Copied to
clipboard

G = D5×D13order 260 = 22·5·13

Direct product of D5 and D13

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D5×D13, D65⋊C2, C51D26, C65⋊C22, C131D10, (D5×C13)⋊C2, (C5×D13)⋊C2, SmallGroup(260,11)

Series: Derived Chief Lower central Upper central

C1C65 — D5×D13
C1C13C65C5×D13 — D5×D13
C65 — D5×D13
C1

Generators and relations for D5×D13
 G = < a,b,c,d | a5=b2=c13=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

5C2
13C2
65C2
65C22
13C10
13D5
5C26
5D13
13D10
5D26

Smallest permutation representation of D5×D13
On 65 points
Generators in S65
(1 56 40 33 23)(2 57 41 34 24)(3 58 42 35 25)(4 59 43 36 26)(5 60 44 37 14)(6 61 45 38 15)(7 62 46 39 16)(8 63 47 27 17)(9 64 48 28 18)(10 65 49 29 19)(11 53 50 30 20)(12 54 51 31 21)(13 55 52 32 22)
(1 23)(2 24)(3 25)(4 26)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(13 22)(27 63)(28 64)(29 65)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)
(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 41 42 43 44 45 46 47 48 49 50 51 52)(53 54 55 56 57 58 59 60 61 62 63 64 65)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 18)(15 17)(19 26)(20 25)(21 24)(22 23)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(53 58)(54 57)(55 56)(59 65)(60 64)(61 63)

G:=sub<Sym(65)| (1,56,40,33,23)(2,57,41,34,24)(3,58,42,35,25)(4,59,43,36,26)(5,60,44,37,14)(6,61,45,38,15)(7,62,46,39,16)(8,63,47,27,17)(9,64,48,28,18)(10,65,49,29,19)(11,53,50,30,20)(12,54,51,31,21)(13,55,52,32,22), (1,23)(2,24)(3,25)(4,26)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(13,22)(27,63)(28,64)(29,65)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62), (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)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,18)(15,17)(19,26)(20,25)(21,24)(22,23)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,58)(54,57)(55,56)(59,65)(60,64)(61,63)>;

G:=Group( (1,56,40,33,23)(2,57,41,34,24)(3,58,42,35,25)(4,59,43,36,26)(5,60,44,37,14)(6,61,45,38,15)(7,62,46,39,16)(8,63,47,27,17)(9,64,48,28,18)(10,65,49,29,19)(11,53,50,30,20)(12,54,51,31,21)(13,55,52,32,22), (1,23)(2,24)(3,25)(4,26)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(13,22)(27,63)(28,64)(29,65)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62), (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)(40,41,42,43,44,45,46,47,48,49,50,51,52)(53,54,55,56,57,58,59,60,61,62,63,64,65), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,18)(15,17)(19,26)(20,25)(21,24)(22,23)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,58)(54,57)(55,56)(59,65)(60,64)(61,63) );

G=PermutationGroup([[(1,56,40,33,23),(2,57,41,34,24),(3,58,42,35,25),(4,59,43,36,26),(5,60,44,37,14),(6,61,45,38,15),(7,62,46,39,16),(8,63,47,27,17),(9,64,48,28,18),(10,65,49,29,19),(11,53,50,30,20),(12,54,51,31,21),(13,55,52,32,22)], [(1,23),(2,24),(3,25),(4,26),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21),(13,22),(27,63),(28,64),(29,65),(30,53),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62)], [(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),(40,41,42,43,44,45,46,47,48,49,50,51,52),(53,54,55,56,57,58,59,60,61,62,63,64,65)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,18),(15,17),(19,26),(20,25),(21,24),(22,23),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(53,58),(54,57),(55,56),(59,65),(60,64),(61,63)]])

32 conjugacy classes

class 1 2A2B2C5A5B10A10B13A···13F26A···26F65A···65L
order122255101013···1326···2665···65
size1513652226262···210···104···4

32 irreducible representations

dim111122224
type+++++++++
imageC1C2C2C2D5D10D13D26D5×D13
kernelD5×D13D5×C13C5×D13D65D13C13D5C5C1
# reps1111226612

Matrix representation of D5×D13 in GL4(𝔽131) generated by

1000
0100
0001
00130119
,
1000
0100
0001
0010
,
103100
304100
0010
0001
,
1067500
862500
0010
0001
G:=sub<GL(4,GF(131))| [1,0,0,0,0,1,0,0,0,0,0,130,0,0,1,119],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[103,30,0,0,1,41,0,0,0,0,1,0,0,0,0,1],[106,86,0,0,75,25,0,0,0,0,1,0,0,0,0,1] >;

D5×D13 in GAP, Magma, Sage, TeX

D_5\times D_{13}
% in TeX

G:=Group("D5xD13");
// GroupNames label

G:=SmallGroup(260,11);
// by ID

G=gap.SmallGroup(260,11);
# by ID

G:=PCGroup([4,-2,-2,-5,-13,102,3843]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^13=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of D5×D13 in TeX

׿
×
𝔽