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G = C2×C22.32C24  order 128 = 27

Direct product of C2 and C22.32C24

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

Aliases: C2×C22.32C24, C42⋊11C23, C25.74C22, C22.39C25, C24.481C23, C23.272C24, C22.1022+ 1+4, C4⋊C4⋊5C23, (C2×D4)⋊7C23, (C2×Q8)⋊5C23, (C4×D4)⋊93C22, C22⋊C4⋊6C23, (C2×C4).42C24, C4⋊D4⋊64C22, (C2×C42)⋊46C22, (C23×C4)⋊32C22, (C22×C4)⋊15C23, C22⋊Q8⋊75C22, C22≀C2⋊28C22, C4.4D4⋊65C22, (C22×D4)⋊31C22, (C22×Q8)⋊26C22, C42⋊2C2⋊22C22, C2.7(C2×2+ 1+4), C23.259(C4○D4), C22.D4⋊33C22, (C2×C4×D4)⋊72C2, (C2×C4⋊D4)⋊56C2, (C2×C4⋊C4)⋊64C22, (C2×C22⋊Q8)⋊63C2, (C2×C22≀C2)⋊22C2, C22.8(C2×C4○D4), (C2×C4.4D4)⋊47C2, C2.16(C22×C4○D4), (C2×C42⋊2C2)⋊31C2, (C22×C22⋊C4)⋊31C2, (C2×C22⋊C4)⋊85C22, (C2×C22.D4)⋊50C2, SmallGroup(128,2182)

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

Derived series
C1 — C22 — C2×C22.32C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C2×C22.32C24
Lower central
C1 — C22 — C2×C22.32C24
Upper central
C1 — C23 — C2×C22.32C24
Jennings
C1 — C22 — C2×C22.32C24

Generators and relations for C2×C22.32C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=g2=1, e2=c, 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: 1196 in 668 conjugacy classes, 396 normal (20 characteristic)
C1, C2, C2, C2, 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, C24, C24, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C25, C22×C22⋊C4, C2×C4×D4, C2×C22≀C2, C2×C4⋊D4, C2×C4⋊D4, C2×C22⋊Q8, C2×C22.D4, C2×C4.4D4, C2×C42⋊2C2, C22.32C24, C2×C22.32C24
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, 2+ 1+4, C25, C22.32C24, C22×C4○D4, C2×2+ 1+4, C2×C22.32C24

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

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

(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)

(2 22)(4 24)(5 12)(7 10)(13 15)(14 25)(16 27)(17 30)(18 20)(19 32)(26 28)(29 31)

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

44 conjugacy classes

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

44 irreducible representations

dim111111111124
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C2C4○D42+ 1+4
kernelC2×C22.32C24C22×C22⋊C4C2×C4×D4C2×C22≀C2C2×C4⋊D4C2×C22⋊Q8C2×C22.D4C2×C4.4D4C2×C42⋊2C2C22.32C24C23C22
# reps1122312221684

Matrix representation of C2×C22.32C24 ►in GL8(𝔽5)

40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
01000000
10000000
00010000
00100000
00000010
00001414
00001000
00000001
,
20000000
02000000
00200000
00020000
00000100
00001000
00004141
00003201
,
10000000
04000000
00400000
00010000
00001000
00000400
00000010
00002024
,
40000000
04000000
00100000
00010000
00001000
00000100
00000040
00002304

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

C2×C22.32C24 in GAP, Magma, Sage, TeX 

C_2\times C_2^2._{32}C_2^4
 
% in TeX
 
G:=Group("C2xC2^2.32C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2182);
 
// by ID
 
G=gap.SmallGroup(128,2182);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,387,1123]);
 
// 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=c,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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