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## G = C25⋊C22order 128 = 27

### 2nd semidirect product of C25 and C22 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C25⋊C22
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C2×C22≀C2 — C25⋊C22
 Lower central C1 — C23 — C25⋊C22
 Upper central C1 — C23 — C25⋊C22
 Jennings C1 — C23 — C25⋊C22

Generators and relations for C25⋊C22
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, gag=acd, gbg=bc=cb, bd=db, fbf=be=eb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1268 in 500 conjugacy classes, 112 normal (6 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×12], C22, C22 [×6], C22 [×92], C2×C4 [×6], C2×C4 [×24], D4 [×36], C23, C23 [×12], C23 [×92], C42, C22⋊C4 [×36], C4⋊C4 [×3], C22×C4 [×9], C2×D4 [×12], C2×D4 [×36], C24 [×6], C24 [×20], C2.C42 [×2], C2×C42, C2×C22⋊C4 [×18], C2×C4⋊C4 [×3], C22≀C2 [×24], C22×D4 [×9], C25 [×2], C24.3C22 [×3], C23.10D4 [×6], C2×C22≀C2 [×6], C25⋊C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22×D4 [×3], 2+ 1+4 [×4], C233D4 [×3], D42 [×3], C24⋊C22, C25⋊C22

Smallest permutation representation of C25⋊C22
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 8)(2 7)(3 17)(4 18)(5 20)(6 19)(9 28)(10 27)(11 30)(12 29)(13 25)(14 26)(15 24)(16 23)(21 32)(22 31)
(1 3)(2 4)(5 23)(6 24)(7 18)(8 17)(9 13)(10 14)(11 22)(12 21)(15 19)(16 20)(25 28)(26 27)(29 32)(30 31)
(1 5)(2 6)(3 23)(4 24)(7 19)(8 20)(9 31)(10 32)(11 25)(12 26)(13 30)(14 29)(15 18)(16 17)(21 27)(22 28)
(1 32)(2 31)(3 29)(4 30)(5 10)(6 9)(7 22)(8 21)(11 18)(12 17)(13 24)(14 23)(15 25)(16 26)(19 28)(20 27)
(1 5)(3 23)(7 22)(8 27)(10 32)(11 18)(12 16)(14 29)(15 25)(17 26)(19 28)(20 21)
(1 29)(2 9)(3 32)(4 13)(5 14)(6 31)(7 25)(8 21)(10 23)(11 19)(12 17)(15 22)(16 26)(18 28)(20 27)(24 30)```

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

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

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

32 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 4A ··· 4F 4G ··· 4L order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 4 ··· 4 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 2 4 type + + + + + + image C1 C2 C2 C2 D4 2+ 1+4 kernel C25⋊C22 C24.3C22 C23.10D4 C2×C22≀C2 C2×D4 C22 # reps 1 3 6 6 12 4

Matrix representation of C25⋊C22 in GL6(ℤ)

 0 1 0 0 0 0 1 0 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 -1 0
,
 1 0 0 0 0 0 0 1 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 1 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 -1 0 0 0 0 0 0 -1
,
 -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 1 0 0 0 0 0 0 1
,
 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 1 0 0 0 0 0 0 1
,
 -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 1 0 0 0 0 0 0 1
,
 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 1 0 0 0 0 0 0 -1

`G:=sub<GL(6,Integers())| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[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,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1] >;`

C25⋊C22 in GAP, Magma, Sage, TeX

`C_2^5\rtimes C_2^2`
`% in TeX`

`G:=Group("C2^5:C2^2");`
`// GroupNames label`

`G:=SmallGroup(128,1411);`
`// by ID`

`G=gap.SmallGroup(128,1411);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,758,723,1571,346]);`
`// Polycyclic`

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

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