Copied to
clipboard

G = C42.691C23  order 128 = 27

106th non-split extension by C42 of C23 acting via C23/C22=C2

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

Aliases: C42.691C23, C4.1662+ 1+4, D4⋊7(C2×C8), (C2×D4)⋊9C8, (C8×D4)⋊3C2, C23⋊3(C2×C8), (C4×C8)⋊4C22, C4⋊C8⋊95C22, D4○2(C22⋊C8), (C4×D4).32C4, C2.7(C23×C8), C4.19(C22×C8), C24.83(C2×C4), (C22×C8)⋊5C22, C22⋊C8⋊83C22, (C2×C4).669C24, (C2×C8).481C23, C42.220(C2×C4), (C22×D4).41C4, C22.2(C22×C8), (C4×D4).362C22, C2.3(Q8○M4(2)), C22.43(C23×C4), C42.12C4⋊23C2, (C23×C4).186C22, C23.149(C22×C4), (C2×C42).779C22, (C22×C4).1280C23, C2.4(C22.11C24), C4⋊C8○(C4⋊C8), (C2×C4)⋊4(C2×C8), C22⋊C8○(C4×D4), C4⋊C4○(C22⋊C8), (C2×C4×D4).75C2, (C2×C4⋊C4).75C4, (C2×D4)○(C22⋊C8), C4⋊C4.250(C2×C4), (C2×C22⋊C8)⋊17C2, C22⋊C8○(C22⋊C8), C22⋊C4○(C22⋊C8), (C2×D4).252(C2×C4), C22⋊C4.93(C2×C4), (C2×C22⋊C4).33C4, (C2×C4).499(C22×C4), (C22×C4).137(C2×C4), SmallGroup(128,1704)

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

Derived series
C1 — C2 — C42.691C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C2×C4×D4 — C42.691C23
Lower central
C1 — C2 — C42.691C23
Upper central
C1 — C2×C4 — C42.691C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.691C23

Generators and relations for C42.691C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=b, e2=a2, ab=ba, ac=ca, dad=a-1, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2c, ede-1=a2d >

Subgroups: 388 in 250 conjugacy classes, 174 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C24, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C22×C8, C23×C4, C22×D4, C2×C22⋊C8, C42.12C4, C8×D4, C2×C4×D4, C42.691C23
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, C24, C22×C8, C23×C4, 2+ 1+4, C22.11C24, C23×C8, Q8○M4(2), C42.691C23

Smallest permutation representation of C42.691C23
►On 32 points
Generators in S32
(1 23 31 12)(2 24 32 13)(3 17 25 14)(4 18 26 15)(5 19 27 16)(6 20 28 9)(7 21 29 10)(8 22 30 11)

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

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

68 conjugacy classes

class 1 2A2B2C2D···2M4A4B4C4D4E···4V8A···8AF
order12222···244444···48···8
size11112···211112···22···2

68 irreducible representations

dim111111111144
type++++++
imageC1C2C2C2C2C4C4C4C4C82+ 1+4Q8○M4(2)
kernelC42.691C23C2×C22⋊C8C42.12C4C8×D4C2×C4×D4C2×C22⋊C4C2×C4⋊C4C4×D4C22×D4C2×D4C4C2
# reps1428142823222

Matrix representation of C42.691C23 ►in GL5(𝔽17)

10000
00100
016000
00001
000160
,
130000
01000
00100
00010
00001
,
90000
00010
00001
01000
00100
,
160000
016000
00100
000160
00001
,
10000
001600
01000
00001
000160

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0],[13,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[9,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,1,0] >;
 

C42.691C23 in GAP, Magma, Sage, TeX 

C_4^2._{691}C_2^3
 
% in TeX
 
G:=Group("C4^2.691C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1704);
 
// by ID
 
G=gap.SmallGroup(128,1704);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,219,675,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=b,e^2=a^2,a*b=b*a,a*c=c*a,d*a*d=a^-1,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^2*c,e*d*e^-1=a^2*d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁