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G = C2×C22.11C24  order 128 = 27

Direct product of C2 and C22.11C24

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

Aliases: C2×C22.11C24, C42⋊8C23, C22.11C25, C25.67C22, C24.472C23, C23.105C24, C22.972+ 1+4, C4⋊C4⋊24C23, C24⋊14(C2×C4), D4⋊8(C22×C4), C2.7(C24×C4), (C4×D4)⋊85C22, (C22×D4)⋊30C4, C4.37(C23×C4), (C22×C4)⋊9C23, C23⋊5(C22×C4), C22⋊C4⋊22C23, (C2×C4).157C24, (C2×C42)⋊39C22, (C23×C4)⋊15C22, (D4×C23).18C2, C22.2(C23×C4), (C2×D4).496C23, C42⋊C2⋊80C22, C2.1(C2×2+ 1+4), (C22×D4).578C22, (C2×C4×D4)⋊66C2, D4○(C2×C22⋊C4), C22⋊C4○2(C2×D4), (C2×D4)⋊52(C2×C4), (C2×C4)⋊6(C22×C4), C22⋊C4○(C22×D4), (C22×C4)⋊40(C2×C4), (C2×C4⋊C4)⋊145C22, (C2×C42⋊C2)⋊48C2, (C22×C22⋊C4)⋊14C2, (C2×C22⋊C4)⋊77C22, C4⋊C4○4(C2×C4⋊C4), (C2×D4)○2(C2×C22⋊C4), C22⋊C4○4(C2×C22⋊C4), (C2×C4⋊C4)○(C2×C4⋊C4), (C2×C22⋊C4)○(C2×C22⋊C4), SmallGroup(128,2157)

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

Derived series
C1 — C2 — C2×C22.11C24
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C2×C22.11C24
Lower central
C1 — C2 — C2×C22.11C24
Upper central
C1 — C23 — C2×C22.11C24
Jennings
C1 — C22 — C2×C22.11C24

Generators and relations for C2×C22.11C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=g2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=gdg=bd=db, fef=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, df=fd, eg=ge, fg=gf >

Subgroups: 1500 in 960 conjugacy classes, 692 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, C22×C4, C2×D4, C24, C24, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C23×C4, C23×C4, C22×D4, C25, C22×C22⋊C4, C2×C42⋊C2, C2×C4×D4, C22.11C24, D4×C23, C2×C22.11C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, 2+ 1+4, C25, C22.11C24, C24×C4, C2×2+ 1+4, C2×C22.11C24

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

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

(5 21)(6 22)(7 23)(8 24)(17 29)(18 30)(19 31)(20 32)

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

68 conjugacy classes

class 1 2A···2G2H···2AA4A···4AN
order12···22···24···4
size11···12···22···2

68 irreducible representations

dim11111114
type+++++++
imageC1C2C2C2C2C2C42+ 1+4
kernelC2×C22.11C24C22×C22⋊C4C2×C42⋊C2C2×C4×D4C22.11C24D4×C23C22×D4C22
# reps1428161324

Matrix representation of C2×C22.11C24 ►in GL6(𝔽5)

400000
040000
004000
000400
000040
000004
,
100000
010000
004000
000400
000040
000004
,
400000
010000
001000
000100
000010
000001
,
300000
010000
000010
001413
001000
000001
,
400000
040000
000100
001000
004142
004101
,
100000
010000
001000
000400
000010
001014
,
100000
040000
001000
000100
000040
001404

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,4,0,0,0,0,1,1,0,0,0,0,0,3,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,4,4,0,0,1,0,1,1,0,0,0,0,4,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,4,0,0,0,0,0,0,1,1,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,4,0,0,0,0,4,0,0,0,0,0,0,4] >;
 

C2×C22.11C24 in GAP, Magma, Sage, TeX 

C_2\times C_2^2._{11}C_2^4
 
% in TeX
 
G:=Group("C2xC2^2.11C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2157);
 
// by ID
 
G=gap.SmallGroup(128,2157);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,387,1123]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=e^2=f^2=g^2=1,d^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=g*d*g=b*d=d*b,f*e*f=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,d*f=f*d,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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