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G = C40.6C4order 160 = 25·5

1st non-split extension by C40 of C4 acting via C4/C2=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C40.6C4, C4.18D20, C20.34D4, C8.1Dic5, C22.2Dic10, (C2×C8).5D5, (C2×C40).7C2, (C2×C10).3Q8, C53(C8.C4), C20.56(C2×C4), (C2×C4).70D10, C10.14(C4⋊C4), C4.8(C2×Dic5), C2.5(C4⋊Dic5), C4.Dic5.1C2, (C2×C20).97C22, SmallGroup(160,26)

Series: Derived Chief Lower central Upper central

C1C20 — C40.6C4
C1C5C10C20C2×C20C4.Dic5 — C40.6C4
C5C10C20 — C40.6C4
C1C4C2×C4C2×C8

Generators and relations for C40.6C4
 G = < a,b | a40=1, b4=a20, bab-1=a19 >

2C2
2C10
10C8
10C8
5M4(2)
5M4(2)
2C52C8
2C52C8
5C8.C4

Smallest permutation representation of C40.6C4
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 51 31 61 21 71 11 41)(2 70 32 80 22 50 12 60)(3 49 33 59 23 69 13 79)(4 68 34 78 24 48 14 58)(5 47 35 57 25 67 15 77)(6 66 36 76 26 46 16 56)(7 45 37 55 27 65 17 75)(8 64 38 74 28 44 18 54)(9 43 39 53 29 63 19 73)(10 62 40 72 30 42 20 52)

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

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

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

C40.6C4 is a maximal subgroup of
C8.Dic10  C40.7Q8  C80.6C4  D40.3C4  C40.Q8  D40.4C4  C20.4D8  D8.Dic5  Q16.Dic5  C20.58D8  D4017C4  D4010C4  D5×C8.C4  M4(2).25D10  M4(2).Dic5  D4.3D20  D4.4D20  D4.5D20  C40.23D4  C40.44D4  C40.29D4  D85Dic5  D84Dic5  C12.59D20  C4.18D60
C40.6C4 is a maximal quotient of
C406C8  C405C8  C20.40C42  C12.59D20  C4.18D60

46 conjugacy classes

class 1 2A2B4A4B4C5A5B8A8B8C8D8E8F8G8H10A···10F20A···20H40A···40P
order122444558888888810···1020···2040···40
size112112222222202020202···22···22···2

46 irreducible representations

dim1111222222222
type++++-+-++-
imageC1C2C2C4D4Q8D5Dic5D10C8.C4D20Dic10C40.6C4
kernelC40.6C4C4.Dic5C2×C40C40C20C2×C10C2×C8C8C2×C4C5C4C22C1
# reps12141124244416

Matrix representation of C40.6C4 in GL2(𝔽41) generated by

300
015
,
032
10
G:=sub<GL(2,GF(41))| [30,0,0,15],[0,1,32,0] >;

C40.6C4 in GAP, Magma, Sage, TeX

C_{40}._6C_4
% in TeX

G:=Group("C40.6C4");
// GroupNames label

G:=SmallGroup(160,26);
// by ID

G=gap.SmallGroup(160,26);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,55,86,579,69,4613]);
// Polycyclic

G:=Group<a,b|a^40=1,b^4=a^20,b*a*b^-1=a^19>;
// generators/relations

Export

Subgroup lattice of C40.6C4 in TeX

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