Copied to
clipboard

?

G = C2×C22.19C24  order 128 = 27

Direct product of C2 and C22.19C24

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

Aliases: C2×C22.19C24, C42⋊9C23, C22.24C25, C23.12C24, C25.96C22, C24.608C23, (C24×C4)⋊9C2, C4⋊C4⋊16C23, (C22×C4)⋊56D4, C2.8(D4×C23), (C4×D4)⋊89C22, (C2×D4)⋊15C23, C23⋊7(C4○D4), (C2×C4).29C24, (C2×Q8)⋊14C23, C4⋊D4⋊95C22, C22⋊C4⋊17C23, (C23×C4)⋊28C22, (C22×C4)⋊12C23, (C2×C42)⋊42C22, C23.707(C2×D4), C4.169(C22×D4), C22≀C2⋊41C22, C4○(C22.19C24), (C22×D4)⋊58C22, C22⋊Q8⋊103C22, (C22×Q8)⋊56C22, C22.46(C22×D4), C42⋊C2⋊83C22, C22.D4⋊66C22, (C2×C4×D4)⋊69C2, C4○(C2×C4⋊D4), C4○(C2×C22⋊Q8), C4○(C2×C22≀C2), (C2×C4)⋊22(C2×D4), (C2×C4)○2C22≀C2, C22⋊1(C2×C4○D4), (C2×C4)○2(C4⋊D4), (C2×C4⋊D4)⋊76C2, (C22×C4)○C22≀C2, (C2×C4)○2(C22⋊Q8), (C2×C22⋊Q8)⋊86C2, (C2×C22≀C2)⋊30C2, C2.8(C22×C4○D4), C4○(C2×C22.D4), (C2×C4⋊C4)⋊125C22, (C2×C4○D4)⋊65C22, (C22×C4○D4)⋊11C2, (C2×C42⋊C2)⋊51C2, (C22×C4)○(C22⋊Q8), (C2×C22⋊C4)⋊83C22, (C2×C4)○(C22.19C24), (C2×C4)○2(C22.D4), (C2×C22.D4)⋊67C2, (C2×C4)○2(C2×C22⋊Q8), (C2×C4)○2(C2×C22≀C2), (C22×C4)○(C2×C22⋊Q8), (C22×C4)○(C2×C22≀C2), SmallGroup(128,2167)

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

Derived series
C1 — C22 — C2×C22.19C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C24×C4 — C2×C22.19C24
Lower central
C1 — C22 — C2×C22.19C24
Upper central
C1 — C22×C4 — C2×C22.19C24
Jennings
C1 — C22 — C2×C22.19C24

Subgroups: 1452 in 940 conjugacy classes, 452 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×16], C4 [×8], C4 [×16], C22, C22 [×18], C22 [×88], C2×C4 [×44], C2×C4 [×72], D4 [×56], Q8 [×8], C23, C23 [×22], C23 [×80], C42 [×8], C22⋊C4 [×40], C4⋊C4 [×24], C22×C4 [×2], C22×C4 [×48], C22×C4 [×60], C2×D4 [×28], C2×D4 [×28], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×32], C24, C24 [×8], C24 [×12], C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×6], C42⋊C2 [×8], C4×D4 [×32], C22≀C2 [×16], C4⋊D4 [×16], C22⋊Q8 [×16], C22.D4 [×16], C23×C4 [×2], C23×C4 [×14], C23×C4 [×8], C22×D4, C22×D4 [×6], C22×Q8, C2×C4○D4 [×8], C2×C4○D4 [×8], C25, C2×C42⋊C2, C2×C4×D4 [×4], C2×C22≀C2 [×2], C2×C4⋊D4 [×2], C2×C22⋊Q8 [×2], C2×C22.D4 [×2], C22.19C24 [×16], C24×C4, C22×C4○D4, C2×C22.19C24

Quotients:
C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C4○D4 [×8], C24 [×31], C22×D4 [×14], C2×C4○D4 [×12], C25, C22.19C24 [×4], D4×C23, C22×C4○D4 [×2], C2×C22.19C24

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

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

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

400000
040000
001000
000100
000040
000004
,
100000
010000
001000
000100
000040
000004
,
400000
040000
004000
000400
000040
000004
,
010000
100000
000400
004000
000012
000004
,
400000
040000
004000
000400
000040
000011
,
100000
040000
001000
000400
000040
000011
,
400000
040000
004000
000400
000030
000003

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

56 conjugacy classes

class 1 2A···2G2H···2S2T2U2V2W4A···4H4I···4T4U···4AF
order12···22···222224···44···44···4
size11···12···244441···12···24···4

56 irreducible representations

dim111111111122
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4C4○D4
kernelC2×C22.19C24C2×C42⋊C2C2×C4×D4C2×C22≀C2C2×C4⋊D4C2×C22⋊Q8C2×C22.D4C22.19C24C24×C4C22×C4○D4C22×C4C23
# reps11422221611816

In GAP, Magma, Sage, TeX 

C_2\times C_2^2._{19}C_2^4
 
% in TeX
 
G:=Group("C2xC2^2.19C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,2167);
 
// by ID
 
G=gap.SmallGroup(128,2167);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,136]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=1,g^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=b*d=d*b,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,d*g=g*d,e*f=f*e,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀