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G = C22.153C25  order 128 = 27

134th central stem extension by C22 of C25

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

Aliases: C23.93C24, C24.158C23, C42.135C23, C22.153C25, C22.142- 1+4, C4⋊Q8⋊47C22, D4⋊3Q8⋊45C2, (C4×D4)⋊75C22, (C4×Q8)⋊71C22, C4⋊C4.336C23, (C2×C4).143C24, C23⋊2Q8⋊10C2, C22⋊Q8⋊57C22, (C2×D4).341C23, C4.4D4⋊94C22, C22⋊C4.63C23, (C2×Q8).318C23, C42.C2⋊70C22, C42⋊C2⋊69C22, C42⋊2C2⋊49C22, C22≀C2.17C22, C4⋊D4.124C22, (C2×C42).978C22, (C22×C4).412C23, C22.32C24.4C2, C22.45C24⋊24C2, C2.56(C2×2- 1+4), C2.64(C2.C25), C22.D4⋊27C22, C22.36C24⋊41C2, C23.36C23⋊61C2, C22.57C24⋊20C2, C22.50C24⋊40C2, C22.46C24⋊40C2, C22.35C24⋊25C2, C22.33C24⋊24C2, C23.41C23⋊26C2, C23.37C23⋊57C2, (C2×C4⋊C4)⋊94C22, (C2×C42⋊2C2)⋊42C2, (C2×C22⋊C4).396C22, SmallGroup(128,2296)

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

Derived series
C1 — C22 — C22.153C25
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C42⋊2C2 — C22.153C25
Lower central
C1 — C22 — C22.153C25
Upper central
C1 — C22 — C22.153C25
Jennings
C1 — C22 — C22.153C25

Generators and relations for C22.153C25
 G = < a,b,c,d,e,f,g | a2=b2=d2=g2=1, c2=b, e2=a, f2=ba=ab, dcd=gcg=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: 636 in 468 conjugacy classes, 380 normal (58 characteristic)
C1, C2, C2, C4, C22, 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×Q8, C2×Q8, C24, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊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.C2, C42⋊2C2, C4⋊Q8, C4⋊Q8, C2×C42⋊2C2, C23.36C23, C23.37C23, C22.32C24, C22.33C24, C22.33C24, C22.35C24, C22.36C24, C23⋊2Q8, C23.41C23, C22.45C24, C22.46C24, D4⋊3Q8, C22.50C24, C22.57C24, C22.153C25
Quotients: C1, C2, C22, C23, C24, 2- 1+4, C25, C2×2- 1+4, C2.C25, C22.153C25

Smallest permutation representation of C22.153C25
►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 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)

(2 28)(4 26)(5 7)(6 19)(8 17)(9 13)(11 15)(18 20)(21 23)(22 32)(24 30)(29 31)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4AB
order122222222244444···4
size111122444422224···4

38 irreducible representations

dim11111111111111144
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C22- 1+4C2.C25
kernelC22.153C25C2×C42⋊2C2C23.36C23C23.37C23C22.32C24C22.33C24C22.35C24C22.36C24C23⋊2Q8C23.41C23C22.45C24C22.46C24D4⋊3Q8C22.50C24C22.57C24C22C2
# reps11111342114422424

Matrix representation of C22.153C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
00100000
00010000
10000000
01000000
00000010
00000001
00004000
00000400
,
10000000
04000000
00400000
00010000
00001000
00000400
00000040
00000001
,
20000000
02000000
00200000
00020000
00000300
00003000
00000002
00000020
,
01000000
40000000
00010000
00400000
00000100
00001000
00000004
00000040
,
10000000
01000000
00400000
00040000
00001000
00000100
00000040
00000004

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

C22.153C25 in GAP, Magma, Sage, TeX 

C_2^2._{153}C_2^5
 
% in TeX
 
G:=Group("C2^2.153C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2296);
 
// by ID
 
G=gap.SmallGroup(128,2296);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,1430,723,184,2019,570,360,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=g^2=1,c^2=b,e^2=a,f^2=b*a=a*b,d*c*d=g*c*g=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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