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G = C5×SD16order 80 = 24·5

Direct product of C5 and SD16

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

Aliases: C5×SD16, Q8⋊C10, C406C2, C82C10, D4.C10, C10.15D4, C20.18C22, (C5×Q8)⋊4C2, C2.4(C5×D4), C4.2(C2×C10), (C5×D4).2C2, SmallGroup(80,26)

Series: Derived Chief Lower central Upper central

C1C4 — C5×SD16
C1C2C4C20C5×Q8 — C5×SD16
C1C2C4 — C5×SD16
C1C10C20 — C5×SD16

Generators and relations for C5×SD16
 G = < a,b,c | a5=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

4C2
2C4
2C22
4C10
2C20
2C2×C10

Smallest permutation representation of C5×SD16
On 40 points
Generators in S40
(1 18 31 35 14)(2 19 32 36 15)(3 20 25 37 16)(4 21 26 38 9)(5 22 27 39 10)(6 23 28 40 11)(7 24 29 33 12)(8 17 30 34 13)
(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 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 23)(19 21)(20 24)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)

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

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

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

C5×SD16 is a maximal subgroup of   D40⋊C2  SD16⋊D5  SD163D5

35 conjugacy classes

class 1 2A2B4A4B5A5B5C5D8A8B10A10B10C10D10E10F10G10H20A20B20C20D20E20F20G20H40A···40H
order122445555881010101010101010202020202020202040···40
size1142411112211114444222244442···2

35 irreducible representations

dim111111112222
type+++++
imageC1C2C2C2C5C10C10C10D4SD16C5×D4C5×SD16
kernelC5×SD16C40C5×D4C5×Q8SD16C8D4Q8C10C5C2C1
# reps111144441248

Matrix representation of C5×SD16 in GL2(𝔽11) generated by

40
04
,
85
90
,
107
01
G:=sub<GL(2,GF(11))| [4,0,0,4],[8,9,5,0],[10,0,7,1] >;

C5×SD16 in GAP, Magma, Sage, TeX

C_5\times {\rm SD}_{16}
% in TeX

G:=Group("C5xSD16");
// GroupNames label

G:=SmallGroup(80,26);
// by ID

G=gap.SmallGroup(80,26);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-2,200,221,1203,608,58]);
// 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^3>;
// generators/relations

Export

Subgroup lattice of C5×SD16 in TeX

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