Copied to
clipboard

G = C22.110C25  order 128 = 27

91st central stem extension by C22 of C25

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

Aliases: C22.110C25, C42.101C23, C24.141C23, C23.145C24, C4⋊Q8⋊36C22, D4⋊6D4⋊29C2, D4⋊5D4⋊26C2, D4⋊3Q8⋊30C2, (C4×D4)⋊54C22, D4○(C42⋊2C2), (C4×Q8)⋊53C22, D4.42(C4○D4), C4⋊C4.501C23, C4⋊D4⋊31C22, (C2×C4).100C24, (C2×C42)⋊67C22, C22⋊Q8⋊39C22, C22.32C24⋊9C2, (C2×D4).482C23, C4.4D4⋊87C22, C22⋊C4.33C23, (C2×Q8).459C23, C42.C2⋊61C22, C22.11C24⋊24C2, C42⋊2C2⋊53C22, C42⋊C2⋊47C22, C22≀C2.11C22, (C22×C4).379C23, C22.45C24⋊11C2, C2.39(C2.C25), C22.33C24⋊9C2, (C22×D4).431C22, C22.D4⋊56C22, C22.36C24⋊21C2, C23.36C23⋊41C2, C22.47C24⋊24C2, C22.50C24⋊29C2, C22.46C24⋊24C2, C22.35C24⋊14C2, C23.33C23⋊30C2, (C4×C4○D4)⋊37C2, (C2×C4⋊C4)⋊80C22, C4.283(C2×C4○D4), (C2×D4)○(C42⋊2C2), C22.46(C2×C4○D4), C2.66(C22×C4○D4), (C2×C42⋊2C2)⋊38C2, C22⋊C4○(C42⋊2C2), (C2×C4○D4).333C22, (C2×C22⋊C4).386C22, SmallGroup(128,2253)

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

Derived series
C1 — C22 — C22.110C25
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C4×C4○D4 — C22.110C25
Lower central
C1 — C22 — C22.110C25
Upper central
C1 — C22 — C22.110C25
Jennings
C1 — C22 — C22.110C25

Generators and relations for C22.110C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=a, g2=ba=ab, dcd=gcg-1=ac=ca, fdf=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 >

Subgroups: 748 in 521 conjugacy classes, 388 normal (122 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C42⋊2C2, C4⋊Q8, C22×D4, C2×C4○D4, C4×C4○D4, C22.11C24, C23.33C23, C2×C42⋊2C2, C23.36C23, C23.36C23, C22.32C24, C22.33C24, C22.35C24, C22.36C24, D4⋊5D4, D4⋊6D4, C22.45C24, C22.46C24, C22.46C24, C22.47C24, C22.47C24, D4⋊3Q8, C22.50C24, C22.110C25
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, C25, C22×C4○D4, C2.C25, C22.110C25

Smallest permutation representation of C22.110C25
►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 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)

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

(5 7)(6 8)(9 11)(10 12)(17 19)(18 20)(29 31)(30 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)

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2L4A···4N4O···4AE
order122222222···24···44···4
size111122224···42···24···4

44 irreducible representations

dim1111111111111111124
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C4○D4C2.C25
kernelC22.110C25C4×C4○D4C22.11C24C23.33C23C2×C42⋊2C2C23.36C23C22.32C24C22.33C24C22.35C24C22.36C24D4⋊5D4D4⋊6D4C22.45C24C22.46C24C22.47C24D4⋊3Q8C22.50C24D4C2
# reps1111242211214331284

Matrix representation of C22.110C25 ►in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
440000
000010
001111
001000
003034
,
400000
040000
001000
000100
000040
003304
,
120000
040000
003000
000300
000030
000003
,
100000
010000
000010
004444
001000
000001
,
200000
020000
000100
004000
004444
002021

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

C22.110C25 in GAP, Magma, Sage, TeX 

C_2^2._{110}C_2^5
 
% in TeX
 
G:=Group("C2^2.110C2^5");
 
// GroupNames label
 
G:=SmallGroup(128,2253);
 
// by ID
 
G=gap.SmallGroup(128,2253);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,184,570,1684,242]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=a,g^2=b*a=a*b,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^-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
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁