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G = C22.127C25  order 128 = 27

108th central stem extension by C22 of C25

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

Aliases: C23.68C24, C42.110C23, C22.127C25, C24.516C23, C4.862+ 1+4, C22.112- 1+4, (D4×Q8)⋊26C2, C4⋊Q8⋊40C22, (C4×Q8)⋊58C22, C4⋊C4.315C23, (C2×C4).117C24, C22⋊Q8⋊45C22, (C4×D4).246C22, (C2×D4).319C23, (C2×Q8).303C23, C42.C2⋊21C22, (C22×Q8)⋊40C22, C42⋊2C2⋊14C22, C42⋊C2⋊55C22, C22≀C2.31C22, C4⋊D4.233C22, C22⋊C4.113C23, (C23×C4).618C22, (C22×C4).387C23, C22.45C24⋊16C2, C2.41(C2×2- 1+4), C2.56(C2×2+ 1+4), C22.19C24.23C2, C22.57C24⋊6C2, C4.4D4.101C22, C22.D4⋊17C22, C22.46C24⋊29C2, C22.33C24⋊14C2, C23.41C23⋊19C2, C22.35C24⋊16C2, C23.38C23⋊27C2, (C2×C22⋊Q8)⋊84C2, (C2×C4⋊C4).718C22, (C2×C4○D4).237C22, (C2×C22⋊C4).391C22, SmallGroup(128,2270)

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

Derived series
C1 — C22 — C22.127C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22.19C24 — C22.127C25
Lower central
C1 — C22 — C22.127C25
Upper central
C1 — C22 — C22.127C25
Jennings
C1 — C22 — C22.127C25

Generators and relations for C22.127C25
 G = < a,b,c,d,e,f,g | a2=b2=e2=f2=1, c2=d2=g2=a, ab=ba, dcd-1=gcg-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece=fcf=bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 716 in 507 conjugacy classes, 384 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C24, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C23×C4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, C22.19C24, C23.38C23, C22.33C24, C22.35C24, C23.41C23, D4×Q8, C22.45C24, C22.46C24, C22.57C24, C22.127C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, 2- 1+4, C25, C2×2+ 1+4, C2×2- 1+4, C22.127C25

Smallest permutation representation of C22.127C25
►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 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)

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

(1 27)(3 25)(6 17)(8 19)(10 30)(12 32)(13 21)(15 23)

(2 28)(4 26)(5 7)(6 19)(8 17)(9 11)(10 32)(12 30)(14 22)(16 24)(18 20)(29 31)

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C···4AA
order12222222222444···4
size11112222444224···4

38 irreducible representations

dim1111111111144
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C22+ 1+42- 1+4
kernelC22.127C25C2×C22⋊Q8C22.19C24C23.38C23C22.33C24C22.35C24C23.41C23D4×Q8C22.45C24C22.46C24C22.57C24C4C22
# reps1212424444424

Matrix representation of C22.127C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
30000000
02000000
00200000
00030000
00004030
00003103
00000010
00000024
,
00100000
00010000
40000000
04000000
00000100
00001000
00001401
00001410
,
10000000
01000000
00100000
00010000
00004000
00000100
00001010
00000104
,
10000000
01000000
00400000
00040000
00001000
00000100
00004040
00003104
,
01000000
40000000
00010000
00400000
00004000
00000400
00000040
00000004

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

C22.127C25 in GAP, Magma, Sage, TeX 

C_2^2._{127}C_2^5
 
% in TeX
 
G:=Group("C2^2.127C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2270);
 
// by ID
 
G=gap.SmallGroup(128,2270);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,232,1430,723,2019,570,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=e^2=f^2=1,c^2=d^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=f*c*f=b*c=c*b,e*d*e=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
 

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