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G = C22.126C25  order 128 = 27

107th central stem extension by C22 of C25

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

Aliases: C23.67C24, C24.146C23, C42.109C23, C22.126C25, C4.852+ 1+4, C22.222+ 1+4, D42⋊20C2, (C4×D4)⋊61C22, C4⋊D4⋊91C22, C4⋊C4.314C23, C23⋊3D4⋊12C2, C4⋊1D4⋊23C22, (C2×C4).116C24, (C23×C4)⋊50C22, C22≀C2⋊13C22, (C2×D4).318C23, C4.4D4⋊37C22, (C22×D4)⋊43C22, C22⋊C4.44C23, C22⋊Q8⋊100C22, (C2×Q8).302C23, C42.C2⋊20C22, C22.19C24⋊40C2, C22.32C24⋊15C2, C42⋊2C2⋊13C22, C42⋊C2⋊54C22, C22.29C24⋊28C2, C22.54C24⋊5C2, (C22×C4).386C23, C2.55(C2×2+ 1+4), C22.D4⋊16C22, C22.34C24⋊18C2, C22.47C24⋊28C2, (C2×C4⋊D4)⋊74C2, (C2×C4⋊C4)⋊86C22, (C2×C4○D4)⋊45C22, (C2×C22⋊C4)⋊59C22, SmallGroup(128,2269)

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

Derived series
C1 — C22 — C22.126C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22.19C24 — C22.126C25
Lower central
C1 — C22 — C22.126C25
Upper central
C1 — C22 — C22.126C25
Jennings
C1 — C22 — C22.126C25

Generators and relations for C22.126C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, 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: 1228 in 635 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×D4, C2×Q8, C4○D4, C24, C24, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C23×C4, C22×D4, C2×C4○D4, C2×C4⋊D4, C22.19C24, C23⋊3D4, C22.29C24, C22.32C24, C22.34C24, D42, C22.47C24, C22.54C24, C22.126C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C25, C2×2+ 1+4, C22.126C25

Smallest permutation representation of C22.126C25
►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 23)(2 22)(3 21)(4 24)(5 29)(6 32)(7 31)(8 30)(9 20)(10 19)(11 18)(12 17)(13 25)(14 28)(15 27)(16 26)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2R4A4B4C···4S
order122222222···2444···4
size111122224···4224···4

38 irreducible representations

dim111111111144
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C22+ 1+42+ 1+4
kernelC22.126C25C2×C4⋊D4C22.19C24C23⋊3D4C22.29C24C22.32C24C22.34C24D42C22.47C24C22.54C24C4C22
# reps121424284424

Matrix representation of C22.126C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
01000000
10000000
01040000
10400000
00000100
00001000
00000204
00002040
,
10300000
01030000
10400000
01040000
00001040
00000104
00002040
00000204
,
03000000
20000000
03020000
20300000
00004000
00000400
00000040
00000004
,
03000000
20000000
03020000
20300000
00004000
00000400
00003010
00000301
,
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],[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],[0,1,0,1,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,1,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0],[1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,3,0,4,0,0,0,0,0,0,3,0,4,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,1,0,2,0,0,0,0,4,0,4,0,0,0,0,0,0,4,0,4],[0,2,0,2,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,2,0,2,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,4,0,3,0,0,0,0,0,0,4,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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] >;
 

C22.126C25 in GAP, Magma, Sage, TeX 

C_2^2._{126}C_2^5
 
% in TeX
 
G:=Group("C2^2.126C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2269);
 
// by ID
 
G=gap.SmallGroup(128,2269);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,723,2019,570,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,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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