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G = C22.83C25  order 128 = 27

64th central stem extension by C22 of C25

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

Aliases: C22.83C25, C23.133C24, C42.572C23, C24.505C23, C22.182+ (1+4), C4⋊Q8⋊33C22, D4⋊5D4⋊17C2, D4⋊6D4⋊20C2, (C4×D4)⋊41C22, (C2×C4).74C24, (C4×Q8)⋊42C22, C4⋊1D4⋊49C22, C4⋊C4.486C23, C4⋊D4⋊25C22, (C2×C42)⋊57C22, C22⋊Q8⋊29C22, C22≀C2.8C22, (C2×D4).301C23, C4.4D4⋊26C22, C22⋊C4.99C23, (C2×Q8).286C23, C42.C2⋊54C22, C42⋊2C2⋊35C22, C22.45C24⋊4C2, C22.29C24⋊21C2, C22.11C24⋊17C2, C42⋊C2⋊38C22, C22.19C24⋊29C2, (C22×C4).356C23, (C23×C4).609C22, C2.30(C2×2+ (1+4)), C2.20(C2.C25), C22.26C24⋊33C2, C22.34C24⋊9C2, (C22×D4).425C22, C22.D4⋊51C22, C23.36C23⋊27C2, C22.36C24⋊12C2, C22.49C24⋊12C2, C23.41C23⋊13C2, C22.47C24⋊14C2, C23.33C23⋊19C2, C4⋊C4○(C4⋊D4), (C2×C4)⋊12(C4○D4), (C2×C4⋊D4)⋊66C2, C22⋊C4○(C4⋊D4), C4.139(C2×C4○D4), (C2×C4⋊C4)⋊142C22, (C2×C4○D4)⋊28C22, C2.48(C22×C4○D4), C22.28(C2×C4○D4), (C2×C42⋊C2)⋊67C2, (C2×C22⋊C4).380C22, SmallGroup(128,2226)

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

Derived series
C1 — C22 — C22.83C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C42⋊C2 — C22.83C25
Lower central
C1 — C22 — C22.83C25
Upper central
C1 — C22 — C22.83C25
Jennings
C1 — C22 — C22.83C25

Subgroups: 900 in 571 conjugacy classes, 390 normal (50 characteristic)
C1, C2 [×3], C2 [×11], C4 [×4], C4 [×20], C22, C22 [×4], C22 [×33], C2×C4 [×4], C2×C4 [×22], C2×C4 [×34], D4 [×40], Q8 [×6], C23 [×3], C23 [×6], C23 [×13], C42 [×14], C22⋊C4 [×46], C4⋊C4 [×6], C4⋊C4 [×28], C22×C4 [×6], C22×C4 [×20], C22×C4 [×4], C2×D4, C2×D4 [×26], C2×D4 [×8], C2×Q8, C2×Q8 [×4], C4○D4 [×12], C24, C24 [×2], C2×C42 [×2], C2×C22⋊C4 [×8], C2×C4⋊C4 [×3], C2×C4⋊C4 [×2], C42⋊C2 [×5], C42⋊C2 [×10], C4×D4 [×26], C4×Q8 [×2], C22≀C2 [×6], C4⋊D4 [×2], C4⋊D4 [×24], C22⋊Q8 [×2], C22⋊Q8 [×8], C22.D4 [×18], C4.4D4 [×10], C42.C2 [×4], C42⋊2C2 [×8], C4⋊1D4 [×2], C4⋊Q8 [×4], C23×C4, C22×D4, C22×D4 [×2], C2×C4○D4, C2×C4○D4 [×4], C2×C42⋊C2, C22.11C24, C23.33C23, C2×C4⋊D4, C22.19C24, C23.36C23 [×2], C22.26C24 [×2], C22.29C24, C22.34C24 [×2], C22.36C24 [×2], C23.41C23, D4⋊5D4 [×4], D4⋊6D4 [×2], C22.45C24 [×4], C22.47C24 [×4], C22.49C24 [×2], C22.83C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], 2+ (1+4) [×2], C25, C22×C4○D4, C2×2+ (1+4), C2.C25, C22.83C25

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

Smallest permutation representation
►On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 30)(6 31)(7 32)(8 29)(9 15)(10 16)(11 13)(12 14)(17 23)(18 24)(19 21)(20 22)

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

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

(2 28)(4 26)(5 30)(7 32)(10 16)(12 14)(18 24)(20 22)

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
100000
040000
000120
001002
000004
000040
,
300000
030000
000300
003000
002002
000220
,
010000
100000
000310
002004
000003
000020
,
400000
040000
001002
000120
000040
000004
,
400000
040000
000120
004003
000001
000040

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,0,2,4,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,3,2,0,0,0,3,0,0,2,0,0,0,0,0,2,0,0,0,0,2,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,1,0,0,2,0,0,0,4,3,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,4,0,0,0,2,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,2,0,0,4,0,0,0,3,1,0] >;
 

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2N4A···4N4O···4AC
order122222222···24···44···4
size111122224···42···24···4

44 irreducible representations

dim11111111111111111244
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C4○D42+ (1+4)C2.C25
kernelC22.83C25C2×C42⋊C2C22.11C24C23.33C23C2×C4⋊D4C22.19C24C23.36C23C22.26C24C22.29C24C22.34C24C22.36C24C23.41C23D4⋊5D4D4⋊6D4C22.45C24C22.47C24C22.49C24C2×C4C22C2
# reps11111122122142442822

In GAP, Magma, Sage, TeX 

C_2^2._{83}C_2^5
 
% in TeX
 
G:=Group("C2^2.83C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2226);
 
// by ID
 
G=gap.SmallGroup(128,2226);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,570,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=1,d^2=b,g^2=a,a*b=b*a,d*c*d^-1=g*c*g^-1=a*c=c*a,f*d*f=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*c*e=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*f=f*c,d*e=e*d,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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