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G = D5×C10order 100 = 22·52

Direct product of C10 and D5

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

Aliases: D5×C10, C10⋊C10, C522C22, C5⋊(C2×C10), (C5×C10)⋊1C2, SmallGroup(100,14)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C10
C1C5C52C5×D5 — D5×C10
C5 — D5×C10
C1C10

Generators and relations for D5×C10
 G = < a,b,c | a10=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C2
2C5
2C5
5C22
2C10
2C10
5C10
5C10
5C2×C10

Permutation representations of D5×C10
On 20 points - transitive group 20T24
Generators in S20
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 5 9 3 7)(2 6 10 4 8)(11 17 13 19 15)(12 18 14 20 16)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 11)(8 12)(9 13)(10 14)

G:=sub<Sym(20)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,5,9,3,7)(2,6,10,4,8)(11,17,13,19,15)(12,18,14,20,16), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,11)(8,12)(9,13)(10,14)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,5,9,3,7)(2,6,10,4,8)(11,17,13,19,15)(12,18,14,20,16), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,11)(8,12)(9,13)(10,14) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,5,9,3,7),(2,6,10,4,8),(11,17,13,19,15),(12,18,14,20,16)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,11),(8,12),(9,13),(10,14)])

G:=TransitiveGroup(20,24);

D5×C10 is a maximal subgroup of   C522D4  C5⋊D20

40 conjugacy classes

class 1 2A2B2C5A5B5C5D5E···5N10A10B10C10D10E···10N10O···10V
order122255555···51010101010···1010···10
size115511112···211112···25···5

40 irreducible representations

dim1111112222
type+++++
imageC1C2C2C5C10C10D5D10C5×D5D5×C10
kernelD5×C10C5×D5C5×C10D10D5C10C10C5C2C1
# reps1214842288

Matrix representation of D5×C10 in GL2(𝔽11) generated by

20
02
,
30
54
,
78
54
G:=sub<GL(2,GF(11))| [2,0,0,2],[3,5,0,4],[7,5,8,4] >;

D5×C10 in GAP, Magma, Sage, TeX

D_5\times C_{10}
% in TeX

G:=Group("D5xC10");
// GroupNames label

G:=SmallGroup(100,14);
// by ID

G=gap.SmallGroup(100,14);
# by ID

G:=PCGroup([4,-2,-2,-5,-5,1283]);
// Polycyclic

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

Export

Subgroup lattice of D5×C10 in TeX

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