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G = C25order 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

C1 — C25
C1C2C22C23C24 — C25
C1 — C25
C1 — C25
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 12)(2 11)(3 22)(4 21)(5 9)(6 10)(7 15)(8 16)(13 30)(14 29)(17 20)(18 19)(23 31)(24 32)(25 27)(26 28)
(1 9)(2 10)(3 14)(4 13)(5 12)(6 11)(7 28)(8 27)(15 26)(16 25)(17 23)(18 24)(19 32)(20 31)(21 30)(22 29)
(1 3)(2 4)(5 29)(6 30)(7 23)(8 24)(9 14)(10 13)(11 21)(12 22)(15 31)(16 32)(17 28)(18 27)(19 25)(20 26)
(1 23)(2 24)(3 7)(4 8)(5 20)(6 19)(9 17)(10 18)(11 32)(12 31)(13 27)(14 28)(15 22)(16 21)(25 30)(26 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), (1,12)(2,11)(3,22)(4,21)(5,9)(6,10)(7,15)(8,16)(13,30)(14,29)(17,20)(18,19)(23,31)(24,32)(25,27)(26,28), (1,9)(2,10)(3,14)(4,13)(5,12)(6,11)(7,28)(8,27)(15,26)(16,25)(17,23)(18,24)(19,32)(20,31)(21,30)(22,29), (1,3)(2,4)(5,29)(6,30)(7,23)(8,24)(9,14)(10,13)(11,21)(12,22)(15,31)(16,32)(17,28)(18,27)(19,25)(20,26), (1,23)(2,24)(3,7)(4,8)(5,20)(6,19)(9,17)(10,18)(11,32)(12,31)(13,27)(14,28)(15,22)(16,21)(25,30)(26,29)>;

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

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

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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