metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊2D26, D8⋊2D13, D4⋊2D26, C104⋊4C22, D26.14D4, C52.2C23, D52.1C22, Dic13.16D4, Dic26⋊1C22, D4⋊D13⋊2C2, (C13×D8)⋊4C2, (D4×D13)⋊2C2, C104⋊C2⋊3C2, C8⋊D13⋊3C2, C13⋊2(C8⋊C22), D4.D13⋊1C2, C2.16(D4×D13), C26.28(C2×D4), D4⋊2D13⋊1C2, C13⋊2C8⋊1C22, (D4×C13)⋊2C22, C4.2(C22×D13), (C4×D13).1C22, SmallGroup(416,132)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊D13
G = < a,b,c,d | a4=b2=c26=d2=1, bab=cac-1=dad=a-1, cbc-1=ab, dbd=a-1b, dcd=c-1 >
Subgroups: 584 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C8, C8, C2×C4, D4, D4, Q8, C23, C13, M4(2), D8, D8, SD16, C2×D4, C4○D4, D13, C26, C26, C8⋊C22, Dic13, Dic13, C52, D26, D26, C2×C26, C13⋊2C8, C104, Dic26, C4×D13, D52, C2×Dic13, C13⋊D4, D4×C13, C22×D13, C8⋊D13, C104⋊C2, D4⋊D13, D4.D13, C13×D8, D4×D13, D4⋊2D13, D8⋊D13
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, C8⋊C22, D26, C22×D13, D4×D13, D8⋊D13
►On 104 points
Generators in S104
(1 31 19 44)(2 45 20 32)(3 33 21 46)(4 47 22 34)(5 35 23 48)(6 49 24 36)(7 37 25 50)(8 51 26 38)(9 39 14 52)(10 27 15 40)(11 41 16 28)(12 29 17 42)(13 43 18 30)(53 66 89 102)(54 103 90 67)(55 68 91 104)(56 79 92 69)(57 70 93 80)(58 81 94 71)(59 72 95 82)(60 83 96 73)(61 74 97 84)(62 85 98 75)(63 76 99 86)(64 87 100 77)(65 78 101 88)
(1 80)(2 94)(3 82)(4 96)(5 84)(6 98)(7 86)(8 100)(9 88)(10 102)(11 90)(12 104)(13 92)(14 78)(15 66)(16 54)(17 68)(18 56)(19 70)(20 58)(21 72)(22 60)(23 74)(24 62)(25 76)(26 64)(27 89)(28 67)(29 91)(30 69)(31 93)(32 71)(33 95)(34 73)(35 97)(36 75)(37 99)(38 77)(39 101)(40 53)(41 103)(42 55)(43 79)(44 57)(45 81)(46 59)(47 83)(48 61)(49 85)(50 63)(51 87)(52 65)
(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 22)(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 26)(11 25)(12 24)(13 23)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(79 84)(80 83)(81 82)(85 104)(86 103)(87 102)(88 101)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)
G:=sub<Sym(104)| (1,31,19,44)(2,45,20,32)(3,33,21,46)(4,47,22,34)(5,35,23,48)(6,49,24,36)(7,37,25,50)(8,51,26,38)(9,39,14,52)(10,27,15,40)(11,41,16,28)(12,29,17,42)(13,43,18,30)(53,66,89,102)(54,103,90,67)(55,68,91,104)(56,79,92,69)(57,70,93,80)(58,81,94,71)(59,72,95,82)(60,83,96,73)(61,74,97,84)(62,85,98,75)(63,76,99,86)(64,87,100,77)(65,78,101,88), (1,80)(2,94)(3,82)(4,96)(5,84)(6,98)(7,86)(8,100)(9,88)(10,102)(11,90)(12,104)(13,92)(14,78)(15,66)(16,54)(17,68)(18,56)(19,70)(20,58)(21,72)(22,60)(23,74)(24,62)(25,76)(26,64)(27,89)(28,67)(29,91)(30,69)(31,93)(32,71)(33,95)(34,73)(35,97)(36,75)(37,99)(38,77)(39,101)(40,53)(41,103)(42,55)(43,79)(44,57)(45,81)(46,59)(47,83)(48,61)(49,85)(50,63)(51,87)(52,65), (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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,26)(11,25)(12,24)(13,23)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,84)(80,83)(81,82)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)>;
G:=Group( (1,31,19,44)(2,45,20,32)(3,33,21,46)(4,47,22,34)(5,35,23,48)(6,49,24,36)(7,37,25,50)(8,51,26,38)(9,39,14,52)(10,27,15,40)(11,41,16,28)(12,29,17,42)(13,43,18,30)(53,66,89,102)(54,103,90,67)(55,68,91,104)(56,79,92,69)(57,70,93,80)(58,81,94,71)(59,72,95,82)(60,83,96,73)(61,74,97,84)(62,85,98,75)(63,76,99,86)(64,87,100,77)(65,78,101,88), (1,80)(2,94)(3,82)(4,96)(5,84)(6,98)(7,86)(8,100)(9,88)(10,102)(11,90)(12,104)(13,92)(14,78)(15,66)(16,54)(17,68)(18,56)(19,70)(20,58)(21,72)(22,60)(23,74)(24,62)(25,76)(26,64)(27,89)(28,67)(29,91)(30,69)(31,93)(32,71)(33,95)(34,73)(35,97)(36,75)(37,99)(38,77)(39,101)(40,53)(41,103)(42,55)(43,79)(44,57)(45,81)(46,59)(47,83)(48,61)(49,85)(50,63)(51,87)(52,65), (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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,26)(11,25)(12,24)(13,23)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)(79,84)(80,83)(81,82)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95) );
G=PermutationGroup([[(1,31,19,44),(2,45,20,32),(3,33,21,46),(4,47,22,34),(5,35,23,48),(6,49,24,36),(7,37,25,50),(8,51,26,38),(9,39,14,52),(10,27,15,40),(11,41,16,28),(12,29,17,42),(13,43,18,30),(53,66,89,102),(54,103,90,67),(55,68,91,104),(56,79,92,69),(57,70,93,80),(58,81,94,71),(59,72,95,82),(60,83,96,73),(61,74,97,84),(62,85,98,75),(63,76,99,86),(64,87,100,77),(65,78,101,88)], [(1,80),(2,94),(3,82),(4,96),(5,84),(6,98),(7,86),(8,100),(9,88),(10,102),(11,90),(12,104),(13,92),(14,78),(15,66),(16,54),(17,68),(18,56),(19,70),(20,58),(21,72),(22,60),(23,74),(24,62),(25,76),(26,64),(27,89),(28,67),(29,91),(30,69),(31,93),(32,71),(33,95),(34,73),(35,97),(36,75),(37,99),(38,77),(39,101),(40,53),(41,103),(42,55),(43,79),(44,57),(45,81),(46,59),(47,83),(48,61),(49,85),(50,63),(51,87),(52,65)], [(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)], [(1,22),(2,21),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,26),(11,25),(12,24),(13,23),(27,51),(28,50),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40),(53,64),(54,63),(55,62),(56,61),(57,60),(58,59),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,72),(79,84),(80,83),(81,82),(85,104),(86,103),(87,102),(88,101),(89,100),(90,99),(91,98),(92,97),(93,96),(94,95)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 8A | 8B | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26R | 52A | ··· | 52F | 104A | ··· | 104L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 | 104 | ··· | 104 |
| size | 1 | 1 | 4 | 4 | 26 | 52 | 2 | 26 | 52 | 4 | 52 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D13 | D26 | D26 | C8⋊C22 | D4×D13 | D8⋊D13 |
| kernel | D8⋊D13 | C8⋊D13 | C104⋊C2 | D4⋊D13 | D4.D13 | C13×D8 | D4×D13 | D4⋊2D13 | Dic13 | D26 | D8 | C8 | D4 | C13 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 1 | 6 | 12 |
Matrix representation of D8⋊D13 ►in GL6(𝔽313)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 289 | 0 | 0 |
| 0 | 0 | 287 | 312 | 0 | 0 |
| 0 | 0 | 29 | 278 | 312 | 259 |
| 0 | 0 | 98 | 0 | 58 | 1 |
| 312 | 0 | 0 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 | 0 | 0 |
| 0 | 0 | 237 | 0 | 70 | 24 |
| 0 | 0 | 98 | 0 | 58 | 1 |
| 0 | 0 | 226 | 54 | 76 | 35 |
| 0 | 0 | 287 | 312 | 0 | 0 |
| 14 | 13 | 0 | 0 | 0 | 0 |
| 272 | 275 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 289 | 0 | 0 |
| 0 | 0 | 0 | 312 | 0 | 0 |
| 0 | 0 | 29 | 278 | 312 | 0 |
| 0 | 0 | 98 | 0 | 58 | 1 |
| 84 | 37 | 0 | 0 | 0 | 0 |
| 190 | 229 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 289 | 0 | 0 |
| 0 | 0 | 0 | 312 | 0 | 0 |
| 0 | 0 | 0 | 278 | 1 | 0 |
| 0 | 0 | 215 | 0 | 255 | 312 |
G:=sub<GL(6,GF(313))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,287,29,98,0,0,289,312,278,0,0,0,0,0,312,58,0,0,0,0,259,1],[312,0,0,0,0,0,0,312,0,0,0,0,0,0,237,98,226,287,0,0,0,0,54,312,0,0,70,58,76,0,0,0,24,1,35,0],[14,272,0,0,0,0,13,275,0,0,0,0,0,0,1,0,29,98,0,0,289,312,278,0,0,0,0,0,312,58,0,0,0,0,0,1],[84,190,0,0,0,0,37,229,0,0,0,0,0,0,1,0,0,215,0,0,289,312,278,0,0,0,0,0,1,255,0,0,0,0,0,312] >;
D8⋊D13 in GAP, Magma, Sage, TeX
D_8\rtimes D_{13} % in TeX
G:=Group("D8:D13"); // GroupNames label
G:=SmallGroup(416,132);
// by ID
G=gap.SmallGroup(416,132);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,362,116,297,159,69,13829]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^26=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations