Copied to
clipboard

G = C20.4C8order 160 = 25·5

1st non-split extension by C20 of C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C20.4C8, C40.9C4, C54M5(2), C8.22D10, C8.2Dic5, C40.22C22, C4.(C52C8), (C2×C8).7D5, C52C165C2, (C2×C10).5C8, (C2×C40).10C2, C20.60(C2×C4), (C2×C20).19C4, C10.18(C2×C8), C22.(C52C8), (C2×C4).5Dic5, C4.11(C2×Dic5), C2.4(C2×C52C8), SmallGroup(160,19)

Series: Derived Chief Lower central Upper central

C1C10 — C20.4C8
C1C5C10C20C40C52C16 — C20.4C8
C5C10 — C20.4C8
C1C8C2×C8

Generators and relations for C20.4C8
 G = < a,b | a40=1, b4=a10, bab-1=a29 >

2C2
2C10
5C16
5C16
5M5(2)

Smallest permutation representation of C20.4C8
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 58 26 63 11 68 36 73 21 78 6 43 31 48 16 53)(2 47 27 52 12 57 37 62 22 67 7 72 32 77 17 42)(3 76 28 41 13 46 38 51 23 56 8 61 33 66 18 71)(4 65 29 70 14 75 39 80 24 45 9 50 34 55 19 60)(5 54 30 59 15 64 40 69 25 74 10 79 35 44 20 49)

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

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

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

C20.4C8 is a maximal subgroup of
C40.7C8  C20.45C42  C8.Dic10  D4014C4  C40.6Q8  D40.6C4  C40.8D4  D20.3C8  C40.9Q8  C80⋊C4  C8.25D20  C40.D4  C40.92D4  D8.Dic5  Q16.Dic5  D82Dic5  D20.6C8  D5×M5(2)  C40.70C23  D8.D10  Q16.D10  D8⋊D10  C40.31C23  C40.52D6  C60.7C8
C20.4C8 is a maximal quotient of
C40.10C8  C203C16  C40.91D4  C40.52D6  C60.7C8

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)C52C8C52C8C20.4C8
kernelC20.4C8C52C16C2×C40C40C2×C20C20C2×C10C2×C8C8C8C2×C4C5C4C22C1
# reps1212244222244416

Matrix representation of C20.4C8 in GL2(𝔽41) generated by

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

C20.4C8 in GAP, Magma, Sage, TeX

C_{20}._4C_8
% in TeX

G:=Group("C20.4C8");
// 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

Export

Subgroup lattice of C20.4C8 in TeX

׿
×
𝔽