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G = C23.333C24  order 128 = 27

50th central stem extension by C23 of C24

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

Aliases: C25.40C22, C24.265C23, C23.333C24, C22.1442+ 1+4, C2.14D42, C4⋊1C22≀C2, (C2×D4)⋊46D4, C22⋊C4⋊23D4, (D4×C23)⋊5C2, (C22×C4)⋊27D4, C23⋊2D4⋊13C2, C22⋊2(C4⋊1D4), (C2×C42)⋊22C22, C23.162(C2×D4), (C22×D4)⋊6C22, (C22×C4).800C23, (C23×C4).346C22, C22.213(C22×D4), C24.3C22⋊37C2, C2.C42⋊65C22, C2.13(C22.29C24), (C2×C4)⋊3(C2×D4), (C2×C4⋊1D4)⋊5C2, C2.7(C2×C4⋊1D4), (C2×C22≀C2)⋊7C2, (C2×C4⋊D4)⋊11C2, (C4×C22⋊C4)⋊57C2, (C2×C4⋊C4)⋊15C22, C2.21(C2×C22≀C2), (C2×C22⋊C4)⋊17C22, SmallGroup(128,1165)

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

Derived series
C1 — C23 — C23.333C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C23.333C24
Lower central
C1 — C23 — C23.333C24
Upper central
C1 — C23 — C23.333C24
Jennings
C1 — C23 — C23.333C24

Generators and relations for C23.333C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=a, f2=b, ab=ba, ac=ca, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 1572 in 674 conjugacy classes, 140 normal (16 characteristic)
C1, C2, C2, C2, C4, 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×C22⋊C4, C2×C4⋊C4, C22≀C2, C4⋊D4, C4⋊1D4, C23×C4, C22×D4, C22×D4, C22×D4, C25, C4×C22⋊C4, C24.3C22, C23⋊2D4, C2×C22≀C2, C2×C4⋊D4, C2×C4⋊1D4, D4×C23, C23.333C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C4⋊1D4, C22×D4, 2+ 1+4, C2×C22≀C2, C2×C4⋊1D4, C22.29C24, D42, C23.333C24

Smallest permutation representation of C23.333C24
►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 20)(2 17)(3 18)(4 19)(5 13)(6 14)(7 15)(8 16)(9 32)(10 29)(11 30)(12 31)(21 26)(22 27)(23 28)(24 25)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S2T2U4A4B4C4D4E···4N4O4P
order12···222222···22244444···444
size11···122224···48822224···488

38 irreducible representations

dim111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D42+ 1+4
kernelC23.333C24C4×C22⋊C4C24.3C22C23⋊2D4C2×C22≀C2C2×C4⋊D4C2×C4⋊1D4D4×C23C22⋊C4C22×C4C2×D4C22
# reps112441218482

Matrix representation of C23.333C24 ►in GL6(ℤ)

100000
010000
001000
000100
0000-10
00000-1
,
100000
010000
00-1000
000-100
000010
000001
,
-100000
0-10000
001000
000100
000010
000001
,
-100000
0-10000
001000
000-100
000012
00000-1
,
100000
0-10000
00-1000
000-100
000012
0000-1-1
,
010000
100000
000100
00-1000
000010
000001
,
100000
010000
001000
000100
000012
0000-1-1

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],[-1,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,1,0,0,0,0,0,2,-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,-1,0,0,0,0,2,-1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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,-1,0,0,0,0,2,-1] >;
 

C23.333C24 in GAP, Magma, Sage, TeX 

C_2^3._{333}C_2^4
 
% in TeX
 
G:=Group("C2^3.333C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1165);
 
// by ID
 
G=gap.SmallGroup(128,1165);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,184,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=a,f^2=b,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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