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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

 Derived series C1 — C4 — C5×SD16
 Chief series C1 — C2 — C4 — C20 — C5×Q8 — C5×SD16
 Lower central C1 — C2 — C4 — C5×SD16
 Upper central C1 — C10 — C20 — C5×SD16

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

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 2A 2B 4A 4B 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 4 4 5 5 5 5 8 8 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 4 2 4 1 1 1 1 2 2 1 1 1 1 4 4 4 4 2 2 2 2 4 4 4 4 2 ··· 2

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 SD16 C5×D4 C5×SD16 kernel C5×SD16 C40 C5×D4 C5×Q8 SD16 C8 D4 Q8 C10 C5 C2 C1 # reps 1 1 1 1 4 4 4 4 1 2 4 8

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

 4 0 0 4
,
 8 5 9 0
,
 10 7 0 1
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

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