Copied to
clipboard

G = C40.9C4  order 160 = 25·5

4th non-split extension by C40 of C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C40.9C4, C20.4C8, C5⋊4M5(2), C8.22D10, C8.2Dic5, C40.22C22, C4.(C5⋊2C8), (C2×C8).7D5, C5⋊2C16⋊5C2, (C2×C10).5C8, (C2×C40).10C2, C20.60(C2×C4), (C2×C20).19C4, C10.18(C2×C8), C22.(C5⋊2C8), (C2×C4).5Dic5, C4.11(C2×Dic5), C2.4(C2×C5⋊2C8), SmallGroup(160,19)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C40.9C4
Chief series
C1 — C5 — C10 — C20 — C40 — C5⋊2C16 — C40.9C4
Lower central
C5 — C10 — C40.9C4
Upper central
C1 — C8 — C2×C8

Generators and relations for C40.9C4
 G = < a,b | a40=1, b4=a10, bab-1=a29 >

C1
C2
2C2
C5
C22
C4
C4
C10
2C10
C8
C2×C4
C8
C20
C2×C10
C20
C2×C8
5C16
5C16
C2×C20
C40
C40
5M5(2)
C5⋊2C16
C2×C40
C5⋊2C16
C40.9C4

Smallest permutation representation of C40.9C4
►On 80 points
Generators in S80
(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 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)

(1 64 26 69 11 74 36 79 21 44 6 49 31 54 16 59)(2 53 27 58 12 63 37 68 22 73 7 78 32 43 17 48)(3 42 28 47 13 52 38 57 23 62 8 67 33 72 18 77)(4 71 29 76 14 41 39 46 24 51 9 56 34 61 19 66)(5 60 30 65 15 70 40 75 25 80 10 45 35 50 20 55)
 
G:=sub<Sym(80)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,64,26,69,11,74,36,79,21,44,6,49,31,54,16,59)(2,53,27,58,12,63,37,68,22,73,7,78,32,43,17,48)(3,42,28,47,13,52,38,57,23,62,8,67,33,72,18,77)(4,71,29,76,14,41,39,46,24,51,9,56,34,61,19,66)(5,60,30,65,15,70,40,75,25,80,10,45,35,50,20,55)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,64,26,69,11,74,36,79,21,44,6,49,31,54,16,59)(2,53,27,58,12,63,37,68,22,73,7,78,32,43,17,48)(3,42,28,47,13,52,38,57,23,62,8,67,33,72,18,77)(4,71,29,76,14,41,39,46,24,51,9,56,34,61,19,66)(5,60,30,65,15,70,40,75,25,80,10,45,35,50,20,55) );
 
G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,64,26,69,11,74,36,79,21,44,6,49,31,54,16,59),(2,53,27,58,12,63,37,68,22,73,7,78,32,43,17,48),(3,42,28,47,13,52,38,57,23,62,8,67,33,72,18,77),(4,71,29,76,14,41,39,46,24,51,9,56,34,61,19,66),(5,60,30,65,15,70,40,75,25,80,10,45,35,50,20,55)])
 

52 conjugacy classes

class 1 2A2B4A4B4C5A5B8A8B8C8D8E8F10A···10F16A···16H20A···20H40A···40P
order1224445588888810···1016···1620···2040···40
size112112221111222···210···102···22···2

52 irreducible representations

dim111111122222222
type++++-+-
imageC1C2C2C4C4C8C8D5Dic5D10Dic5M5(2)C5⋊2C8C5⋊2C8C40.9C4
kernelC40.9C4C5⋊2C16C2×C40C40C2×C20C20C2×C10C2×C8C8C8C2×C4C5C4C22C1
# reps1212244222244416

Matrix representation of C40.9C4 ►in GL2(𝔽41) generated by

110
029
,
03
10
G:=sub<GL(2,GF(41))| [11,0,0,29],[0,1,3,0] >;
 

C40.9C4 in GAP, Magma, Sage, TeX 

C_{40}._9C_4
 
% in TeX
 
G:=Group("C40.9C4");
 
// GroupNames label
 
G:=SmallGroup(160,19);
 
// by ID
 
G=gap.SmallGroup(160,19);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,217,50,69,4613]);
 
// Polycyclic
 
G:=Group<a,b|a^40=1,b^4=a^10,b*a*b^-1=a^29>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁