direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×D13, D91⋊C2, C7⋊1D26, C91⋊C22, C13⋊1D14, (C13×D7)⋊C2, (C7×D13)⋊C2, SmallGroup(364,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C91 — D7×D13 |
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 >
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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