Copied to
clipboard

G = C22.129C25  order 128 = 27

110th central stem extension by C22 of C25

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

Aliases: C23.70C24, C24.147C23, C22.129C25, C42.112C23, C4.882+ 1+4, C4⋊Q8⋊42C22, D4⋊5D4⋊32C2, Q8⋊5D4⋊27C2, (C4×D4)⋊63C22, (C4×Q8)⋊60C22, C4⋊D4⋊37C22, C4⋊C4.317C23, (C2×C4).119C24, (C23×C4)⋊51C22, C22⋊Q8⋊47C22, C22≀C2⋊14C22, C24⋊C22⋊5C2, (C2×D4).321C23, C4.4D4⋊38C22, (C22×D4)⋊44C22, (C2×Q8).304C23, (C22×Q8)⋊41C22, C22.19C24⋊42C2, C22.29C24⋊29C2, C22.32C24⋊16C2, C42⋊2C2⋊16C22, C22.54C24⋊7C2, C42⋊C2⋊57C22, C4⋊1D4.117C22, C22⋊C4.114C23, (C22×C4).389C23, C22.45C24⋊17C2, C2.58(C2×2+ 1+4), C2.47(C2.C25), C22.D4⋊61C22, C22.36C24⋊27C2, C22.50C24⋊31C2, C22.49C24⋊19C2, C23.38C23⋊28C2, C22.53C24⋊20C2, (C2×C22⋊C4)⋊60C22, (C2×C4○D4).238C22, SmallGroup(128,2272)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C22.129C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22.19C24 — C22.129C25
Lower central
C1 — C22 — C22.129C25
Upper central
C1 — C22 — C22.129C25
Jennings
C1 — C22 — C22.129C25

Generators and relations for C22.129C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=e2=g2=a, ab=ba, dcd-1=gcg-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece-1=fcf=bc=cb, ede-1=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 860 in 529 conjugacy classes, 380 normal (36 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C24, C24, C2×C22⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C42⋊2C2, C4⋊1D4, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C22.19C24, C22.19C24, C22.29C24, C23.38C23, C22.32C24, C22.36C24, D4⋊5D4, Q8⋊5D4, C22.45C24, C22.49C24, C22.50C24, C22.53C24, C22.54C24, C24⋊C22, C22.129C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C25, C2×2+ 1+4, C2.C25, C22.129C25

Smallest permutation representation of C22.129C25
►On 32 points
Generators in S32
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)

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

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D···2L4A···4F4G···4Y
order12222···24···44···4
size11114···42···24···4

38 irreducible representations

dim1111111111111144
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C22+ 1+4C2.C25
kernelC22.129C25C22.19C24C22.29C24C23.38C23C22.32C24C22.36C24D4⋊5D4Q8⋊5D4C22.45C24C22.49C24C22.50C24C22.53C24C22.54C24C24⋊C22C4C2
# reps1311462222222224

Matrix representation of C22.129C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
01000000
10000000
00010000
00100000
00000100
00001000
00000004
00000040
,
04000000
10000000
00010000
00400000
00000400
00001000
00000001
00000040
,
20000000
02000000
00200000
00020000
00000030
00000003
00003000
00000300
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100
,
01000000
40000000
00010000
00400000
00000100
00004000
00000001
00000040

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0],[0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0] >;
 

C22.129C25 in GAP, Magma, Sage, TeX 

C_2^2._{129}C_2^5
 
% in TeX
 
G:=Group("C2^2.129C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2272);
 
// by ID
 
G=gap.SmallGroup(128,2272);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,723,184,2019,570,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=1,d^2=e^2=g^2=a,a*b=b*a,d*c*d^-1=g*c*g^-1=a*c=c*a,f*d*f=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e^-1=f*c*f=b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁