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G = D4×C10order 80 = 24·5

Direct product of C10 and D4

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

Aliases: D4×C10, C23⋊C10, C204C22, C10.11C23, C4⋊(C2×C10), (C2×C20)⋊6C2, (C2×C4)⋊2C10, C22⋊(C2×C10), (C2×C10)⋊2C22, (C22×C10)⋊1C2, C2.1(C22×C10), SmallGroup(80,46)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C10
C1C2C10C2×C10C5×D4 — D4×C10
C1C2 — D4×C10
C1C2×C10 — D4×C10

Generators and relations for D4×C10
 G = < a,b,c | a10=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 70 in 54 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C22, C22 [×4], C22 [×4], C5, C2×C4, D4 [×4], C23 [×2], C10, C10 [×2], C10 [×4], C2×D4, C20 [×2], C2×C10, C2×C10 [×4], C2×C10 [×4], C2×C20, C5×D4 [×4], C22×C10 [×2], D4×C10
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], C2×D4, C2×C10 [×7], C5×D4 [×2], C22×C10, D4×C10

Smallest permutation representation of D4×C10
On 40 points
Generators in S40
(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 38 24 19)(2 39 25 20)(3 40 26 11)(4 31 27 12)(5 32 28 13)(6 33 29 14)(7 34 30 15)(8 35 21 16)(9 36 22 17)(10 37 23 18)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)

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

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

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

D4×C10 is a maximal subgroup of
D4⋊Dic5  C20.D4  C23⋊Dic5  D4.D10  C23.18D10  C20.17D4  C23⋊D10  C202D4  Dic5⋊D4  C20⋊D4  D46D10

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B5A5B5C5D10A···10L10M···10AB20A···20H
order1222222244555510···1010···1020···20
size111122222211111···12···22···2

50 irreducible representations

dim1111111122
type+++++
imageC1C2C2C2C5C10C10C10D4C5×D4
kernelD4×C10C2×C20C5×D4C22×C10C2×D4C2×C4D4C23C10C2
# reps11424416828

Matrix representation of D4×C10 in GL3(𝔽41) generated by

4000
0180
0018
,
100
001
0400
,
100
0040
0400
G:=sub<GL(3,GF(41))| [40,0,0,0,18,0,0,0,18],[1,0,0,0,0,40,0,1,0],[1,0,0,0,0,40,0,40,0] >;

D4×C10 in GAP, Magma, Sage, TeX

D_4\times C_{10}
% in TeX

G:=Group("D4xC10");
// GroupNames label

G:=SmallGroup(80,46);
// by ID

G=gap.SmallGroup(80,46);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-2,421]);
// Polycyclic

G:=Group<a,b,c|a^10=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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