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G = C22.135C25  order 128 = 27

116th central stem extension by C22 of C25

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

Aliases: C23.76C24, C42.118C23, C22.135C25, C24.151C23, C4.912+ 1+4, D42⋊23C2, C4⋊Q8⋊99C22, D4⋊5D4⋊36C2, D4⋊6D4⋊38C2, Q8⋊6D4⋊28C2, (C4×D4)⋊68C22, (C4×Q8)⋊65C22, C4⋊D4⋊41C22, C4⋊C4.322C23, C4⋊1D4⋊27C22, (C2×C4).125C24, (C2×C42)⋊73C22, C22⋊Q8⋊51C22, C22≀C2⋊18C22, (C2×D4).327C23, C4.4D4⋊92C22, C22⋊C4.50C23, (C2×Q8).466C23, C42.C2⋊67C22, C22.32C24⋊20C2, C42⋊2C2⋊45C22, C42⋊C2⋊62C22, C22.29C24⋊32C2, (C22×C4).395C23, C22.54C24⋊10C2, C2.64(C2×2+ 1+4), C2.51(C2.C25), C22.26C24⋊48C2, (C22×D4).438C22, C22.D4⋊22C22, C22.50C24⋊33C2, C22.36C24⋊32C2, C22.47C24⋊32C2, C23.36C23⋊49C2, C22.31C24⋊24C2, (C2×C4⋊C4)⋊90C22, (C2×C4○D4)⋊50C22, (C2×C22⋊C4)⋊64C22, SmallGroup(128,2278)

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

Derived series
C1 — C22 — C22.135C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C22.26C24 — C22.135C25
Lower central
C1 — C22 — C22.135C25
Upper central
C1 — C22 — C22.135C25
Jennings
C1 — C22 — C22.135C25

Generators and relations for C22.135C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=b, g2=a, ab=ba, dcd=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: 1012 in 575 conjugacy classes, 382 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊1D4, C4⋊Q8, C22×D4, C2×C4○D4, C23.36C23, C22.26C24, C22.29C24, C22.31C24, C22.32C24, C22.36C24, D42, D4⋊5D4, D4⋊6D4, Q8⋊6D4, Q8⋊6D4, C22.47C24, C22.50C24, C22.54C24, C22.135C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C25, C2×2+ 1+4, C2.C25, C22.135C25

Smallest permutation representation of C22.135C25
►On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 13)(10 14)(11 15)(12 16)(21 29)(22 30)(23 31)(24 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 10)(2 9)(3 12)(4 11)(5 21)(6 24)(7 23)(8 22)(13 28)(14 27)(15 26)(16 25)(17 32)(18 31)(19 30)(20 29)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D···2N4A···4F4G···4W
order12222···24···44···4
size11114···42···24···4

38 irreducible representations

dim1111111111111144
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C22+ 1+4C2.C25
kernelC22.135C25C23.36C23C22.26C24C22.29C24C22.31C24C22.32C24C22.36C24D42D4⋊5D4D4⋊6D4Q8⋊6D4C22.47C24C22.50C24C22.54C24C4C2
# reps1124242141321442

Matrix representation of C22.135C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
04000000
40000000
00040000
00400000
00000002
00000020
00000300
00003000
,
00010000
00400000
04000000
10000000
00000001
00000040
00000400
00001000
,
40000000
04000000
00400000
00040000
00000020
00000002
00002000
00000200
,
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],[0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0],[0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0],[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] >;
 

C22.135C25 in GAP, Magma, Sage, TeX 

C_2^2._{135}C_2^5
 
% in TeX
 
G:=Group("C2^2.135C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2278);
 
// by ID
 
G=gap.SmallGroup(128,2278);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,352,2019,570,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=b,g^2=a,a*b=b*a,d*c*d=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
 

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