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## G = C4.22C25order 128 = 27

### 4th central extension by C4 of C25

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C4.22C25
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2.C25 — C4.22C25
 Lower central C1 — C2 — C4.22C25
 Upper central C1 — C8 — C4.22C25
 Jennings C1 — C2 — C2 — C4 — C4.22C25

Generators and relations for C4.22C25
G = < a,b,c,d,e,f | a4=c2=d2=e2=f2=1, b2=a, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf=a2c, ede=a2d, df=fd, ef=fe >

Subgroups: 772 in 712 conjugacy classes, 682 normal (6 characteristic)
C1, C2, C2 [×15], C4, C4 [×15], C22 [×15], C22 [×15], C8, C8 [×15], C2×C4 [×60], D4 [×60], Q8 [×20], C23 [×15], C2×C8 [×60], M4(2) [×60], C22×C4 [×15], C2×D4 [×45], C2×Q8 [×15], C4○D4 [×80], C22×C8 [×15], C2×M4(2) [×45], C8○D4 [×80], C2×C4○D4 [×15], 2+ 1+4 [×10], 2- 1+4 [×6], C2×C8○D4 [×15], Q8○M4(2) [×15], C2.C25, C4.22C25
Quotients: C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C24 [×31], C23×C4 [×30], C25, C24×C4, C4.22C25

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

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

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

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

68 conjugacy classes

 class 1 2A 2B ··· 2P 4A 4B 4C ··· 4Q 8A 8B 8C 8D 8E ··· 8AH order 1 2 2 ··· 2 4 4 4 ··· 4 8 8 8 8 8 ··· 8 size 1 1 2 ··· 2 1 1 2 ··· 2 1 1 1 1 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 1 4 type + + + + image C1 C2 C2 C2 C4 C4 C4.22C25 kernel C4.22C25 C2×C8○D4 Q8○M4(2) C2.C25 2+ 1+4 2- 1+4 C1 # reps 1 15 15 1 20 12 4

Matrix representation of C4.22C25 in GL4(𝔽17) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 15 0 0 0 0 15 0 0 0 0 15 0 0 0 0 15
,
 13 0 0 13 4 0 13 0 13 4 0 13 8 0 0 4
,
 0 1 16 16 1 0 1 1 0 0 16 0 0 0 2 1
,
 0 1 1 0 0 16 0 0 1 1 0 0 0 15 0 1
,
 0 1 1 0 1 0 16 0 0 0 1 0 0 0 15 16
`G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[13,4,13,8,0,0,4,0,0,13,0,0,13,0,13,4],[0,1,0,0,1,0,0,0,16,1,16,2,16,1,0,1],[0,0,1,0,1,16,1,15,1,0,0,0,0,0,0,1],[0,1,0,0,1,0,0,0,1,16,1,15,0,0,0,16] >;`

C4.22C25 in GAP, Magma, Sage, TeX

`C_4._{22}C_2^5`
`% in TeX`

`G:=Group("C4.22C2^5");`
`// GroupNames label`

`G:=SmallGroup(128,2305);`
`// by ID`

`G=gap.SmallGroup(128,2305);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,2,-2,-2,224,521,1411,124]);`
`// Polycyclic`

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

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