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G = C22.118C25  order 128 = 27

99th central stem extension by C22 of C25

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

Aliases: C23.60C24, C24.515C23, C22.118C25, C42.582C23, C4.1882+ 1+4, C4⋊Q8⋊98C22, (C4×D4)⋊55C22, (C4×Q8)⋊54C22, C4⋊D4⋊89C22, C4⋊C4.307C23, (C2×C4).108C24, (C23×C4)⋊47C22, (C2×C42)⋊70C22, C22⋊Q8⋊99C22, C22≀C2⋊38C22, C4○(C24⋊C22), (C2×D4).312C23, C4.4D4⋊90C22, C22⋊C4.38C23, (C2×Q8).297C23, C42.C2⋊64C22, C4○(C22.54C24), C42⋊C2⋊48C22, C42⋊2C2⋊42C22, C24⋊C22⋊10C2, C22.19C24⋊35C2, C4⋊1D4.190C22, C22.54C24⋊15C2, C2.49(C2×2+ 1+4), C2.41(C2.C25), C4○(C22.57C24), C4○(C22.56C24), (C22×C4).1216C23, C22.26C24⋊47C2, C22.D4⋊59C22, C22.57C24⋊24C2, C23.36C23⋊44C2, C22.56C24⋊20C2, (C2×C4○D4)⋊39C22, (C2×C4)○(C24⋊C22), (C2×C4)○(C22.54C24), (C2×C4)○(C22.56C24), (C2×C4)○(C22.57C24), SmallGroup(128,2261)

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

Derived series
C1 — C22 — C22.118C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22.19C24 — C22.118C25
Lower central
C1 — C22 — C22.118C25
Upper central
C1 — C2×C4 — C22.118C25
Jennings
C1 — C22 — C22.118C25

Generators and relations for C22.118C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=a, ab=ba, dcd=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, fcf=abc, cg=gc, ede=abd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 844 in 529 conjugacy classes, 380 normal (8 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C2×C42, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊Q8, C23×C4, C2×C4○D4, C2×C4○D4, C22.19C24, C23.36C23, C22.26C24, C22.54C24, C24⋊C22, C22.56C24, C22.57C24, C22.118C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C25, C2×2+ 1+4, C2.C25, C22.118C25

Smallest permutation representation of C22.118C25
►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 11)(2 12)(3 9)(4 10)(5 29)(6 30)(7 31)(8 32)(13 18)(14 19)(15 20)(16 17)(21 26)(22 27)(23 28)(24 25)

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D···2L4A4B4C4D4E···4Y
order12222···244444···4
size11114···411114···4

38 irreducible representations

dim1111111144
type+++++++++
imageC1C2C2C2C2C2C2C22+ 1+4C2.C25
kernelC22.118C25C22.19C24C23.36C23C22.26C24C22.54C24C24⋊C22C22.56C24C22.57C24C4C2
# reps1663616324

Matrix representation of C22.118C25 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
40020000
40110000
41010000
00010000
00000002
00000030
00000200
00003000
,
13000000
04000000
14040000
14400000
00000200
00003000
00000003
00000020
,
10300000
00410000
00400000
01400000
00000010
00000001
00001000
00000100
,
13000000
04000000
04010000
04100000
00000100
00001000
00000001
00000010
,
10000000
01000000
00100000
00010000
00002000
00000200
00000020
00000002

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

C22.118C25 in GAP, Magma, Sage, TeX 

C_2^2._{118}C_2^5
 
% in TeX
 
G:=Group("C2^2.118C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2261);
 
// by ID
 
G=gap.SmallGroup(128,2261);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,1059,2915,570,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=1,g^2=a,a*b=b*a,d*c*d=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=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,f*c*f=a*b*c,c*g=g*c,e*d*e=a*b*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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