Copied to
clipboard

G = C5×M4(2)  order 80 = 24·5

Direct product of C5 and M4(2)

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

Aliases: C5×M4(2), C4.C20, C40⋊7C2, C8⋊3C10, C20.7C4, C22.C20, C20.22C22, (C2×C20).8C2, C2.3(C2×C20), (C2×C4).2C10, C4.6(C2×C10), (C2×C10).3C4, C10.19(C2×C4), SmallGroup(80,24)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C5×M4(2)
Chief series
C1 — C2 — C4 — C20 — C40 — C5×M4(2)
Lower central
C1 — C2 — C5×M4(2)
Upper central
C1 — C20 — C5×M4(2)

Generators and relations for C5×M4(2)
 G = < a,b,c | a5=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

C1
C2
2C2
C5
C4
C22
C4
C10
2C10
C8
C2×C4
C8
C2×C10
C20
C20
M4(2)
C40
C40
C2×C20
C5×M4(2)

Smallest permutation representation of C5×M4(2)
►On 40 points
Generators in S40
(1 18 31 35 11)(2 19 32 36 12)(3 20 25 37 13)(4 21 26 38 14)(5 22 27 39 15)(6 23 28 40 16)(7 24 29 33 9)(8 17 30 34 10)

(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)

(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)(34 38)(36 40)
 
G:=sub<Sym(40)| (1,18,31,35,11)(2,19,32,36,12)(3,20,25,37,13)(4,21,26,38,14)(5,22,27,39,15)(6,23,28,40,16)(7,24,29,33,9)(8,17,30,34,10), (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40)>;
 
G:=Group( (1,18,31,35,11)(2,19,32,36,12)(3,20,25,37,13)(4,21,26,38,14)(5,22,27,39,15)(6,23,28,40,16)(7,24,29,33,9)(8,17,30,34,10), (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), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32)(34,38)(36,40) );
 
G=PermutationGroup([[(1,18,31,35,11),(2,19,32,36,12),(3,20,25,37,13),(4,21,26,38,14),(5,22,27,39,15),(6,23,28,40,16),(7,24,29,33,9),(8,17,30,34,10)], [(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)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(26,30),(28,32),(34,38),(36,40)]])
 

C5×M4(2) is a maximal subgroup of   C20.53D4  C20.46D4  C4.12D20  D20⋊7C4  D20.2C4  C8⋊D10  C8.D10

50 conjugacy classes

class 1 2A2B4A4B4C5A5B5C5D8A8B8C8D10A10B10C10D10E10F10G10H20A···20H20I20J20K20L40A···40P
order12244455558888101010101010101020···202020202040···40
size11211211112222111122221···122222···2

50 irreducible representations

dim111111111122
type+++
imageC1C2C2C4C4C5C10C10C20C20M4(2)C5×M4(2)
kernelC5×M4(2)C40C2×C20C20C2×C10M4(2)C8C2×C4C4C22C5C1
# reps121224848828

Matrix representation of C5×M4(2) ►in GL2(𝔽41) generated by

180
018
,
4036
231
,
19
040
G:=sub<GL(2,GF(41))| [18,0,0,18],[40,23,36,1],[1,0,9,40] >;
 

C5×M4(2) in GAP, Magma, Sage, TeX 

C_5\times M_4(2)
 
% in TeX
 
G:=Group("C5xM4(2)");
 
// GroupNames label
 
G:=SmallGroup(80,24);
 
// by ID
 
G=gap.SmallGroup(80,24);
 
# by ID
 
G:=PCGroup([5,-2,-2,-5,-2,-2,100,421,58]);
 
// Polycyclic
 
G:=Group<a,b,c|a^5=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
 
// generators/relations
 

Export

Subgroup lattice of C5×M4(2) in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁