metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊2D13, C4.5D26, Dic26⋊3C2, C52.5C22, C26.6C23, C22.1D26, D26.2C22, Dic13.8C22, (C4×D13)⋊2C2, (D4×C13)⋊3C2, C13⋊2(C4○D4), C13⋊D4⋊2C2, (C2×C26).C22, (C2×Dic13)⋊3C2, C2.7(C22×D13), SmallGroup(208,40)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊2D13
G = < a,b,c,d | a4=b2=c13=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Smallest permutation representation of D4⋊2D13
►On 104 points
Generators in S104
(1 74 14 60)(2 75 15 61)(3 76 16 62)(4 77 17 63)(5 78 18 64)(6 66 19 65)(7 67 20 53)(8 68 21 54)(9 69 22 55)(10 70 23 56)(11 71 24 57)(12 72 25 58)(13 73 26 59)(27 99 49 80)(28 100 50 81)(29 101 51 82)(30 102 52 83)(31 103 40 84)(32 104 41 85)(33 92 42 86)(34 93 43 87)(35 94 44 88)(36 95 45 89)(37 96 46 90)(38 97 47 91)(39 98 48 79)
(1 80)(2 81)(3 82)(4 83)(5 84)(6 85)(7 86)(8 87)(9 88)(10 89)(11 90)(12 91)(13 79)(14 99)(15 100)(16 101)(17 102)(18 103)(19 104)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(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 78)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 77)
(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 26)(15 25)(16 24)(17 23)(18 22)(19 21)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 52)(37 51)(38 50)(39 49)(54 65)(55 64)(56 63)(57 62)(58 61)(59 60)(66 68)(69 78)(70 77)(71 76)(72 75)(73 74)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 104)(88 103)(89 102)(90 101)(91 100)
G:=sub<Sym(104)| (1,74,14,60)(2,75,15,61)(3,76,16,62)(4,77,17,63)(5,78,18,64)(6,66,19,65)(7,67,20,53)(8,68,21,54)(9,69,22,55)(10,70,23,56)(11,71,24,57)(12,72,25,58)(13,73,26,59)(27,99,49,80)(28,100,50,81)(29,101,51,82)(30,102,52,83)(31,103,40,84)(32,104,41,85)(33,92,42,86)(34,93,43,87)(35,94,44,88)(36,95,45,89)(37,96,46,90)(38,97,47,91)(39,98,48,79), (1,80)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,79)(14,99)(15,100)(16,101)(17,102)(18,103)(19,104)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(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)(40,78)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77), (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)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,52)(37,51)(38,50)(39,49)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)(66,68)(69,78)(70,77)(71,76)(72,75)(73,74)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,104)(88,103)(89,102)(90,101)(91,100)>;
G:=Group( (1,74,14,60)(2,75,15,61)(3,76,16,62)(4,77,17,63)(5,78,18,64)(6,66,19,65)(7,67,20,53)(8,68,21,54)(9,69,22,55)(10,70,23,56)(11,71,24,57)(12,72,25,58)(13,73,26,59)(27,99,49,80)(28,100,50,81)(29,101,51,82)(30,102,52,83)(31,103,40,84)(32,104,41,85)(33,92,42,86)(34,93,43,87)(35,94,44,88)(36,95,45,89)(37,96,46,90)(38,97,47,91)(39,98,48,79), (1,80)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,79)(14,99)(15,100)(16,101)(17,102)(18,103)(19,104)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(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)(40,78)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77), (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)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,52)(37,51)(38,50)(39,49)(54,65)(55,64)(56,63)(57,62)(58,61)(59,60)(66,68)(69,78)(70,77)(71,76)(72,75)(73,74)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,104)(88,103)(89,102)(90,101)(91,100) );
G=PermutationGroup([[(1,74,14,60),(2,75,15,61),(3,76,16,62),(4,77,17,63),(5,78,18,64),(6,66,19,65),(7,67,20,53),(8,68,21,54),(9,69,22,55),(10,70,23,56),(11,71,24,57),(12,72,25,58),(13,73,26,59),(27,99,49,80),(28,100,50,81),(29,101,51,82),(30,102,52,83),(31,103,40,84),(32,104,41,85),(33,92,42,86),(34,93,43,87),(35,94,44,88),(36,95,45,89),(37,96,46,90),(38,97,47,91),(39,98,48,79)], [(1,80),(2,81),(3,82),(4,83),(5,84),(6,85),(7,86),(8,87),(9,88),(10,89),(11,90),(12,91),(13,79),(14,99),(15,100),(16,101),(17,102),(18,103),(19,104),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(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),(40,78),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,73),(49,74),(50,75),(51,76),(52,77)], [(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),(14,26),(15,25),(16,24),(17,23),(18,22),(19,21),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43),(33,42),(34,41),(35,40),(36,52),(37,51),(38,50),(39,49),(54,65),(55,64),(56,63),(57,62),(58,61),(59,60),(66,68),(69,78),(70,77),(71,76),(72,75),(73,74),(79,99),(80,98),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,104),(88,103),(89,102),(90,101),(91,100)]])
D4⋊2D13 is a maximal subgroup of
Dic26⋊C4 D8⋊D13 D8⋊3D13 D4.D26 D26.6D4 Dic26.C4 D4⋊6D26 C4○D4×D13 D4.10D26
D4⋊2D13 is a maximal quotient of
C23.11D26 C22⋊Dic26 C23.D26 Dic13⋊4D4 D26.12D4 C23.6D26 C22.D52 Dic13⋊3Q8 Dic13.Q8 C4.Dic26 C4⋊C4⋊7D13 D26⋊2Q8 C4⋊C4⋊D13 D4×Dic13 C23.18D26 C52.17D4 C52⋊2D4 Dic13⋊D4
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26R | 52A | ··· | 52F |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 2 | 26 | 2 | 13 | 13 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4○D4 | D13 | D26 | D26 | D4⋊2D13 |
| kernel | D4⋊2D13 | Dic26 | C4×D13 | C2×Dic13 | C13⋊D4 | D4×C13 | C13 | D4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 6 | 6 | 12 | 6 |
Matrix representation of D4⋊2D13 ►in GL4(𝔽53) generated by
| 30 | 0 | 0 | 0 |
| 0 | 23 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
| 0 | 23 | 0 | 0 |
| 30 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 44 | 1 |
| 0 | 0 | 31 | 20 |
| 1 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 46 | 11 |
| 0 | 0 | 39 | 7 |
G:=sub<GL(4,GF(53))| [30,0,0,0,0,23,0,0,0,0,52,0,0,0,0,52],[0,30,0,0,23,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,44,31,0,0,1,20],[1,0,0,0,0,52,0,0,0,0,46,39,0,0,11,7] >;
D4⋊2D13 in GAP, Magma, Sage, TeX
D_4\rtimes_2D_{13} % in TeX
G:=Group("D4:2D13"); // GroupNames label
G:=SmallGroup(208,40);
// by ID
G=gap.SmallGroup(208,40);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,46,182,97,4804]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^13=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
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