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G = D7×D13order 364 = 22·7·13

Direct product of D7 and D13

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

Aliases: D7×D13, D91⋊C2, C71D26, C91⋊C22, C131D14, (C13×D7)⋊C2, (C7×D13)⋊C2, SmallGroup(364,7)

Series: Derived Chief Lower central Upper central

C1C91 — D7×D13
C1C13C91C7×D13 — D7×D13
C91 — D7×D13
C1

Generators and relations for D7×D13
 G = < a,b,c,d | a7=b2=c13=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

7C2
13C2
91C2
91C22
13C14
13D7
7C26
7D13
13D14
7D26

Smallest permutation representation of D7×D13
On 91 points
Generators in S91
(1 91 72 55 48 30 17)(2 79 73 56 49 31 18)(3 80 74 57 50 32 19)(4 81 75 58 51 33 20)(5 82 76 59 52 34 21)(6 83 77 60 40 35 22)(7 84 78 61 41 36 23)(8 85 66 62 42 37 24)(9 86 67 63 43 38 25)(10 87 68 64 44 39 26)(11 88 69 65 45 27 14)(12 89 70 53 46 28 15)(13 90 71 54 47 29 16)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 14)(12 15)(13 16)(27 88)(28 89)(29 90)(30 91)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 77)(41 78)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)
(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)(66 67 68 69 70 71 72 73 74 75 76 77 78)(79 80 81 82 83 84 85 86 87 88 89 90 91)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 19)(15 18)(16 17)(20 26)(21 25)(22 24)(27 32)(28 31)(29 30)(33 39)(34 38)(35 37)(40 42)(43 52)(44 51)(45 50)(46 49)(47 48)(53 56)(54 55)(57 65)(58 64)(59 63)(60 62)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(79 89)(80 88)(81 87)(82 86)(83 85)(90 91)

G:=sub<Sym(91)| (1,91,72,55,48,30,17)(2,79,73,56,49,31,18)(3,80,74,57,50,32,19)(4,81,75,58,51,33,20)(5,82,76,59,52,34,21)(6,83,77,60,40,35,22)(7,84,78,61,41,36,23)(8,85,66,62,42,37,24)(9,86,67,63,43,38,25)(10,87,68,64,44,39,26)(11,88,69,65,45,27,14)(12,89,70,53,46,28,15)(13,90,71,54,47,29,16), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,14)(12,15)(13,16)(27,88)(28,89)(29,90)(30,91)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,77)(41,78)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76), (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)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,19)(15,18)(16,17)(20,26)(21,25)(22,24)(27,32)(28,31)(29,30)(33,39)(34,38)(35,37)(40,42)(43,52)(44,51)(45,50)(46,49)(47,48)(53,56)(54,55)(57,65)(58,64)(59,63)(60,62)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,89)(80,88)(81,87)(82,86)(83,85)(90,91)>;

G:=Group( (1,91,72,55,48,30,17)(2,79,73,56,49,31,18)(3,80,74,57,50,32,19)(4,81,75,58,51,33,20)(5,82,76,59,52,34,21)(6,83,77,60,40,35,22)(7,84,78,61,41,36,23)(8,85,66,62,42,37,24)(9,86,67,63,43,38,25)(10,87,68,64,44,39,26)(11,88,69,65,45,27,14)(12,89,70,53,46,28,15)(13,90,71,54,47,29,16), (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,14)(12,15)(13,16)(27,88)(28,89)(29,90)(30,91)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,77)(41,78)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74)(51,75)(52,76), (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)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,19)(15,18)(16,17)(20,26)(21,25)(22,24)(27,32)(28,31)(29,30)(33,39)(34,38)(35,37)(40,42)(43,52)(44,51)(45,50)(46,49)(47,48)(53,56)(54,55)(57,65)(58,64)(59,63)(60,62)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,89)(80,88)(81,87)(82,86)(83,85)(90,91) );

G=PermutationGroup([[(1,91,72,55,48,30,17),(2,79,73,56,49,31,18),(3,80,74,57,50,32,19),(4,81,75,58,51,33,20),(5,82,76,59,52,34,21),(6,83,77,60,40,35,22),(7,84,78,61,41,36,23),(8,85,66,62,42,37,24),(9,86,67,63,43,38,25),(10,87,68,64,44,39,26),(11,88,69,65,45,27,14),(12,89,70,53,46,28,15),(13,90,71,54,47,29,16)], [(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,14),(12,15),(13,16),(27,88),(28,89),(29,90),(30,91),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,77),(41,78),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,73),(50,74),(51,75),(52,76)], [(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),(66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,19),(15,18),(16,17),(20,26),(21,25),(22,24),(27,32),(28,31),(29,30),(33,39),(34,38),(35,37),(40,42),(43,52),(44,51),(45,50),(46,49),(47,48),(53,56),(54,55),(57,65),(58,64),(59,63),(60,62),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(79,89),(80,88),(81,87),(82,86),(83,85),(90,91)]])

40 conjugacy classes

class 1 2A2B2C7A7B7C13A···13F14A14B14C26A···26F91A···91R
order122277713···1314141426···2691···91
size1713912222···226262614···144···4

40 irreducible representations

dim111122224
type+++++++++
imageC1C2C2C2D7D13D14D26D7×D13
kernelD7×D13C13×D7C7×D13D91D13D7C13C7C1
# reps1111363618

Matrix representation of D7×D13 in GL4(𝔽547) generated by

16300
22231200
0010
0001
,
16300
054600
0010
0001
,
1000
0100
003661
00508245
,
1000
0100
00245546
00401302
G:=sub<GL(4,GF(547))| [1,222,0,0,63,312,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,63,546,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,366,508,0,0,1,245],[1,0,0,0,0,1,0,0,0,0,245,401,0,0,546,302] >;

D7×D13 in GAP, Magma, Sage, TeX

D_7\times D_{13}
% in TeX

G:=Group("D7xD13");
// GroupNames label

G:=SmallGroup(364,7);
// by ID

G=gap.SmallGroup(364,7);
# by ID

G:=PCGroup([4,-2,-2,-7,-13,150,5379]);
// Polycyclic

G:=Group<a,b,c,d|a^7=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 D7×D13 in TeX

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