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## G = D7×D13order 364 = 22·7·13

### Direct product of D7 and D13

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

 Derived series C1 — C91 — D7×D13
 Chief series C1 — C13 — C91 — C7×D13 — D7×D13
 Lower central C91 — D7×D13
 Upper central 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 2A 2B 2C 7A 7B 7C 13A ··· 13F 14A 14B 14C 26A ··· 26F 91A ··· 91R order 1 2 2 2 7 7 7 13 ··· 13 14 14 14 26 ··· 26 91 ··· 91 size 1 7 13 91 2 2 2 2 ··· 2 26 26 26 14 ··· 14 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 D7 D13 D14 D26 D7×D13 kernel D7×D13 C13×D7 C7×D13 D91 D13 D7 C13 C7 C1 # reps 1 1 1 1 3 6 3 6 18

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

 1 63 0 0 222 312 0 0 0 0 1 0 0 0 0 1
,
 1 63 0 0 0 546 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 366 1 0 0 508 245
,
 1 0 0 0 0 1 0 0 0 0 245 546 0 0 401 302
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

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