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G = C25order 32 = 25

Elementary abelian group of type [2,2,2,2,2]

Aliases: C25, SmallGroup(32,51)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C25
G = < a,b,c,d,e | a2=b2=c2=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 374, all normal (2 characteristic)
C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], C25

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

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

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

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

C25 is a maximal subgroup of   C243C4
C25 is a maximal quotient of   C2.C25

32 conjugacy classes

 class 1 2A ··· 2AE order 1 2 ··· 2 size 1 1 ··· 1

32 irreducible representations

 dim 1 1 type + + image C1 C2 kernel C25 C24 # reps 1 31

Matrix representation of C25 in GL5(ℤ)

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

`G:=sub<GL(5,Integers())| [1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1],[-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1],[1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1] >;`

C25 in GAP, Magma, Sage, TeX

`C_2^5`
`% in TeX`

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

`G:=SmallGroup(32,51);`
`// by ID`

`G=gap.SmallGroup(32,51);`
`# by ID`

`G:=PCGroup([5,-2,2,2,2,2]);`
`// Polycyclic`

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

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