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G = C22.102C25  order 128 = 27

83rd central stem extension by C22 of C25

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

Aliases: C23.51C24, C42.93C23, C22.102C25, C24.509C23, C22.92+ (1+4), (D42)⋊15C2, C4⋊Q8⋊35C22, D4⋊13(C4○D4), D4⋊6D4⋊25C2, D4⋊5D4⋊22C2, D4⋊3Q8⋊25C2, (C4×D4)⋊50C22, (C2×C4).92C24, (C4×Q8)⋊49C22, C4⋊C4.300C23, C4⋊D4⋊85C22, (C23×C4)⋊44C22, (C2×C42)⋊64C22, C22⋊Q8⋊36C22, C22≀C2⋊10C22, C22.32C24⋊6C2, (C2×D4).475C23, D4○2(C22.D4), C4.4D4⋊29C22, C22⋊C4.26C23, (C2×Q8).293C23, C42.C2⋊58C22, C42⋊2C2⋊38C22, C42⋊C2⋊43C22, C22.19C24⋊32C2, C22.45C24⋊8C2, C22.11C24⋊21C2, C4⋊1D4.114C22, (C22×C4).372C23, C2.39(C2×2+ (1+4)), C2.33(C2.C25), C22.33C24⋊6C2, (C22×D4).429C22, C22.D4⋊10C22, C22.34C24⋊12C2, C22.53C24⋊15C2, C22.47C24⋊19C2, C23.33C23⋊25C2, C22.46C24⋊20C2, C22.36C24⋊16C2, C23.36C23⋊33C2, (C2×C4×D4)⋊93C2, (C2×C4⋊C4)⋊77C22, C4.275(C2×C4○D4), (C2×C4○D4)⋊35C22, C2.58(C22×C4○D4), C22.41(C2×C4○D4), (C2×C22⋊C4)⋊51C22, (C2×C22.D4)⋊61C2, SmallGroup(128,2245)

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

Derived series
C1 — C22 — C22.102C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C2×C4×D4 — C22.102C25
Lower central
C1 — C22 — C22.102C25
Upper central
C1 — C22 — C22.102C25
Jennings
C1 — C22 — C22.102C25

Subgroups: 916 in 576 conjugacy classes, 390 normal (122 characteristic)
C1, C2 [×3], C2 [×12], C4 [×2], C4 [×21], C22, C22 [×6], C22 [×36], C2×C4 [×12], C2×C4 [×10], C2×C4 [×35], D4 [×4], D4 [×35], Q8 [×5], C23 [×3], C23 [×6], C23 [×18], C42 [×6], C42 [×6], C22⋊C4 [×10], C22⋊C4 [×38], C4⋊C4 [×14], C4⋊C4 [×22], C22×C4 [×5], C22×C4 [×22], C22×C4 [×4], C2×D4 [×6], C2×D4 [×18], C2×D4 [×12], C2×Q8 [×2], C2×Q8 [×2], C4○D4 [×8], C24 [×4], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4 [×2], C2×C4⋊C4 [×4], C42⋊C2 [×3], C42⋊C2 [×6], C4×D4 [×11], C4×D4 [×18], C4×Q8 [×3], C22≀C2 [×8], C4⋊D4 [×5], C4⋊D4 [×12], C22⋊Q8 [×3], C22⋊Q8 [×10], C22.D4 [×2], C22.D4 [×24], C4.4D4 [×3], C4.4D4 [×4], C42.C2 [×3], C42.C2 [×2], C42⋊2C2 [×2], C42⋊2C2 [×8], C4⋊1D4, C4⋊Q8, C23×C4 [×2], C22×D4 [×2], C22×D4 [×2], C2×C4○D4, C2×C4○D4 [×2], C2×C4×D4, C22.11C24, C23.33C23, C2×C22.D4 [×2], C22.19C24 [×2], C23.36C23 [×2], C22.32C24 [×2], C22.33C24 [×2], C22.34C24, C22.36C24, D42, D4⋊5D4 [×2], D4⋊5D4 [×2], D4⋊6D4, C22.45C24 [×4], C22.46C24, C22.47C24, C22.47C24 [×2], D4⋊3Q8, C22.53C24, C22.102C25

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, C22.102C25

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

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

(5 30)(6 31)(7 32)(8 29)(13 18)(14 19)(15 20)(16 17)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
004000
000400
000040
000004
,
400000
010000
000010
000024
001000
002400
,
400000
040000
001000
000100
000040
000004
,
040000
400000
001400
000400
000014
000004
,
400000
040000
000010
000001
001000
000100
,
200000
020000
002300
000300
000023
000003

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

44 conjugacy classes

class 1 2A2B2C2D···2I2J···2O4A···4L4M···4AB
order12222···22···24···44···4
size11112···24···42···24···4

44 irreducible representations

dim1111111111111111111244
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C4○D42+ (1+4)C2.C25
kernelC22.102C25C2×C4×D4C22.11C24C23.33C23C2×C22.D4C22.19C24C23.36C23C22.32C24C22.33C24C22.34C24C22.36C24D42D4⋊5D4D4⋊6D4C22.45C24C22.46C24C22.47C24D4⋊3Q8C22.53C24D4C22C2
# reps1111222221114141311822

In GAP, Magma, Sage, TeX 

C_2^2._{102}C_2^5
 
% in TeX
 
G:=Group("C2^2.102C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2245);
 
// by ID
 
G=gap.SmallGroup(128,2245);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,570,1684,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=1,g^2=b,a*b=b*a,d*c*d=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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ℤ
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