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G = C22.82C25  order 128 = 27

63rd central stem extension by C22 of C25

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

Aliases: C22.82C25, C23.39C24, C24.504C23, C42.571C23, C4⋊Q8⋊32C22, (C4×D4)⋊40C22, (C4×Q8)⋊41C22, C4⋊C4.519C23, C4⋊D4⋊24C22, (C2×C4).602C24, C22⋊Q8⋊28C22, C22.32C24⋊3C2, (C2×D4).300C23, C4.4D4⋊79C22, C22⋊C4.98C23, (C2×Q8).285C23, C42.C2⋊53C22, C22.19C24⋊28C2, C42⋊C2⋊37C22, C42⋊2C2⋊34C22, C22≀C2.27C22, (C23×C4).608C22, (C2×C42).940C22, (C22×C4).355C23, C2.19(C2.C25), C23.36C23⋊26C2, C22.47C24⋊13C2, C22.46C24⋊14C2, C22.50C24⋊18C2, C23.41C23⋊12C2, C22.31C24⋊14C2, C23.33C23⋊18C2, C22.49C24⋊11C2, C22.D4.43C22, (C2×C4⋊C4)⋊71C22, C4.138(C2×C4○D4), C4⋊C4○(C22.D4), C22.15(C2×C4○D4), C2.47(C22×C4○D4), (C2×C42⋊C2)⋊66C2, (C2×C4).491(C4○D4), (C2×C4○D4).225C22, (C2×C22⋊C4).546C22, SmallGroup(128,2225)

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

Derived series
C1 — C22 — C22.82C25
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C22.82C25
Lower central
C1 — C22 — C22.82C25
Upper central
C1 — C22 — C22.82C25
Jennings
C1 — C22 — C22.82C25

Subgroups: 732 in 520 conjugacy classes, 388 normal (18 characteristic)
C1, C2 [×3], C2 [×8], C4 [×4], C4 [×22], C22, C22 [×2], C22 [×24], C2×C4 [×2], C2×C4 [×26], C2×C4 [×30], D4 [×24], Q8 [×8], C23, C23 [×6], C23 [×4], C42 [×20], C22⋊C4 [×44], C4⋊C4 [×44], C22×C4 [×2], C22×C4 [×20], C22×C4 [×4], C2×D4 [×18], C2×Q8 [×6], C4○D4 [×8], C24, C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×4], C42⋊C2 [×22], C4×D4 [×24], C4×Q8 [×8], C22≀C2 [×4], C4⋊D4 [×20], C22⋊Q8 [×12], C22.D4 [×12], C4.4D4 [×12], C42.C2 [×8], C42⋊2C2 [×16], C4⋊Q8 [×4], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, C23.33C23 [×2], C22.19C24 [×2], C23.36C23 [×4], C22.31C24, C22.32C24 [×4], C23.41C23, C22.46C24 [×4], C22.47C24 [×4], C22.49C24 [×4], C22.50C24 [×4], C22.82C25

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×4], C24 [×31], C2×C4○D4 [×6], C25, C22×C4○D4, C2.C25 [×2], C22.82C25

Generators and relations
 G = < a,b,c,d,e,f,g | a2=b2=c2=g2=1, d2=b, e2=f2=a, ab=ba, dcd-1=gcg=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece-1=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 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 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 31)(2 27)(3 29)(4 25)(5 21)(6 11)(7 23)(8 9)(10 18)(12 20)(13 28)(14 30)(15 26)(16 32)(17 24)(19 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 8 15 17)(2 5 16 18)(3 6 13 19)(4 7 14 20)(9 28 24 29)(10 25 21 30)(11 26 22 31)(12 27 23 32)

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
010000
100000
003030
000001
004020
000100
,
300000
030000
003300
000200
004224
000003
,
400000
010000
002000
000200
001030
000003
,
400000
040000
003000
004200
000030
002022
,
100000
010000
004000
000400
002010
000001

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

44 conjugacy classes

class 1 2A2B2C2D2E2F···2K4A···4P4Q···4AF
order1222222···24···44···4
size1111224···42···24···4

44 irreducible representations

dim11111111111124
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C4○D4C2.C25
kernelC22.82C25C2×C42⋊C2C23.33C23C22.19C24C23.36C23C22.31C24C22.32C24C23.41C23C22.46C24C22.47C24C22.49C24C22.50C24C2×C4C2
# reps11224141444484

In GAP, Magma, Sage, TeX 

C_2^2._{82}C_2^5
 
% in TeX
 
G:=Group("C2^2.82C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2225);
 
// by ID
 
G=gap.SmallGroup(128,2225);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,184,570,136,1684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=g^2=1,d^2=b,e^2=f^2=a,a*b=b*a,d*c*d^-1=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=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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