direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8×D13, C8⋊4D26, D4⋊1D26, D104⋊4C2, C104⋊2C22, D26.23D4, D52⋊1C22, C52.1C23, Dic13.7D4, C13⋊2(C2×D8), D4⋊D13⋊1C2, (C13×D8)⋊2C2, (C8×D13)⋊1C2, (D4×D13)⋊1C2, C26.27(C2×D4), C2.15(D4×D13), C13⋊2C8⋊5C22, (D4×C13)⋊1C22, C4.1(C22×D13), (C4×D13).15C22, SmallGroup(416,131)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×D13
G = < a,b,c,d | a8=b2=c13=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 752 in 76 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, C23, C13, C2×C8, D8, D8, C2×D4, D13, D13, C26, C26, C2×D8, Dic13, C52, D26, D26, C2×C26, C13⋊2C8, C104, C4×D13, D52, C13⋊D4, D4×C13, C22×D13, C8×D13, D104, D4⋊D13, C13×D8, D4×D13, D8×D13
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, D13, C2×D8, D26, C22×D13, D4×D13, D8×D13
►On 104 points
Generators in S104
(1 84 31 62 23 93 51 72)(2 85 32 63 24 94 52 73)(3 86 33 64 25 95 40 74)(4 87 34 65 26 96 41 75)(5 88 35 53 14 97 42 76)(6 89 36 54 15 98 43 77)(7 90 37 55 16 99 44 78)(8 91 38 56 17 100 45 66)(9 79 39 57 18 101 46 67)(10 80 27 58 19 102 47 68)(11 81 28 59 20 103 48 69)(12 82 29 60 21 104 49 70)(13 83 30 61 22 92 50 71)
(1 62)(2 63)(3 64)(4 65)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 76)(15 77)(16 78)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 79)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 92)(51 93)(52 94)
(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)(92 93 94 95 96 97 98 99 100 101 102 103 104)
(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 34)(28 33)(29 32)(30 31)(35 39)(36 38)(40 48)(41 47)(42 46)(43 45)(49 52)(50 51)(53 57)(54 56)(58 65)(59 64)(60 63)(61 62)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(79 88)(80 87)(81 86)(82 85)(83 84)(89 91)(92 93)(94 104)(95 103)(96 102)(97 101)(98 100)
G:=sub<Sym(104)| (1,84,31,62,23,93,51,72)(2,85,32,63,24,94,52,73)(3,86,33,64,25,95,40,74)(4,87,34,65,26,96,41,75)(5,88,35,53,14,97,42,76)(6,89,36,54,15,98,43,77)(7,90,37,55,16,99,44,78)(8,91,38,56,17,100,45,66)(9,79,39,57,18,101,46,67)(10,80,27,58,19,102,47,68)(11,81,28,59,20,103,48,69)(12,82,29,60,21,104,49,70)(13,83,30,61,22,92,50,71), (1,62)(2,63)(3,64)(4,65)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,76)(15,77)(16,78)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,79)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,92)(51,93)(52,94), (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)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,18)(15,17)(19,26)(20,25)(21,24)(22,23)(27,34)(28,33)(29,32)(30,31)(35,39)(36,38)(40,48)(41,47)(42,46)(43,45)(49,52)(50,51)(53,57)(54,56)(58,65)(59,64)(60,63)(61,62)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,88)(80,87)(81,86)(82,85)(83,84)(89,91)(92,93)(94,104)(95,103)(96,102)(97,101)(98,100)>;
G:=Group( (1,84,31,62,23,93,51,72)(2,85,32,63,24,94,52,73)(3,86,33,64,25,95,40,74)(4,87,34,65,26,96,41,75)(5,88,35,53,14,97,42,76)(6,89,36,54,15,98,43,77)(7,90,37,55,16,99,44,78)(8,91,38,56,17,100,45,66)(9,79,39,57,18,101,46,67)(10,80,27,58,19,102,47,68)(11,81,28,59,20,103,48,69)(12,82,29,60,21,104,49,70)(13,83,30,61,22,92,50,71), (1,62)(2,63)(3,64)(4,65)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,76)(15,77)(16,78)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,79)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,92)(51,93)(52,94), (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)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,18)(15,17)(19,26)(20,25)(21,24)(22,23)(27,34)(28,33)(29,32)(30,31)(35,39)(36,38)(40,48)(41,47)(42,46)(43,45)(49,52)(50,51)(53,57)(54,56)(58,65)(59,64)(60,63)(61,62)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,88)(80,87)(81,86)(82,85)(83,84)(89,91)(92,93)(94,104)(95,103)(96,102)(97,101)(98,100) );
G=PermutationGroup([[(1,84,31,62,23,93,51,72),(2,85,32,63,24,94,52,73),(3,86,33,64,25,95,40,74),(4,87,34,65,26,96,41,75),(5,88,35,53,14,97,42,76),(6,89,36,54,15,98,43,77),(7,90,37,55,16,99,44,78),(8,91,38,56,17,100,45,66),(9,79,39,57,18,101,46,67),(10,80,27,58,19,102,47,68),(11,81,28,59,20,103,48,69),(12,82,29,60,21,104,49,70),(13,83,30,61,22,92,50,71)], [(1,62),(2,63),(3,64),(4,65),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,76),(15,77),(16,78),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,80),(28,81),(29,82),(30,83),(31,84),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,91),(39,79),(40,95),(41,96),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,92),(51,93),(52,94)], [(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),(92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,18),(15,17),(19,26),(20,25),(21,24),(22,23),(27,34),(28,33),(29,32),(30,31),(35,39),(36,38),(40,48),(41,47),(42,46),(43,45),(49,52),(50,51),(53,57),(54,56),(58,65),(59,64),(60,63),(61,62),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(79,88),(80,87),(81,86),(82,85),(83,84),(89,91),(92,93),(94,104),(95,103),(96,102),(97,101),(98,100)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 8A | 8B | 8C | 8D | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26R | 52A | ··· | 52F | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 4 | 4 | 13 | 13 | 52 | 52 | 2 | 26 | 2 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | D13 | D26 | D26 | D4×D13 | D8×D13 |
| kernel | D8×D13 | C8×D13 | D104 | D4⋊D13 | C13×D8 | D4×D13 | Dic13 | D26 | D13 | D8 | C8 | D4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 4 | 6 | 6 | 12 | 6 | 12 |
Matrix representation of D8×D13 ►in GL4(𝔽313) generated by
| 312 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 |
| 0 | 0 | 0 | 193 |
| 0 | 0 | 60 | 193 |
| 312 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 |
| 0 | 0 | 0 | 120 |
| 0 | 0 | 60 | 0 |
| 71 | 1 | 0 | 0 |
| 252 | 127 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 54 | 222 | 0 | 0 |
| 290 | 259 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(313))| [312,0,0,0,0,312,0,0,0,0,0,60,0,0,193,193],[312,0,0,0,0,312,0,0,0,0,0,60,0,0,120,0],[71,252,0,0,1,127,0,0,0,0,1,0,0,0,0,1],[54,290,0,0,222,259,0,0,0,0,1,0,0,0,0,1] >;
D8×D13 in GAP, Magma, Sage, TeX
D_8\times D_{13} % in TeX
G:=Group("D8xD13"); // GroupNames label
G:=SmallGroup(416,131);
// by ID
G=gap.SmallGroup(416,131);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,116,297,159,69,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^8=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