Copied to
clipboard

G = C23.240C24  order 128 = 27

93rd central extension by C23 of C24

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

Aliases: C25.24C22, C24.213C23, C23.240C24, C22.742+ 1+4, C2.3D42, C22⋊2(C4×D4), C22≀C2⋊8C4, C24⋊12(C2×C4), C22⋊C4⋊43D4, (C23×C4)⋊7C22, C2.6(D4⋊5D4), (C2×C42)⋊16C22, C23.417(C2×D4), C23.8Q8⋊15C2, C23.292(C4○D4), C23.23D4⋊12C2, C22.131(C23×C4), (C22×C4).762C23, C23.131(C22×C4), C22.111(C22×D4), C2.C42⋊12C22, C24.3C22⋊19C2, C24.C22⋊17C2, (C22×D4).485C22, C2.4(C22.45C24), C2.29(C22.11C24), (C2×C4×D4)⋊11C2, C2.34(C2×C4×D4), (C2×D4)⋊18(C2×C4), (C4×C22⋊C4)⋊39C2, C22⋊C4⋊29(C2×C4), (C2×C4).886(C2×D4), (C2×C4⋊C4)⋊104C22, (C2×C22≀C2).5C2, C22⋊C4○4(C22⋊C4), (C22×C22⋊C4)⋊9C2, (C2×C4).39(C22×C4), (C2×C22⋊C4)⋊74C22, C22.125(C2×C4○D4), C22⋊C4○3(C2×C22⋊C4), SmallGroup(128,1090)

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

Derived series
C1 — C22 — C23.240C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C23.240C24
Lower central
C1 — C22 — C23.240C24
Upper central
C1 — C23 — C23.240C24
Jennings
C1 — C23 — C23.240C24

Generators and relations for C23.240C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=e2=c, ab=ba, ac=ca, ede-1=ad=da, geg=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, fg=gf >

Subgroups: 940 in 464 conjugacy classes, 164 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C22≀C2, C23×C4, C23×C4, C22×D4, C22×D4, C25, C4×C22⋊C4, C23.8Q8, C23.23D4, C23.23D4, C24.C22, C24.3C22, C22×C22⋊C4, C2×C4×D4, C2×C22≀C2, C23.240C24
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, 2+ 1+4, C2×C4×D4, C22.11C24, D42, D4⋊5D4, C22.45C24, C23.240C24

Smallest permutation representation of C23.240C24
►On 32 points
Generators in S32
(1 11)(2 12)(3 9)(4 10)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 31)(18 32)(19 29)(20 30)

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

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

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

50 conjugacy classes

class 1 2A···2G2H···2O2P2Q2R2S4A···4P4Q···4AD
order12···22···222224···44···4
size11···12···244442···24···4

50 irreducible representations

dim1111111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C4D4C4○D42+ 1+4
kernelC23.240C24C4×C22⋊C4C23.8Q8C23.23D4C24.C22C24.3C22C22×C22⋊C4C2×C4×D4C2×C22≀C2C22≀C2C22⋊C4C23C22
# reps12232122116882

Matrix representation of C23.240C24 ►in GL5(𝔽5)

10000
04000
00400
00010
00001
,
10000
01000
00100
00040
00004
,
40000
04000
00400
00010
00001
,
20000
03000
00200
00032
00012
,
20000
00100
04000
00010
00001
,
10000
04000
00400
00010
00024
,
10000
04000
00100
00040
00004

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

C23.240C24 in GAP, Magma, Sage, TeX 

C_2^3._{240}C_2^4
 
% in TeX
 
G:=Group("C2^3.240C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1090);
 
// by ID
 
G=gap.SmallGroup(128,1090);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=e^2=c,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,g*e*g=a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁