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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, C407C2, C83C10, 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

C1C2 — C5×M4(2)
C1C2C4C20C40 — C5×M4(2)
C1C2 — C5×M4(2)
C1C20 — C5×M4(2)

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

2C2
2C10

Smallest permutation representation of C5×M4(2)
On 40 points
Generators in S40
(1 18 27 35 13)(2 19 28 36 14)(3 20 29 37 15)(4 21 30 38 16)(5 22 31 39 9)(6 23 32 40 10)(7 24 25 33 11)(8 17 26 34 12)
(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)| (1,18,27,35,13)(2,19,28,36,14)(3,20,29,37,15)(4,21,30,38,16)(5,22,31,39,9)(6,23,32,40,10)(7,24,25,33,11)(8,17,26,34,12), (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( (1,18,27,35,13)(2,19,28,36,14)(3,20,29,37,15)(4,21,30,38,16)(5,22,31,39,9)(6,23,32,40,10)(7,24,25,33,11)(8,17,26,34,12), (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([(1,18,27,35,13),(2,19,28,36,14),(3,20,29,37,15),(4,21,30,38,16),(5,22,31,39,9),(6,23,32,40,10),(7,24,25,33,11),(8,17,26,34,12)], [(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  D207C4  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

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