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G = C2×C22.29C24  order 128 = 27

Direct product of C2 and C22.29C24

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

Aliases: C2×C22.29C24, C42⋊10C23, C23.16C24, C25.73C22, C22.35C25, C24.478C23, C22.1002+ 1+4, C4⋊C4⋊18C23, (C2×D4)⋊6C23, (C22×C4)⋊42D4, (D4×C23)⋊13C2, C22⋊C4⋊5C23, (C2×C4).38C24, (C2×Q8)⋊16C23, C2.14(D4×C23), C4⋊D4⋊61C22, C4⋊1D4⋊42C22, (C22×C4)⋊14C23, (C23×C4)⋊30C22, (C2×C42)⋊44C22, C4.170(C22×D4), C23.708(C2×D4), C22≀C2⋊26C22, C4.4D4⋊62C22, (C22×D4)⋊30C22, (C22×Q8)⋊58C22, C22.48(C22×D4), C42⋊C2⋊86C22, C2.5(C2×2+ 1+4), (C2×C4)⋊10(C2×D4), (C2×C4⋊D4)⋊54C2, (C2×C4⋊1D4)⋊22C2, (C2×C22≀C2)⋊21C2, (C2×C4⋊C4)⋊127C22, (C2×C4.4D4)⋊45C2, (C2×C4○D4)⋊67C22, (C22×C4○D4)⋊13C2, (C2×C42⋊C2)⋊53C2, (C2×C22⋊C4)⋊40C22, SmallGroup(128,2178)

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

Derived series
C1 — C22 — C2×C22.29C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C2×C22.29C24
Lower central
C1 — C22 — C2×C22.29C24
Upper central
C1 — C23 — C2×C22.29C24
Jennings
C1 — C22 — C2×C22.29C24

Generators and relations for C2×C22.29C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=g2=1, e2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 1804 in 948 conjugacy classes, 436 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C23×C4, C23×C4, C22×D4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C25, C2×C42⋊C2, C2×C22≀C2, C2×C4⋊D4, C2×C4.4D4, C2×C4⋊1D4, C22.29C24, D4×C23, C22×C4○D4, C2×C22.29C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C25, C22.29C24, D4×C23, C2×2+ 1+4, C2×C22.29C24

Smallest permutation representation of C2×C22.29C24
►On 32 points
Generators in S32
(1 19)(2 20)(3 17)(4 18)(5 15)(6 16)(7 13)(8 14)(9 22)(10 23)(11 24)(12 21)(25 32)(26 29)(27 30)(28 31)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2W4A···4H4I···4T
order12···222222···24···44···4
size11···122224···42···24···4

44 irreducible representations

dim11111111124
type+++++++++++
imageC1C2C2C2C2C2C2C2C2D42+ 1+4
kernelC2×C22.29C24C2×C42⋊C2C2×C22≀C2C2×C4⋊D4C2×C4.4D4C2×C4⋊1D4C22.29C24D4×C23C22×C4○D4C22×C4C22
# reps114422161184

Matrix representation of C2×C22.29C24 ►in GL8(ℤ)

10000000
01000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
12000000
0-1000000
00010000
00100000
00001020
0000000-1
000000-10
00000-100
,
-10000000
0-1000000
00100000
00010000
0000-1200
0000-1100
00001-111
0000-10-2-1
,
10000000
-1-1000000
00-100000
00010000
0000-1200
00000100
00001-111
0000000-1
,
-10000000
0-1000000
00100000
00010000
0000-1000
00000-100
00001010
00000001

G:=sub<GL(8,Integers())| [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,-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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-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,-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,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,2,0,-1,0,0,0,0,0,0,-1,0,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,1,-1,0,0,0,0,2,1,-1,0,0,0,0,0,0,0,1,-2,0,0,0,0,0,0,1,-1],[1,-1,0,0,0,0,0,0,0,-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,-1,0,1,0,0,0,0,0,2,1,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,-1],[-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,-1,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

C2×C22.29C24 in GAP, Magma, Sage, TeX 

C_2\times C_2^2._{29}C_2^4
 
% in TeX
 
G:=Group("C2xC2^2.29C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2178);
 
// by ID
 
G=gap.SmallGroup(128,2178);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,387,1123]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=g^2=1,e^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=g*d*g=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=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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