Copied to
clipboard

G = C8.16C42  order 128 = 27

10th non-split extension by C8 of C42 acting via C42/C2×C4=C2

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

Aliases: C8.16C42, (C4×C8)⋊8C4, C8⋊C4⋊14C4, C8○(C4.9C42), C4.42(C2×C42), C42.20(C2×C4), C4.9C42.6C2, C23.1(C4○D4), C8○(C4.10C42), C2.6(C42⋊4C4), C4.45(C42⋊C2), C4.10C42.5C2, C8○2M4(2).14C2, (C22×C4).649C23, (C22×C8).376C22, C42⋊C2.259C22, C22.19(C42⋊C2), (C2×M4(2)).302C22, (C2×C8).9(C2×C4), (C2×C4).41(C4○D4), (C2×C4).520(C22×C4), SmallGroup(128,479)

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

Derived series
C1 — C4 — C8.16C42
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C22×C8 — C8○2M4(2) — C8.16C42
Lower central
C1 — C4 — C8.16C42
Upper central
C1 — C8 — C8.16C42
Jennings
C1 — C2 — C2 — C22×C4 — C8.16C42

Generators and relations for C8.16C42
 G = < a,b,c | a8=b4=c4=1, bab-1=cac-1=a5, cbc-1=a2b >

Subgroups: 148 in 100 conjugacy classes, 66 normal (9 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C8, C2×C4, C2×C4, C23, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C4.9C42, C4.10C42, C8○2M4(2), C8.16C42
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C4○D4, C2×C42, C42⋊C2, C42⋊4C4, C8.16C42

Smallest permutation representation of C8.16C42
►On 32 points
Generators in S32
(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 27 23 11)(2 32 24 16)(3 29 17 13)(4 26 18 10)(5 31 19 15)(6 28 20 12)(7 25 21 9)(8 30 22 14)

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

44 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F···4Q8A8B8C8D8E···8J8K···8V
order12222444444···488888···88···8
size11222112224···411112···24···4

44 irreducible representations

dim111111224
type++++
imageC1C2C2C2C4C4C4○D4C4○D4C8.16C42
kernelC8.16C42C4.9C42C4.10C42C8○2M4(2)C4×C8C8⋊C4C2×C4C23C1
# reps13131212624

Matrix representation of C8.16C42 ►in GL4(𝔽17) generated by

2000
0200
00150
00015
,
0024
001515
8000
9900
,
0010
0001
161500
0100
G:=sub<GL(4,GF(17))| [2,0,0,0,0,2,0,0,0,0,15,0,0,0,0,15],[0,0,8,9,0,0,0,9,2,15,0,0,4,15,0,0],[0,0,16,0,0,0,15,1,1,0,0,0,0,1,0,0] >;
 

C8.16C42 in GAP, Magma, Sage, TeX 

C_8._{16}C_4^2
 
% in TeX
 
G:=Group("C8.16C4^2");
 
// GroupNames label
 
G:=SmallGroup(128,479);
 
// by ID
 
G=gap.SmallGroup(128,479);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,58,248,1411,4037]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^4=c^4=1,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^2*b>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁