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G = C23.333C24order 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, C41C22≀C2, (C2×D4)⋊46D4, C22⋊C423D4, (D4×C23)⋊5C2, (C22×C4)⋊27D4, C232D413C2, C222(C41D4), (C2×C42)⋊22C22, C23.162(C2×D4), (C22×D4)⋊6C22, (C22×C4).800C23, (C23×C4).346C22, C22.213(C22×D4), C24.3C2237C2, C2.C4265C22, C2.13(C22.29C24), (C2×C4)⋊3(C2×D4), (C2×C41D4)⋊5C2, C2.7(C2×C41D4), (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

C1C23 — C23.333C24
C1C2C22C23C22×C4C2×C42C4×C22⋊C4 — C23.333C24
C1C23 — C23.333C24
C1C23 — C23.333C24
C1C23 — 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 [×3], C2 [×4], C2 [×14], C4 [×4], C4 [×12], C22, C22 [×10], C22 [×98], C2×C4 [×16], C2×C4 [×24], D4 [×76], C23, C23 [×14], C23 [×106], C42 [×4], C22⋊C4 [×8], C22⋊C4 [×20], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×10], C22×C4 [×4], C2×D4 [×8], C2×D4 [×110], C24, C24 [×6], C24 [×24], C2.C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4, C22≀C2 [×16], C4⋊D4 [×4], C41D4 [×8], C23×C4, C22×D4, C22×D4 [×14], C22×D4 [×12], C25 [×2], C4×C22⋊C4, C24.3C22 [×2], C232D4 [×4], C2×C22≀C2 [×4], C2×C4⋊D4, C2×C41D4 [×2], D4×C23, C23.333C24
Quotients: C1, C2 [×15], C22 [×35], D4 [×20], C23 [×15], C2×D4 [×30], C24, C22≀C2 [×4], C41D4 [×4], C22×D4 [×5], 2+ 1+4 [×2], C2×C22≀C2, C2×C41D4, C22.29C24, D42 [×4], 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⋊C4C24.3C22C232D4C2×C22≀C2C2×C4⋊D4C2×C41D4D4×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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