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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

 Derived series C1 — C2 — D4×C10
 Chief series C1 — C2 — C10 — C2×C10 — C5×D4 — D4×C10
 Lower central C1 — C2 — D4×C10
 Upper central C1 — C2×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, C2, C4, C22, C22, C22, C5, C2×C4, D4, C23, C10, C10, C10, C2×D4, C20, C2×C10, C2×C10, C2×C10, C2×C20, C5×D4, C22×C10, D4×C10
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C5×D4, 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 26 19)(2 39 27 20)(3 40 28 11)(4 31 29 12)(5 32 30 13)(6 33 21 14)(7 34 22 15)(8 35 23 16)(9 36 24 17)(10 37 25 18)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 21)(20 22)

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

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

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

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 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AB 20A ··· 20H order 1 2 2 2 2 2 2 2 4 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 2 2 2 2 1 1 1 1 1 ··· 1 2 ··· 2 2 ··· 2

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 C5×D4 kernel D4×C10 C2×C20 C5×D4 C22×C10 C2×D4 C2×C4 D4 C23 C10 C2 # reps 1 1 4 2 4 4 16 8 2 8

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

 40 0 0 0 18 0 0 0 18
,
 1 0 0 0 0 1 0 40 0
,
 1 0 0 0 0 40 0 40 0
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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