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G = C22.77C25  order 128 = 27

58th central stem extension by C22 of C25

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

Aliases: C42.79C23, C23.37C24, C22.77C25, C24.131C23, C22⋊22+ 1+4, D42⋊12C2, C4○D4⋊17D4, D4⋊14(C2×D4), D4○(C4⋊D4), Q8⋊13(C2×D4), Q8○(C22⋊Q8), D4⋊5D4⋊14C2, Q8⋊6D4⋊16C2, Q8⋊5D4⋊14C2, (C4×D4)⋊35C22, (C2×C4).70C24, (C4×Q8)⋊37C22, C2.29(D4×C23), C22≀C2⋊5C22, C4⋊1D4⋊17C22, C4⋊C4.291C23, C4⋊D4⋊81C22, (C23×C4)⋊40C22, C4.118(C22×D4), C22⋊Q8⋊24C22, (C2×D4).465C23, C4.4D4⋊23C22, (C22×D4)⋊34C22, C22⋊C4.17C23, (C2×2+ 1+4)⋊9C2, (C2×Q8).442C23, (C22×Q8)⋊66C22, C22.11(C22×D4), C22.29C24⋊20C2, C22.19C24⋊24C2, C42⋊C2⋊32C22, (C22×C4).351C23, C2.27(C2×2+ 1+4), C2.16(C2.C25), C22.D4⋊49C22, C22.31C24⋊11C2, C23.33C23⋊15C2, (C2×C4)⋊13(C2×D4), (C2×D4)○(C4⋊D4), (C2×C4⋊D4)⋊65C2, (C2×C4⋊C4)⋊68C22, (C22×C4○D4)⋊23C2, (C2×C4○D4)⋊25C22, (C2×C22⋊C4)⋊45C22, SmallGroup(128,2220)

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

Derived series
C1 — C22 — C22.77C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C22.77C25
Lower central
C1 — C22 — C22.77C25
Upper central
C1 — C22 — C22.77C25
Jennings
C1 — C22 — C22.77C25

Generators and relations for C22.77C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=a, 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 >

Subgroups: 1444 in 829 conjugacy classes, 430 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4⋊1D4, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, C23.33C23, C2×C4⋊D4, C22.19C24, C22.29C24, C22.31C24, D42, D4⋊5D4, Q8⋊5D4, Q8⋊6D4, C22×C4○D4, C2×2+ 1+4, C22.77C25
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C25, D4×C23, C2×2+ 1+4, C2.C25, C22.77C25

Smallest permutation representation of C22.77C25
►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 11)(2 12)(3 9)(4 10)(5 29)(6 30)(7 31)(8 32)(13 18)(14 19)(15 20)(16 17)(21 26)(22 27)(23 28)(24 25)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2K2L···2T4A···4J4K···4W
order12222···22···24···44···4
size11112···24···42···24···4

44 irreducible representations

dim111111111111244
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D42+ 1+4C2.C25
kernelC22.77C25C23.33C23C2×C4⋊D4C22.19C24C22.29C24C22.31C24D42D4⋊5D4Q8⋊5D4Q8⋊6D4C22×C4○D4C2×2+ 1+4C4○D4C22C2
# reps113333662211822

Matrix representation of C22.77C25 ►in GL6(ℤ)

100000
010000
00-1000
000-100
0000-10
00000-1
,
-100000
0-10000
001000
000100
000010
000001
,
-100000
110000
001-200
000-100
000-101
000-110
,
-100000
0-10000
00100-2
0010-1-1
001-10-1
00000-1
,
120000
0-10000
001000
000100
000010
000001
,
100000
010000
0010-20
0000-11
0000-10
0001-10
,
100000
010000
001-200
001-100
000-101
001-1-10

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,-2,-1,-1,-1,0,0,0,0,0,1,0,0,0,0,1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,1,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-2,-1,-1,-1],[1,0,0,0,0,0,2,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,-2,-1,-1,-1,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,1,0,0,-2,-1,-1,-1,0,0,0,0,0,-1,0,0,0,0,1,0] >;
 

C22.77C25 in GAP, Magma, Sage, TeX 

C_2^2._{77}C_2^5
 
% in TeX
 
G:=Group("C2^2.77C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2220);
 
// by ID
 
G=gap.SmallGroup(128,2220);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,570,1684,102]);
 
// 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=a,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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