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G = C25  order 32 = 25

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

direct product, p-group, elementary abelian, monomial, rational

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, C22, C23, C24, C25
Quotients: C1, C2, C22, C23, C24, 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 22)(2 21)(3 8)(4 7)(5 19)(6 20)(9 24)(10 23)(11 28)(12 27)(13 30)(14 29)(15 18)(16 17)(25 32)(26 31)

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

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

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

C25 is a maximal subgroup of   C24⋊3C4
C25 is a maximal quotient of   C2.C25

32 conjugacy classes

class 1 2A···2AE
order12···2
size11···1

32 irreducible representations

dim11
type++
imageC1C2
kernelC25C24
# reps131

Matrix representation of C25 ►in GL5(ℤ)

10000
0-1000
00-100
00010
00001
,
-10000
01000
00100
00010
0000-1
,
-10000
0-1000
00-100
000-10
0000-1
,
-10000
0-1000
00-100
000-10
00001
,
10000
0-1000
00100
000-10
00001

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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