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## G = D5×C2×C10order 200 = 23·52

### Direct product of C2×C10 and D5

Aliases: D5×C2×C10, C1024C2, C522C23, C10⋊(C2×C10), C5⋊(C22×C10), (C2×C10)⋊3C10, (C5×C10)⋊2C22, SmallGroup(200,50)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C2×C10
 Chief series C1 — C5 — C52 — C5×D5 — D5×C10 — D5×C2×C10
 Lower central C5 — D5×C2×C10
 Upper central C1 — C2×C10

Generators and relations for D5×C2×C10
G = < a,b,c,d | a2=b10=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 172 in 74 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C22, C22, C5, C5, C23, D5, C10, C10, D10, C2×C10, C2×C10, C52, C22×D5, C22×C10, C5×D5, C5×C10, D5×C10, C102, D5×C2×C10
Quotients: C1, C2, C22, C5, C23, D5, C10, D10, C2×C10, C22×D5, C22×C10, C5×D5, D5×C10, D5×C2×C10

Smallest permutation representation of D5×C2×C10
On 40 points
Generators in S40
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 11)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(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 9 7 5 3)(2 10 8 6 4)(11 19 17 15 13)(12 20 18 16 14)(21 23 25 27 29)(22 24 26 28 30)(31 33 35 37 39)(32 34 36 38 40)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 21)(8 22)(9 23)(10 24)(11 39)(12 40)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)

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

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

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

D5×C2×C10 is a maximal subgroup of   D10⋊Dic5  D10.D10

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 5A 5B 5C 5D 5E ··· 5N 10A ··· 10L 10M ··· 10AP 10AQ ··· 10BF order 1 2 2 2 2 2 2 2 5 5 5 5 5 ··· 5 10 ··· 10 10 ··· 10 10 ··· 10 size 1 1 1 1 5 5 5 5 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 5 ··· 5

80 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C5 C10 C10 D5 D10 C5×D5 D5×C10 kernel D5×C2×C10 D5×C10 C102 C22×D5 D10 C2×C10 C2×C10 C10 C22 C2 # reps 1 6 1 4 24 4 2 6 8 24

Matrix representation of D5×C2×C10 in GL3(𝔽11) generated by

 1 0 0 0 10 0 0 0 10
,
 7 0 0 0 5 0 0 0 5
,
 1 0 0 0 4 6 0 0 3
,
 10 0 0 0 7 5 0 8 4
G:=sub<GL(3,GF(11))| [1,0,0,0,10,0,0,0,10],[7,0,0,0,5,0,0,0,5],[1,0,0,0,4,0,0,6,3],[10,0,0,0,7,8,0,5,4] >;

D5×C2×C10 in GAP, Magma, Sage, TeX

D_5\times C_2\times C_{10}
% in TeX

G:=Group("D5xC2xC10");
// GroupNames label

G:=SmallGroup(200,50);
// by ID

G=gap.SmallGroup(200,50);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,4004]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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