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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

Aliases: C252C22, C23.579C24, C24.387C23, C22.3532+ 1+4, C2.43D42, (C2×D4)⋊16D4, (C2×C42)⋊30C22, C23.205(C2×D4), C23.10D476C2, C2.40(C233D4), (C22×C4).177C23, C22.388(C22×D4), C2.C4236C22, C24.3C2274C2, 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

C1C23 — C25⋊C22
C1C2C22C23C24C25C2×C22≀C2 — C25⋊C22
C1C23 — C25⋊C22
C1C23 — C25⋊C22
C1C23 — 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···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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