Copied to
clipboard

?

G = C22.150C25  order 128 = 27

131st central stem extension by C22 of C25

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

Aliases: C23.91C24, C24.156C23, C22.150C25, C42.132C23, C4.1642+ (1+4), C22.122- (1+4), (D4×Q8)⋊32C2, D4⋊5D4⋊40C2, Q8⋊5D4⋊34C2, (C4×D4)⋊73C22, C4⋊Q8⋊103C22, (C4×Q8)⋊69C22, C23⋊2Q8⋊9C2, C4⋊D4⋊45C22, C4⋊C4.333C23, (C2×C4).140C24, C22⋊Q8⋊55C22, (C2×D4).339C23, C4.4D4⋊45C22, C22⋊C4.62C23, (C2×Q8).316C23, C42.C2⋊69C22, (C22×Q8)⋊46C22, C22.32C24⋊24C2, C42⋊C2⋊67C22, C42⋊2C2⋊20C22, C22≀C2.15C22, (C22×C4).409C23, (C2×C42).976C22, C22.45C24⋊22C2, C2.54(C2×2- (1+4)), C2.75(C2×2+ (1+4)), C2.61(C2.C25), (C22×D4).441C22, C22.D4⋊25C22, C23.36C23⋊59C2, C22.50C24⋊38C2, C22.36C24⋊40C2, C23.37C23⋊55C2, C23.38C23⋊37C2, C22.57C24⋊18C2, C22.56C24⋊15C2, (C2×C4.4D4)⋊61C2, (C2×C4○D4).247C22, (C2×C22⋊C4).394C22, SmallGroup(128,2293)

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

Derived series
C1 — C22 — C22.150C25
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C23.36C23 — C22.150C25
Lower central
C1 — C22 — C22.150C25
Upper central
C1 — C22 — C22.150C25
Jennings
C1 — C22 — C22.150C25

Subgroups: 788 in 516 conjugacy classes, 382 normal (38 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×23], C22, C22 [×2], C22 [×24], C2×C4 [×6], C2×C4 [×18], C2×C4 [×23], D4 [×19], Q8 [×17], C23, C23 [×6], C23 [×8], C42 [×4], C42 [×10], C22⋊C4 [×50], C4⋊C4 [×2], C4⋊C4 [×40], C22×C4 [×3], C22×C4 [×16], C2×D4, C2×D4 [×14], C2×D4 [×2], C2×Q8, C2×Q8 [×12], C2×Q8 [×6], C4○D4 [×4], C24 [×2], C2×C42, C2×C22⋊C4 [×8], C42⋊C2 [×8], C4×D4, C4×D4 [×12], C4×Q8, C4×Q8 [×6], C22≀C2 [×4], C4⋊D4, C4⋊D4 [×10], C22⋊Q8, C22⋊Q8 [×34], C22.D4 [×18], C4.4D4, C4.4D4 [×18], C42.C2, C42.C2 [×2], C42⋊2C2 [×14], C4⋊Q8 [×2], C4⋊Q8 [×6], C22×D4, C22×Q8, C22×Q8 [×2], C2×C4○D4 [×2], C2×C4.4D4, C23.36C23, C23.37C23, C23.38C23 [×2], C22.32C24 [×2], C22.36C24 [×6], C23⋊2Q8 [×2], D4⋊5D4 [×2], Q8⋊5D4 [×2], D4×Q8 [×2], C22.45C24 [×4], C22.50C24 [×2], C22.56C24 [×2], C22.57C24 [×2], C22.150C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ (1+4) [×2], 2- (1+4) [×2], C25, C2×2+ (1+4), C2×2- (1+4), C2.C25, C22.150C25

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

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

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

(2 9)(4 11)(5 28)(7 26)(14 32)(16 30)(17 23)(19 21)

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
00020000
00200000
02000000
20000000
00000010
00000004
00001000
00000400
,
00300000
00030000
20000000
02000000
00000010
00000001
00001000
00000100
,
10000000
01000000
00100000
00010000
00001000
00000100
00000040
00000004
,
00100000
00010000
10000000
01000000
00000100
00004000
00000001
00000040
,
01000000
40000000
00010000
00400000
00001000
00000100
00000010
00000001

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

38 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A4B4C4D4E···4Z
order1222222···244444···4
size1111224···422224···4

38 irreducible representations

dim111111111111111444
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C22+ (1+4)2- (1+4)C2.C25
kernelC22.150C25C2×C4.4D4C23.36C23C23.37C23C23.38C23C22.32C24C22.36C24C23⋊2Q8D4⋊5D4Q8⋊5D4D4×Q8C22.45C24C22.50C24C22.56C24C22.57C24C4C22C2
# reps111122622224222222

In GAP, Magma, Sage, TeX 

C_2^2._{150}C_2^5
 
% in TeX
 
G:=Group("C2^2.150C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2293);
 
// by ID
 
G=gap.SmallGroup(128,2293);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,1430,723,2019,570,248,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=e^2=1,c^2=g^2=a,f^2=b,a*b=b*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=f*c*f^-1=b*c=c*b,e*d*e=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
 

׿
×
⋊
ℤ
𝔽
○
≀