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G = C25⋊C22  order 128 = 27

2nd semidirect product of C25 and C22 acting faithfully

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

Aliases: C25⋊2C22, C23.579C24, C24.387C23, C22.3532+ 1+4, C2.43D42, (C2×D4)⋊16D4, (C2×C42)⋊30C22, C23.205(C2×D4), C23.10D4⋊76C2, C2.40(C23⋊3D4), (C22×C4).177C23, C22.388(C22×D4), C2.C42⋊36C22, C24.3C22⋊74C2, C2.4(C24⋊C22), (C22×D4).218C22, (C2×C4).87(C2×D4), (C2×C4⋊C4)⋊32C22, (C2×C22≀C2)⋊15C2, (C2×C22⋊C4)⋊29C22, SmallGroup(128,1411)

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, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22≀C2, C22×D4, C25, C24.3C22, C23.10D4, C2×C22≀C2, C25⋊C22
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C23⋊3D4, D42, 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 17)(2 18)(3 5)(4 6)(7 15)(8 16)(9 32)(10 31)(11 25)(12 26)(13 30)(14 29)(19 23)(20 24)(21 27)(22 28)

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

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

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

(1 24)(3 10)(4 21)(6 13)(7 20)(8 18)(9 30)(11 15)(17 25)(19 26)(22 29)(27 32)

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

32 conjugacy classes

class 1 2A···2G2H···2S4A···4F4G···4L
order12···22···24···44···4
size11···14···44···48···8

32 irreducible representations

dim111124
type++++++
imageC1C2C2C2D42+ 1+4
kernelC25⋊C22C24.3C22C23.10D4C2×C22≀C2C2×D4C22
# reps1366124

Matrix representation of C25⋊C22 ►in GL6(ℤ)

010000
100000
00-1000
000-100
00000-1
0000-10
,
100000
010000
000100
001000
000001
000010
,
100000
010000
001000
000100
0000-10
00000-1
,
-100000
0-10000
001000
000100
000010
000001
,
100000
010000
00-1000
000-100
000010
000001
,
-100000
010000
00-1000
000100
000010
000001
,
100000
0-10000
00-1000
000-100
000010
00000-1

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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