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G = D5xC2xC10order 200 = 23·52

Direct product of C2xC10 and D5

direct product, metabelian, supersoluble, monomial, A-group

Aliases: D5xC2xC10, C102:4C2, C52:2C23, C10:(C2xC10), C5:(C22xC10), (C2xC10):3C10, (C5xC10):2C22, SmallGroup(200,50)

Series: Derived Chief Lower central Upper central

C1C5 — D5xC2xC10
C1C5C52C5xD5D5xC10 — D5xC2xC10
C5 — D5xC2xC10
C1C2xC10

Generators and relations for D5xC2xC10
 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, C2xC10, C2xC10, C52, C22xD5, C22xC10, C5xD5, C5xC10, D5xC10, C102, D5xC2xC10
Quotients: C1, C2, C22, C5, C23, D5, C10, D10, C2xC10, C22xD5, C22xC10, C5xD5, D5xC10, D5xC2xC10

Smallest permutation representation of D5xC2xC10
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)]])

D5xC2xC10 is a maximal subgroup of   D10:Dic5  D10.D10

80 conjugacy classes

class 1 2A2B2C2D2E2F2G5A5B5C5D5E···5N10A···10L10M···10AP10AQ···10BF
order1222222255555···510···1010···1010···10
size1111555511112···21···12···25···5

80 irreducible representations

dim1111112222
type+++++
imageC1C2C2C5C10C10D5D10C5xD5D5xC10
kernelD5xC2xC10D5xC10C102C22xD5D10C2xC10C2xC10C10C22C2
# reps161424426824

Matrix representation of D5xC2xC10 in GL3(F11) generated by

100
0100
0010
,
700
050
005
,
100
046
003
,
1000
075
084
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] >;

D5xC2xC10 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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