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G = C22.151C25  order 128 = 27

132nd central stem extension by C22 of C25

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

Aliases: C23.92C24, C24.157C23, C22.151C25, C42.133C23, C4.1652+ 1+4, D4⋊6D4⋊45C2, D4⋊5D4⋊41C2, Q8⋊5D4⋊35C2, Q8⋊6D4⋊31C2, (C4×D4)⋊74C22, C4⋊Q8⋊104C22, (C4×Q8)⋊70C22, C4⋊1D4⋊29C22, C4⋊D4⋊94C22, C4⋊C4.334C23, (C2×C4).141C24, (C2×C42)⋊78C22, C22⋊Q8⋊56C22, (C2×D4).340C23, C4.4D4⋊46C22, (C2×Q8).472C23, C42.C2⋊27C22, (C22×Q8)⋊47C22, C42⋊2C2⋊48C22, C42⋊C2⋊68C22, C22.29C24⋊36C2, C22.32C24⋊25C2, C22≀C2.16C22, C22⋊C4.119C23, (C22×C4).410C23, C22.54C24⋊13C2, C22.45C24⋊23C2, C2.76(C2×2+ 1+4), C2.62(C2.C25), C22.26C24⋊57C2, (C22×D4).442C22, C22.D4⋊26C22, C22.34C24⋊27C2, C22.47C24⋊39C2, C22.35C24⋊23C2, C22.56C24⋊16C2, C22.50C24⋊39C2, C22.53C24⋊26C2, C22.33C24⋊23C2, C23.38C23⋊38C2, C23.36C23⋊60C2, (C2×C4⋊C4)⋊93C22, (C2×C4○D4)⋊55C22, (C2×C22⋊C4).395C22, SmallGroup(128,2294)

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

Derived series
C1 — C22 — C22.151C25
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C23.36C23 — C22.151C25
Lower central
C1 — C22 — C22.151C25
Upper central
C1 — C22 — C22.151C25
Jennings
C1 — C22 — C22.151C25

Generators and relations for C22.151C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=a, f2=ba=ab, 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 (122 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊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×D4, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C42⋊2C2, C4⋊1D4, C4⋊1D4, C4⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C23.36C23, C22.26C24, C22.29C24, C23.38C23, C22.32C24, C22.33C24, C22.34C24, C22.34C24, C22.35C24, D4⋊5D4, D4⋊6D4, Q8⋊5D4, Q8⋊6D4, C22.45C24, C22.47C24, C22.50C24, C22.53C24, C22.54C24, C22.56C24, C22.151C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C25, C2×2+ 1+4, C2.C25, C22.151C25

Smallest permutation representation of C22.151C25
►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 16)(3 21)(4 14)(5 29)(6 10)(7 31)(8 12)(9 20)(11 18)(13 25)(15 27)(17 30)(19 32)(22 26)(24 28)

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

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

(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)(2,16)(3,21)(4,14)(5,29)(6,10)(7,31)(8,12)(9,20)(11,18)(13,25)(15,27)(17,30)(19,32)(22,26)(24,28), (1,6)(2,18)(3,8)(4,20)(5,26)(7,28)(9,16)(10,21)(11,14)(12,23)(13,30)(15,32)(17,27)(19,25)(22,31)(24,29), (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), (1,29,25,11)(2,30,26,12)(3,31,27,9)(4,32,28,10)(5,21,18,15)(6,22,19,16)(7,23,20,13)(8,24,17,14), (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)(2,16)(3,21)(4,14)(5,29)(6,10)(7,31)(8,12)(9,20)(11,18)(13,25)(15,27)(17,30)(19,32)(22,26)(24,28), (1,6)(2,18)(3,8)(4,20)(5,26)(7,28)(9,16)(10,21)(11,14)(12,23)(13,30)(15,32)(17,27)(19,25)(22,31)(24,29), (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), (1,29,25,11)(2,30,26,12)(3,31,27,9)(4,32,28,10)(5,21,18,15)(6,22,19,16)(7,23,20,13)(8,24,17,14), (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),(2,16),(3,21),(4,14),(5,29),(6,10),(7,31),(8,12),(9,20),(11,18),(13,25),(15,27),(17,30),(19,32),(22,26),(24,28)], [(1,6),(2,18),(3,8),(4,20),(5,26),(7,28),(9,16),(10,21),(11,14),(12,23),(13,30),(15,32),(17,27),(19,25),(22,31),(24,29)], [(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)], [(1,29,25,11),(2,30,26,12),(3,31,27,9),(4,32,28,10),(5,21,18,15),(6,22,19,16),(7,23,20,13),(8,24,17,14)], [(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 2A2B2C2D···2L4A···4F4G···4Y
order12222···24···44···4
size11114···42···24···4

38 irreducible representations

dim111111111111111111144
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C22+ 1+4C2.C25
kernelC22.151C25C23.36C23C22.26C24C22.29C24C23.38C23C22.32C24C22.33C24C22.34C24C22.35C24D4⋊5D4D4⋊6D4Q8⋊5D4Q8⋊6D4C22.45C24C22.47C24C22.50C24C22.53C24C22.54C24C22.56C24C4C2
# reps121114231212122112224

Matrix representation of C22.151C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
01000000
10000000
41130000
00040000
00000100
00001000
00000004
00000040
,
32210000
00300000
02000000
20320000
00000300
00002000
00000002
00000030
,
20000000
02000000
00200000
00020000
00000030
00000003
00003000
00000300
,
00100000
41130000
40000000
41040000
00000010
00000001
00001000
00000100
,
01000000
40000000
41130000
40140000
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],[0,1,4,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0],[3,0,0,2,0,0,0,0,2,0,2,0,0,0,0,0,2,3,0,3,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0],[0,4,4,4,0,0,0,0,0,1,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,3,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,4,4,4,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0] >;
 

C22.151C25 in GAP, Magma, Sage, TeX 

C_2^2._{151}C_2^5
 
% in TeX
 
G:=Group("C2^2.151C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2294);
 
// by ID
 
G=gap.SmallGroup(128,2294);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,184,2019,570,360,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=a,f^2=b*a=a*b,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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