C23 C22 C22 # reps 1 4 2 1 4 2 2 4 8 2 2

Matrix representation of C25.3C4 in GL6(𝔽17)

 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 0 0 0 16 0 0 0 0 0 0 1
,
 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 16 0 0 0 0 0 0 16
,
 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
,
 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
,
 16 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 0 0 0 0 0 16
,
 0 1 0 0 0 0 4 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))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C25.3C4 in GAP, Magma, Sage, TeX

`C_2^5._3C_4`
`% in TeX`

`G:=Group("C2^5.3C4");`
`// GroupNames label`

`G:=SmallGroup(128,194);`
`// by ID`

`G=gap.SmallGroup(128,194);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,1123,851,172]);`
`// 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*d,b*c=c*b,b*d=d*b,f*b*f^-1=b*e=e*b,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;`
`// generators/relations`

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