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G = C22.149C25  order 128 = 27

130th central stem extension by C22 of C25

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

Aliases: C23.90C24, C22.149C25, C42.131C23, C24.155C23, C22.252+ 1+4, D4⋊5D4⋊39C2, D4⋊6D4⋊44C2, (C4×D4)⋊72C22, C4⋊Q8⋊102C22, (C4×Q8)⋊68C22, C4⋊C4.332C23, C4⋊1D4⋊56C22, C23⋊3D4⋊17C2, C4⋊D4⋊44C22, (C2×C4).139C24, (C2×C42)⋊77C22, C22⋊Q8⋊54C22, C22≀C2⋊21C22, (C2×D4).338C23, C4.4D4⋊93C22, C22⋊C4.61C23, (C2×Q8).315C23, C42.C2⋊26C22, C42⋊C2⋊66C22, C42⋊2C2⋊47C22, C22.32C24⋊23C2, (C22×C4).408C23, C22.54C24⋊12C2, C22.45C24⋊21C2, C2.74(C2×2+ 1+4), C2.60(C2.C25), C22.26C24⋊56C2, (C22×D4).440C22, C22.D4⋊24C22, C22.47C24⋊38C2, C22.57C24⋊17C2, C22.33C24⋊22C2, C22.31C24⋊29C2, C23.36C23⋊58C2, C22.36C24⋊39C2, (C2×C4⋊C4)⋊92C22, (C2×C4○D4)⋊54C22, (C2×C42⋊2C2)⋊41C2, (C2×C22⋊C4)⋊66C22, SmallGroup(128,2292)

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

Derived series
C1 — C22 — C22.149C25
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C42⋊2C2 — C22.149C25
Lower central
C1 — C22 — C22.149C25
Upper central
C1 — C22 — C22.149C25
Jennings
C1 — C22 — C22.149C25

Generators and relations for C22.149C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, 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: 860 in 528 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×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, 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⋊1D4, C4⋊Q8, C4⋊Q8, C22×D4, C2×C4○D4, C2×C42⋊2C2, C23.36C23, C22.26C24, C23⋊3D4, C22.31C24, C22.32C24, C22.32C24, C22.33C24, C22.33C24, C22.36C24, D4⋊5D4, D4⋊6D4, C22.45C24, C22.47C24, C22.54C24, C22.57C24, C22.149C25
Quotients: C1, C2, C22, C23, C24, 2+ 1+4, C25, C2×2+ 1+4, C2.C25, C22.149C25

Smallest permutation representation of C22.149C25
►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 7)(2 19)(3 5)(4 17)(6 26)(8 28)(9 23)(10 16)(11 21)(12 14)(13 31)(15 29)(18 27)(20 25)(22 32)(24 30)

(2 28)(4 26)(5 7)(6 19)(8 17)(9 11)(10 32)(12 30)(14 22)(16 24)(18 20)(29 31)

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

38 conjugacy classes

class 1 2A2B2C2D2E2F···2M4A4B4C4D4E···4X
order1222222···244444···4
size1111224···422224···4

38 irreducible representations

dim11111111111111144
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C22+ 1+4C2.C25
kernelC22.149C25C2×C42⋊2C2C23.36C23C22.26C24C23⋊3D4C22.31C24C22.32C24C22.33C24C22.36C24D4⋊5D4D4⋊6D4C22.45C24C22.47C24C22.54C24C22.57C24C22C2
# reps11111153242242224

Matrix representation of C22.149C25 ►in GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
04030000
10200000
01010000
40400000
00000404
00001010
00000201
00003040
,
10000000
01000000
40400000
04040000
00001000
00000100
00003040
00000304
,
30100000
03010000
00200000
00020000
00003000
00000300
00000030
00000003
,
40300000
04030000
00100000
00010000
00004040
00000404
00002010
00000201
,
01000000
10000000
00010000
00100000
00000100
00001000
00000001
00000010

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],[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],[0,1,0,4,0,0,0,0,4,0,1,0,0,0,0,0,0,2,0,4,0,0,0,0,3,0,1,0,0,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,4,0,2,0,0,0,0,0,0,1,0,4,0,0,0,0,4,0,1,0],[1,0,4,0,0,0,0,0,0,1,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,0,0,1,0,3,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,1,0,0,0,0,0,0,3,0,1,0,0,0,0,0,0,0,0,4,0,2,0,0,0,0,0,0,4,0,2,0,0,0,0,4,0,1,0,0,0,0,0,0,4,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C22.149C25 in GAP, Magma, Sage, TeX 

C_2^2._{149}C_2^5
 
% in TeX
 
G:=Group("C2^2.149C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2292);
 
// by ID
 
G=gap.SmallGroup(128,2292);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,184,2019,570,360,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^2=1,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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