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G = C23.194C24  order 128 = 27

47th central extension by C23 of C24

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

Aliases: C25.19C22, C24.191C23, C23.194C24, C22.332+ 1+4, C24.68(C2×C4), (C2×C42)⋊3C22, C24⋊3C4.4C2, (C23×C4).41C22, C23.81(C22×C4), C22.85(C23×C4), C23.7Q8⋊12C2, C23.220(C4○D4), C24.C22⋊1C2, (C22×C4).459C23, C2.C42⋊8C22, C2.7(C22.11C24), C2.1(C22.32C24), C22.35(C42⋊C2), (C4×C22⋊C4)⋊6C2, (C2×C4⋊C4)⋊5C22, (C2×C22⋊C4)⋊21C4, C22⋊C4.55(C2×C4), C22.82(C2×C4○D4), (C2×C4).217(C22×C4), (C22×C4).131(C2×C4), C2.19(C2×C42⋊C2), (C22×C22⋊C4).17C2, (C2×C22⋊C4).23C22, SmallGroup(128,1044)

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

Derived series
C1 — C22 — C23.194C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C23.194C24
Lower central
C1 — C22 — C23.194C24
Upper central
C1 — C23 — C23.194C24
Jennings
C1 — C23 — C23.194C24

Generators and relations for C23.194C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=f2=g2=1, d2=c, e2=b, 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: 716 in 336 conjugacy classes, 140 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C25, C4×C22⋊C4, C24⋊3C4, C23.7Q8, C24.C22, C22×C22⋊C4, C23.194C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C42⋊C2, C23×C4, C2×C4○D4, 2+ 1+4, C2×C42⋊C2, C22.11C24, C22.32C24, C23.194C24

Smallest permutation representation of C23.194C24
►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 31 27 23)(2 18 28 7)(3 29 25 21)(4 20 26 5)(6 11 17 15)(8 9 19 13)(10 30 14 22)(12 32 16 24)

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

(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), (1,31,27,23)(2,18,28,7)(3,29,25,21)(4,20,26,5)(6,11,17,15)(8,9,19,13)(10,30,14,22)(12,32,16,24), (1,3)(2,26)(4,28)(5,32)(6,21)(7,30)(8,23)(9,11)(10,16)(12,14)(13,15)(17,29)(18,22)(19,31)(20,24)(25,27), (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), (1,31,27,23)(2,18,28,7)(3,29,25,21)(4,20,26,5)(6,11,17,15)(8,9,19,13)(10,30,14,22)(12,32,16,24), (1,3)(2,26)(4,28)(5,32)(6,21)(7,30)(8,23)(9,11)(10,16)(12,14)(13,15)(17,29)(18,22)(19,31)(20,24)(25,27), (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)], [(1,31,27,23),(2,18,28,7),(3,29,25,21),(4,20,26,5),(6,11,17,15),(8,9,19,13),(10,30,14,22),(12,32,16,24)], [(1,3),(2,26),(4,28),(5,32),(6,21),(7,30),(8,23),(9,11),(10,16),(12,14),(13,15),(17,29),(18,22),(19,31),(20,24),(25,27)], [(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···2G2H2I2J2K2L2M2N2O4A···4H4I···4AB
order12···2222222224···44···4
size11···1222244442···24···4

44 irreducible representations

dim111111124
type+++++++
imageC1C2C2C2C2C2C4C4○D42+ 1+4
kernelC23.194C24C4×C22⋊C4C24⋊3C4C23.7Q8C24.C22C22×C22⋊C4C2×C22⋊C4C23C22
# reps1222811684

Matrix representation of C23.194C24 ►in GL8(𝔽5)

10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
10000000
01000000
00400000
00040000
00004000
00000400
00000040
00000004
,
01000000
10000000
00030000
00300000
00000010
00000103
00004000
00000104
,
20000000
02000000
00200000
00020000
00004100
00000100
00000412
00000104
,
10000000
04000000
00400000
00010000
00004000
00003100
00000040
00004011
,
40000000
04000000
00400000
00040000
00004000
00000400
00000010
00000401

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

C23.194C24 in GAP, Magma, Sage, TeX 

C_2^3._{194}C_2^4
 
% in TeX
 
G:=Group("C2^3.194C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1044);
 
// by ID
 
G=gap.SmallGroup(128,1044);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,219,675]);
 
// 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=b,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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