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G = C5×M5(2)  order 160 = 25·5

Direct product of C5 and M5(2)

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×M5(2), C4.C40, C80⋊7C2, C16⋊3C10, C8.2C20, C20.7C8, C22.C40, C40.11C4, C40.29C22, (C2×C10).3C8, (C2×C4).5C20, C8.8(C2×C10), (C2×C8).8C10, C2.3(C2×C40), (C2×C40).18C2, (C2×C20).25C4, C10.22(C2×C8), C4.12(C2×C20), C20.70(C2×C4), SmallGroup(160,60)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C5×M5(2)
Chief series
C1 — C2 — C4 — C8 — C40 — C80 — C5×M5(2)
Lower central
C1 — C2 — C5×M5(2)
Upper central
C1 — C40 — C5×M5(2)

Generators and relations for C5×M5(2)
 G = < a,b,c | a5=b16=c2=1, ab=ba, ac=ca, cbc=b9 >

C1
C2
2C2
C5
C22
C4
C4
C10
2C10
C8
C2×C4
C8
C20
C2×C10
C20
C16
C2×C8
C16
C2×C20
C40
C40
M5(2)
C80
C2×C40
C80
C5×M5(2)

Smallest permutation representation of C5×M5(2)
►On 80 points
Generators in S80
(1 39 56 66 19)(2 40 57 67 20)(3 41 58 68 21)(4 42 59 69 22)(5 43 60 70 23)(6 44 61 71 24)(7 45 62 72 25)(8 46 63 73 26)(9 47 64 74 27)(10 48 49 75 28)(11 33 50 76 29)(12 34 51 77 30)(13 35 52 78 31)(14 36 53 79 32)(15 37 54 80 17)(16 38 55 65 18)

(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)

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

C5×M5(2) is a maximal subgroup of
 C40.9Q8  C80⋊C4  C40.Q8  C8.25D20  D20.4C8  D40.4C4  C20.4D8  D40⋊8C4  D20.5C8  D80⋊C2  C16.D10

100 conjugacy classes

class 1 2A2B4A4B4C5A5B5C5D8A8B8C8D8E8F10A10B10C10D10E10F10G10H16A···16H20A···20H20I20J20K20L40A···40P40Q···40X80A···80AF
order1224445555888888101010101010101016···1620···202020202040···4040···4080···80
size1121121111111122111122222···21···122221···12···22···2

100 irreducible representations

dim1111111111111122
type+++
imageC1C2C2C4C4C5C8C8C10C10C20C20C40C40M5(2)C5×M5(2)
kernelC5×M5(2)C80C2×C40C40C2×C20M5(2)C20C2×C10C16C2×C8C8C2×C4C4C22C5C1
# reps1212244484881616416

Matrix representation of C5×M5(2) ►in GL2(𝔽41) generated by

160
016
,
027
10
,
400
01
G:=sub<GL(2,GF(41))| [16,0,0,16],[0,1,27,0],[40,0,0,1] >;
 

C5×M5(2) in GAP, Magma, Sage, TeX 

C_5\times M_5(2)
 
% in TeX
 
G:=Group("C5xM5(2)");
 
// GroupNames label
 
G:=SmallGroup(160,60);
 
// by ID
 
G=gap.SmallGroup(160,60);
 
# by ID
 
G:=PCGroup([6,-2,-2,-5,-2,-2,-2,120,985,69,88]);
 
// Polycyclic
 
G:=Group<a,b,c|a^5=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^9>;
 
// generators/relations
 

Export

Subgroup lattice of C5×M5(2) in TeX

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