metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊D13, C13⋊2D8, D52⋊2C2, C4.1D26, C26.7D4, C52.1C22, C13⋊2C8⋊1C2, (D4×C13)⋊1C2, C2.4(C13⋊D4), SmallGroup(208,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D13
G = < a,b,c,d | a4=b2=c13=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >
Smallest permutation representation of D4⋊D13
►On 104 points
Generators in S104
(1 42 15 33)(2 43 16 34)(3 44 17 35)(4 45 18 36)(5 46 19 37)(6 47 20 38)(7 48 21 39)(8 49 22 27)(9 50 23 28)(10 51 24 29)(11 52 25 30)(12 40 26 31)(13 41 14 32)(53 83 72 102)(54 84 73 103)(55 85 74 104)(56 86 75 92)(57 87 76 93)(58 88 77 94)(59 89 78 95)(60 90 66 96)(61 91 67 97)(62 79 68 98)(63 80 69 99)(64 81 70 100)(65 82 71 101)
(1 102)(2 103)(3 104)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 101)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 79)(25 80)(26 81)(27 60)(28 61)(29 62)(30 63)(31 64)(32 65)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 66)(50 67)(51 68)(52 69)
(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 15)(16 26)(17 25)(18 24)(19 23)(20 22)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 52)(36 51)(37 50)(38 49)(39 48)(53 82)(54 81)(55 80)(56 79)(57 91)(58 90)(59 89)(60 88)(61 87)(62 86)(63 85)(64 84)(65 83)(66 94)(67 93)(68 92)(69 104)(70 103)(71 102)(72 101)(73 100)(74 99)(75 98)(76 97)(77 96)(78 95)
G:=sub<Sym(104)| (1,42,15,33)(2,43,16,34)(3,44,17,35)(4,45,18,36)(5,46,19,37)(6,47,20,38)(7,48,21,39)(8,49,22,27)(9,50,23,28)(10,51,24,29)(11,52,25,30)(12,40,26,31)(13,41,14,32)(53,83,72,102)(54,84,73,103)(55,85,74,104)(56,86,75,92)(57,87,76,93)(58,88,77,94)(59,89,78,95)(60,90,66,96)(61,91,67,97)(62,79,68,98)(63,80,69,99)(64,81,70,100)(65,82,71,101), (1,102)(2,103)(3,104)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,79)(25,80)(26,81)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,66)(50,67)(51,68)(52,69), (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)(14,15)(16,26)(17,25)(18,24)(19,23)(20,22)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,52)(36,51)(37,50)(38,49)(39,48)(53,82)(54,81)(55,80)(56,79)(57,91)(58,90)(59,89)(60,88)(61,87)(62,86)(63,85)(64,84)(65,83)(66,94)(67,93)(68,92)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)>;
G:=Group( (1,42,15,33)(2,43,16,34)(3,44,17,35)(4,45,18,36)(5,46,19,37)(6,47,20,38)(7,48,21,39)(8,49,22,27)(9,50,23,28)(10,51,24,29)(11,52,25,30)(12,40,26,31)(13,41,14,32)(53,83,72,102)(54,84,73,103)(55,85,74,104)(56,86,75,92)(57,87,76,93)(58,88,77,94)(59,89,78,95)(60,90,66,96)(61,91,67,97)(62,79,68,98)(63,80,69,99)(64,81,70,100)(65,82,71,101), (1,102)(2,103)(3,104)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,79)(25,80)(26,81)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,66)(50,67)(51,68)(52,69), (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)(14,15)(16,26)(17,25)(18,24)(19,23)(20,22)(27,47)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,52)(36,51)(37,50)(38,49)(39,48)(53,82)(54,81)(55,80)(56,79)(57,91)(58,90)(59,89)(60,88)(61,87)(62,86)(63,85)(64,84)(65,83)(66,94)(67,93)(68,92)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95) );
G=PermutationGroup([[(1,42,15,33),(2,43,16,34),(3,44,17,35),(4,45,18,36),(5,46,19,37),(6,47,20,38),(7,48,21,39),(8,49,22,27),(9,50,23,28),(10,51,24,29),(11,52,25,30),(12,40,26,31),(13,41,14,32),(53,83,72,102),(54,84,73,103),(55,85,74,104),(56,86,75,92),(57,87,76,93),(58,88,77,94),(59,89,78,95),(60,90,66,96),(61,91,67,97),(62,79,68,98),(63,80,69,99),(64,81,70,100),(65,82,71,101)], [(1,102),(2,103),(3,104),(4,92),(5,93),(6,94),(7,95),(8,96),(9,97),(10,98),(11,99),(12,100),(13,101),(14,82),(15,83),(16,84),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,79),(25,80),(26,81),(27,60),(28,61),(29,62),(30,63),(31,64),(32,65),(33,53),(34,54),(35,55),(36,56),(37,57),(38,58),(39,59),(40,70),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,66),(50,67),(51,68),(52,69)], [(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),(14,15),(16,26),(17,25),(18,24),(19,23),(20,22),(27,47),(28,46),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,52),(36,51),(37,50),(38,49),(39,48),(53,82),(54,81),(55,80),(56,79),(57,91),(58,90),(59,89),(60,88),(61,87),(62,86),(63,85),(64,84),(65,83),(66,94),(67,93),(68,92),(69,104),(70,103),(71,102),(72,101),(73,100),(74,99),(75,98),(76,97),(77,96),(78,95)]])
D4⋊D13 is a maximal subgroup of
D8×D13 D8⋊D13 Q8⋊D26 D26.6D4 D52⋊6C22 D4⋊D26 C52.C23
D4⋊D13 is a maximal quotient of C26.D8 D52⋊6C4 C13⋊D16 D8.D13 C8.6D26 C13⋊Q32 D4⋊Dic13
37 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26R | 52A | ··· | 52F |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 4 | 52 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
37 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D8 | D13 | D26 | C13⋊D4 | D4⋊D13 |
| kernel | D4⋊D13 | C13⋊2C8 | D52 | D4×C13 | C26 | C13 | D4 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 12 | 6 |
Matrix representation of D4⋊D13 ►in GL4(𝔽313) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 264 |
| 0 | 0 | 230 | 312 |
| 312 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 |
| 0 | 0 | 0 | 190 |
| 0 | 0 | 285 | 0 |
| 289 | 1 | 0 | 0 |
| 216 | 4 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 4 | 312 | 0 | 0 |
| 15 | 309 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 230 | 312 |
G:=sub<GL(4,GF(313))| [1,0,0,0,0,1,0,0,0,0,1,230,0,0,264,312],[312,0,0,0,0,312,0,0,0,0,0,285,0,0,190,0],[289,216,0,0,1,4,0,0,0,0,1,0,0,0,0,1],[4,15,0,0,312,309,0,0,0,0,1,230,0,0,0,312] >;
D4⋊D13 in GAP, Magma, Sage, TeX
D_4\rtimes D_{13} % in TeX
G:=Group("D4:D13"); // GroupNames label
G:=SmallGroup(208,15);
// by ID
G=gap.SmallGroup(208,15);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,61,182,97,42,4804]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^13=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
Export