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G = C23.191C24  order 128 = 27

44th central extension by C23 of C24

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

Aliases: C25.18C22, C23.191C24, C24.540C23, C22.312+ 1+4, C24⋊3C4⋊5C2, (C22×D4)⋊22C4, C24.67(C2×C4), (D4×C23).8C2, (C2×C42)⋊13C22, (C22×C4).358D4, C23.602(C2×D4), C22.82(C23×C4), C23.79(C22×C4), C22.85(C22×D4), (C22×C4).456C23, (C23×C4).285C22, C24.3C22⋊11C2, C2.6(C22.11C24), C2.1(C22.29C24), (C22×D4).473C22, (C2×C4)⋊6(C22⋊C4), (C2×C4).827(C2×D4), C4.63(C2×C22⋊C4), (C2×C4⋊C4)⋊102C22, (C2×C42⋊C2)⋊8C2, (C2×D4).210(C2×C4), (C2×C22⋊C4)⋊4C22, (C2×C4).451(C22×C4), (C22×C4).297(C2×C4), C2.13(C22×C22⋊C4), C22.74(C2×C22⋊C4), SmallGroup(128,1041)

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

Derived series
C1 — C22 — C23.191C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C23.191C24
Lower central
C1 — C22 — C23.191C24
Upper central
C1 — C23 — C23.191C24
Jennings
C1 — C23 — C23.191C24

Generators and relations for C23.191C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=c, e2=a, ab=ba, ac=ca, ede-1=gdg=ad=da, fef=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, eg=ge, fg=gf >

Subgroups: 1196 in 552 conjugacy classes, 180 normal (8 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C22×D4, C22×D4, C25, C24⋊3C4, C24.3C22, C2×C42⋊C2, D4×C23, C23.191C24
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C23×C4, C22×D4, 2+ 1+4, C22×C22⋊C4, C22.11C24, C22.29C24, C23.191C24

Smallest permutation representation of C23.191C24
►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 11 8)(2 5 12 22)(3 23 9 6)(4 7 10 24)(13 17 25 31)(14 32 26 18)(15 19 27 29)(16 30 28 20)

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S4A···4H4I···4X
order12···222222···24···44···4
size11···122224···42···24···4

44 irreducible representations

dim11111124
type+++++++
imageC1C2C2C2C2C4D42+ 1+4
kernelC23.191C24C24⋊3C4C24.3C22C2×C42⋊C2D4×C23C22×D4C22×C4C22
# reps148211684

Matrix representation of C23.191C24 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
01000000
10000000
00210000
00030000
00003010
00000001
00000020
00000400
,
10000000
01000000
00400000
00040000
00004300
00001100
00001112
00002044
,
10000000
04000000
00100000
00140000
00004300
00000100
00001112
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00004040
00000004

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,2,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,1,1,2,0,0,0,0,3,1,1,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,2,4],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,1,0,0,0,0,0,3,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4] >;
 

C23.191C24 in GAP, Magma, Sage, TeX 

C_2^3._{191}C_2^4
 
% in TeX
 
G:=Group("C2^3.191C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1041);
 
// by ID
 
G=gap.SmallGroup(128,1041);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,219,675]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=f^2=g^2=1,d^2=c,e^2=a,a*b=b*a,a*c=c*a,e*d*e^-1=g*d*g=a*d=d*a,f*e*f=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,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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