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G = C5×D8order 80 = 24·5

Direct product of C5 and D8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×D8, D4⋊C10, C81C10, C405C2, C10.14D4, C20.17C22, (C5×D4)⋊4C2, C2.3(C5×D4), C4.1(C2×C10), SmallGroup(80,25)

Series: Derived Chief Lower central Upper central

C1C4 — C5×D8
C1C2C4C20C5×D4 — C5×D8
C1C2C4 — C5×D8
C1C10C20 — C5×D8

Generators and relations for C5×D8
 G = < a,b,c | a5=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

4C2
4C2
2C22
2C22
4C10
4C10
2C2×C10
2C2×C10

Smallest permutation representation of C5×D8
On 40 points
Generators in S40
(1 23 25 39 12)(2 24 26 40 13)(3 17 27 33 14)(4 18 28 34 15)(5 19 29 35 16)(6 20 30 36 9)(7 21 31 37 10)(8 22 32 38 11)
(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)
(1 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 16)(17 20)(18 19)(21 24)(22 23)(25 32)(26 31)(27 30)(28 29)(33 36)(34 35)(37 40)(38 39)

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

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

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

C5×D8 is a maximal subgroup of   C5⋊D16  D8.D5  D8⋊D5  D83D5

35 conjugacy classes

class 1 2A2B2C 4 5A5B5C5D8A8B10A10B10C10D10E···10L20A20B20C20D40A···40H
order122245555881010101010···102020202040···40
size1144211112211114···422222···2

35 irreducible representations

dim1111112222
type+++++
imageC1C2C2C5C10C10D4D8C5×D4C5×D8
kernelC5×D8C40C5×D4D8C8D4C10C5C2C1
# reps1124481248

Matrix representation of C5×D8 in GL2(𝔽31) generated by

160
016
,
030
123
,
231
308
G:=sub<GL(2,GF(31))| [16,0,0,16],[0,1,30,23],[23,30,1,8] >;

C5×D8 in GAP, Magma, Sage, TeX

C_5\times D_8
% in TeX

G:=Group("C5xD8");
// GroupNames label

G:=SmallGroup(80,25);
// by ID

G=gap.SmallGroup(80,25);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-2,221,1203,608,58]);
// 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^-1>;
// generators/relations

Export

Subgroup lattice of C5×D8 in TeX

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