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G = C22.140C25  order 128 = 27

121st central stem extension by C22 of C25

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

Aliases: C23.81C24, C22.140C25, C24.153C23, C42.123C23, C4.962+ 1+4, C4⋊Q8⋊46C22, D4⋊5D4⋊37C2, Q8⋊5D4⋊30C2, (C4×D4)⋊70C22, (C4×Q8)⋊66C22, C4⋊C4.325C23, C4⋊D4⋊42C22, (C2×C4).130C24, (C2×C42)⋊75C22, C22⋊Q8⋊52C22, (C2×D4).332C23, C4.4D4⋊43C22, (C2×Q8).310C23, C42.C2⋊25C22, (C22×Q8)⋊44C22, C42⋊2C2⋊19C22, C22.29C24⋊34C2, C42⋊C2⋊64C22, C22.32C24⋊21C2, C22≀C2.14C22, C4⋊1D4.119C22, C22⋊C4.116C23, (C22×C4).400C23, C22.54C24⋊11C2, C22.45C24⋊20C2, C2.69(C2×2+ 1+4), C2.53(C2.C25), C22.26C24⋊52C2, (C22×D4).439C22, C22.D4⋊63C22, C22.57C24⋊11C2, C22.31C24⋊27C2, C23.38C23⋊34C2, C22.46C24⋊35C2, C22.49C24⋊25C2, C22.34C24⋊24C2, C22.47C24⋊34C2, C23.36C23⋊51C2, (C2×C4⋊C4)⋊91C22, (C2×C4○D4)⋊52C22, (C2×C22⋊C4).393C22, SmallGroup(128,2283)

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

Derived series
C1 — C22 — C22.140C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C22.26C24 — C22.140C25
Lower central
C1 — C22 — C22.140C25
Upper central
C1 — C22 — C22.140C25
Jennings
C1 — C22 — C22.140C25

Generators and relations for C22.140C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=ba=ab, f2=g2=a, dcd=gcg-1=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=fcf-1=bc=cb, ede-1=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 844 in 525 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, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊1D4, C4⋊Q8, C4⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C23.36C23, C22.26C24, C22.29C24, C23.38C23, C22.31C24, C22.32C24, C22.34C24, D4⋊5D4, Q8⋊5D4, C22.45C24, C22.46C24, C22.47C24, C22.49C24, C22.54C24, C22.57C24, C22.140C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C25, C2×2+ 1+4, C2.C25, C22.140C25

Smallest permutation representation of C22.140C25
►On 32 points
Generators in S32
(1 5)(2 6)(3 7)(4 8)(9 31)(10 32)(11 29)(12 30)(13 23)(14 24)(15 21)(16 22)(17 27)(18 28)(19 25)(20 26)

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim111111111111111144
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C22+ 1+4C2.C25
kernelC22.140C25C23.36C23C22.26C24C22.29C24C23.38C23C22.31C24C22.32C24C22.34C24D4⋊5D4Q8⋊5D4C22.45C24C22.46C24C22.47C24C22.49C24C22.54C24C22.57C24C4C2
# reps121112442222222224

Matrix representation of C22.140C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
01000000
10000000
00010000
00100000
00004000
00000100
00000010
00000414
,
20040000
03100000
02200000
30030000
00004142
00000040
00000400
00000001
,
30000000
03000000
00300000
00030000
00000010
00001413
00004000
00004101
,
04300000
40030000
10010000
01100000
00000100
00001000
00001413
00000004
,
01000000
40000000
10010000
04400000
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],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,4,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,4],[2,0,0,3,0,0,0,0,0,3,2,0,0,0,0,0,0,1,2,0,0,0,0,0,4,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,4,0,0,0,0,0,4,4,0,0,0,0,0,0,2,0,0,1],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,4,4,0,0,0,0,0,4,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,3,0,1],[0,4,1,0,0,0,0,0,4,0,0,1,0,0,0,0,3,0,0,1,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,4],[0,4,1,0,0,0,0,0,1,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4] >;
 

C22.140C25 in GAP, Magma, Sage, TeX 

C_2^2._{140}C_2^5
 
% in TeX
 
G:=Group("C2^2.140C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2283);
 
// by ID
 
G=gap.SmallGroup(128,2283);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,520,2019,570,136,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=b*a=a*b,f^2=g^2=a,d*c*d=g*c*g^-1=a*c=c*a,f*d*f^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e^-1=f*c*f^-1=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
 

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