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G = C22.147C25  order 128 = 27

128th central stem extension by C22 of C25

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

Aliases: C23.88C24, C22.147C25, C24.154C23, C42.130C23, C4.1632+ (1+4), C22.232+ (1+4), (D42)⋊25C2, D4⋊5D4⋊38C2, Q8⋊5D4⋊33C2, (C4×D4)⋊71C22, C4⋊Q8⋊101C22, (C4×Q8)⋊67C22, C4⋊D4⋊43C22, C4⋊C4.331C23, C4⋊1D4⋊55C22, C23⋊3D4⋊16C2, (C2×C4).137C24, (C2×C42)⋊76C22, C22⋊Q8⋊53C22, C22≀C2⋊20C22, C24⋊C22⋊8C2, (C2×D4).336C23, C4.4D4⋊44C22, (C22×D4)⋊49C22, (C2×Q8).313C23, C42.C2⋊68C22, (C22×Q8)⋊45C22, C42⋊2C2⋊46C22, C22.29C24⋊35C2, C22.32C24⋊22C2, C42⋊C2⋊65C22, C22⋊C4.117C23, (C22×C4).406C23, C2.72(C2×2+ (1+4)), C2.58(C2.C25), C22.26C24⋊54C2, C22.D4⋊23C22, C23.36C23⋊56C2, C22.53C24⋊25C2, C22.56C24⋊13C2, C22.36C24⋊38C2, (C2×C4.4D4)⋊60C2, (C2×C4○D4)⋊53C22, (C2×C22⋊C4)⋊65C22, SmallGroup(128,2290)

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

Derived series
C1 — C22 — C22.147C25
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4.4D4 — C22.147C25
Lower central
C1 — C22 — C22.147C25
Upper central
C1 — C22 — C22.147C25
Jennings
C1 — C22 — C22.147C25

Subgroups: 1044 in 580 conjugacy classes, 382 normal (38 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×19], C22, C22 [×2], C22 [×44], C2×C4 [×6], C2×C4 [×14], C2×C4 [×23], D4 [×47], Q8 [×9], C23, C23 [×10], C23 [×20], C42 [×4], C42 [×6], C22⋊C4 [×54], C4⋊C4 [×2], C4⋊C4 [×20], C22×C4 [×3], C22×C4 [×16], C2×D4, C2×D4 [×36], C2×D4 [×14], C2×Q8, C2×Q8 [×6], C2×Q8 [×2], C4○D4 [×8], C24 [×6], C2×C42, C2×C22⋊C4 [×8], C42⋊C2 [×2], C4×D4, C4×D4 [×16], C4×Q8, C4×Q8 [×2], C22≀C2 [×20], C4⋊D4, C4⋊D4 [×26], C22⋊Q8, C22⋊Q8 [×10], C22.D4 [×18], C4.4D4, C4.4D4 [×24], C42.C2, C42⋊2C2 [×6], C4⋊1D4, C4⋊1D4 [×2], C4⋊Q8, C22×D4, C22×D4 [×6], C22×Q8, C2×C4○D4 [×4], C2×C4.4D4, C23.36C23, C22.26C24, C23⋊3D4 [×2], C22.29C24 [×2], C22.32C24 [×6], C22.36C24 [×2], D42 [×2], D4⋊5D4 [×6], Q8⋊5D4 [×2], C22.53C24 [×2], C24⋊C22 [×2], C22.56C24 [×2], C22.147C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ (1+4) [×4], C25, C2×2+ (1+4) [×2], C2.C25, C22.147C25

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g2=1, e2=a, f2=b, ab=ba, 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 >

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100
,
02000000
30000000
00030000
00200000
00001000
00000400
00000040
00000001
,
20000000
02000000
00200000
00020000
00000400
00001000
00000001
00000040
,
01000000
10000000
00010000
00100000
00000100
00004000
00000004
00000010
,
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],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0],[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] >;
 

38 conjugacy classes

class 1 2A2B2C2D2E2F···2O4A4B4C4D4E···4V
order1222222···244444···4
size1111224···422224···4

38 irreducible representations

dim11111111111111444
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C22+ (1+4)2+ (1+4)C2.C25
kernelC22.147C25C2×C4.4D4C23.36C23C22.26C24C23⋊3D4C22.29C24C22.32C24C22.36C24D42D4⋊5D4Q8⋊5D4C22.53C24C24⋊C22C22.56C24C4C22C2
# reps11112262262222222

In GAP, Magma, Sage, TeX 

C_2^2._{147}C_2^5
 
% in TeX
 
G:=Group("C2^2.147C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2290);
 
// by ID
 
G=gap.SmallGroup(128,2290);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,723,184,2019,570,248,1684]);
 
// 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*b=b*a,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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