metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊D13, C4.3D26, C26.9D4, C13⋊3SD16, D52.2C2, C52.3C22, C13⋊2C8⋊3C2, (Q8×C13)⋊1C2, C2.6(C13⋊D4), SmallGroup(208,17)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊D13
G = < a,b,c,d | a4=c13=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >
Smallest permutation representation of Q8⋊D13
►On 104 points
Generators in S104
(1 47 20 39)(2 48 21 27)(3 49 22 28)(4 50 23 29)(5 51 24 30)(6 52 25 31)(7 40 26 32)(8 41 14 33)(9 42 15 34)(10 43 16 35)(11 44 17 36)(12 45 18 37)(13 46 19 38)(53 83 72 101)(54 84 73 102)(55 85 74 103)(56 86 75 104)(57 87 76 92)(58 88 77 93)(59 89 78 94)(60 90 66 95)(61 91 67 96)(62 79 68 97)(63 80 69 98)(64 81 70 99)(65 82 71 100)
(1 73 20 54)(2 74 21 55)(3 75 22 56)(4 76 23 57)(5 77 24 58)(6 78 25 59)(7 66 26 60)(8 67 14 61)(9 68 15 62)(10 69 16 63)(11 70 17 64)(12 71 18 65)(13 72 19 53)(27 103 48 85)(28 104 49 86)(29 92 50 87)(30 93 51 88)(31 94 52 89)(32 95 40 90)(33 96 41 91)(34 97 42 79)(35 98 43 80)(36 99 44 81)(37 100 45 82)(38 101 46 83)(39 102 47 84)
(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 25)(15 24)(16 23)(17 22)(18 21)(19 20)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 52)(34 51)(35 50)(36 49)(37 48)(38 47)(39 46)(53 84)(54 83)(55 82)(56 81)(57 80)(58 79)(59 91)(60 90)(61 89)(62 88)(63 87)(64 86)(65 85)(66 95)(67 94)(68 93)(69 92)(70 104)(71 103)(72 102)(73 101)(74 100)(75 99)(76 98)(77 97)(78 96)
G:=sub<Sym(104)| (1,47,20,39)(2,48,21,27)(3,49,22,28)(4,50,23,29)(5,51,24,30)(6,52,25,31)(7,40,26,32)(8,41,14,33)(9,42,15,34)(10,43,16,35)(11,44,17,36)(12,45,18,37)(13,46,19,38)(53,83,72,101)(54,84,73,102)(55,85,74,103)(56,86,75,104)(57,87,76,92)(58,88,77,93)(59,89,78,94)(60,90,66,95)(61,91,67,96)(62,79,68,97)(63,80,69,98)(64,81,70,99)(65,82,71,100), (1,73,20,54)(2,74,21,55)(3,75,22,56)(4,76,23,57)(5,77,24,58)(6,78,25,59)(7,66,26,60)(8,67,14,61)(9,68,15,62)(10,69,16,63)(11,70,17,64)(12,71,18,65)(13,72,19,53)(27,103,48,85)(28,104,49,86)(29,92,50,87)(30,93,51,88)(31,94,52,89)(32,95,40,90)(33,96,41,91)(34,97,42,79)(35,98,43,80)(36,99,44,81)(37,100,45,82)(38,101,46,83)(39,102,47,84), (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)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,91)(60,90)(61,89)(62,88)(63,87)(64,86)(65,85)(66,95)(67,94)(68,93)(69,92)(70,104)(71,103)(72,102)(73,101)(74,100)(75,99)(76,98)(77,97)(78,96)>;
G:=Group( (1,47,20,39)(2,48,21,27)(3,49,22,28)(4,50,23,29)(5,51,24,30)(6,52,25,31)(7,40,26,32)(8,41,14,33)(9,42,15,34)(10,43,16,35)(11,44,17,36)(12,45,18,37)(13,46,19,38)(53,83,72,101)(54,84,73,102)(55,85,74,103)(56,86,75,104)(57,87,76,92)(58,88,77,93)(59,89,78,94)(60,90,66,95)(61,91,67,96)(62,79,68,97)(63,80,69,98)(64,81,70,99)(65,82,71,100), (1,73,20,54)(2,74,21,55)(3,75,22,56)(4,76,23,57)(5,77,24,58)(6,78,25,59)(7,66,26,60)(8,67,14,61)(9,68,15,62)(10,69,16,63)(11,70,17,64)(12,71,18,65)(13,72,19,53)(27,103,48,85)(28,104,49,86)(29,92,50,87)(30,93,51,88)(31,94,52,89)(32,95,40,90)(33,96,41,91)(34,97,42,79)(35,98,43,80)(36,99,44,81)(37,100,45,82)(38,101,46,83)(39,102,47,84), (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)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(27,45)(28,44)(29,43)(30,42)(31,41)(32,40)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,91)(60,90)(61,89)(62,88)(63,87)(64,86)(65,85)(66,95)(67,94)(68,93)(69,92)(70,104)(71,103)(72,102)(73,101)(74,100)(75,99)(76,98)(77,97)(78,96) );
G=PermutationGroup([[(1,47,20,39),(2,48,21,27),(3,49,22,28),(4,50,23,29),(5,51,24,30),(6,52,25,31),(7,40,26,32),(8,41,14,33),(9,42,15,34),(10,43,16,35),(11,44,17,36),(12,45,18,37),(13,46,19,38),(53,83,72,101),(54,84,73,102),(55,85,74,103),(56,86,75,104),(57,87,76,92),(58,88,77,93),(59,89,78,94),(60,90,66,95),(61,91,67,96),(62,79,68,97),(63,80,69,98),(64,81,70,99),(65,82,71,100)], [(1,73,20,54),(2,74,21,55),(3,75,22,56),(4,76,23,57),(5,77,24,58),(6,78,25,59),(7,66,26,60),(8,67,14,61),(9,68,15,62),(10,69,16,63),(11,70,17,64),(12,71,18,65),(13,72,19,53),(27,103,48,85),(28,104,49,86),(29,92,50,87),(30,93,51,88),(31,94,52,89),(32,95,40,90),(33,96,41,91),(34,97,42,79),(35,98,43,80),(36,99,44,81),(37,100,45,82),(38,101,46,83),(39,102,47,84)], [(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),(14,25),(15,24),(16,23),(17,22),(18,21),(19,20),(27,45),(28,44),(29,43),(30,42),(31,41),(32,40),(33,52),(34,51),(35,50),(36,49),(37,48),(38,47),(39,46),(53,84),(54,83),(55,82),(56,81),(57,80),(58,79),(59,91),(60,90),(61,89),(62,88),(63,87),(64,86),(65,85),(66,95),(67,94),(68,93),(69,92),(70,104),(71,103),(72,102),(73,101),(74,100),(75,99),(76,98),(77,97),(78,96)]])
Q8⋊D13 is a maximal subgroup of
SD16×D13 Q8⋊D26 Q16⋊D13 D104⋊C2 Q8.D26 D4⋊D26 C52.C23
Q8⋊D13 is a maximal quotient of C52.Q8 D52⋊6C4 Q8⋊Dic13
37 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 13A | ··· | 13F | 26A | ··· | 26F | 52A | ··· | 52R |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 52 | 2 | 4 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
37 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | SD16 | D13 | D26 | C13⋊D4 | Q8⋊D13 |
| kernel | Q8⋊D13 | C13⋊2C8 | D52 | Q8×C13 | C26 | C13 | Q8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 6 | 6 | 12 | 6 |
Matrix representation of Q8⋊D13 ►in GL4(𝔽313) generated by
| 312 | 0 | 0 | 0 |
| 0 | 312 | 0 | 0 |
| 0 | 0 | 1 | 311 |
| 0 | 0 | 1 | 312 |
| 62 | 86 | 0 | 0 |
| 221 | 251 | 0 | 0 |
| 0 | 0 | 130 | 183 |
| 0 | 0 | 65 | 183 |
| 20 | 1 | 0 | 0 |
| 79 | 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 | 1 | 312 |
G:=sub<GL(4,GF(313))| [312,0,0,0,0,312,0,0,0,0,1,1,0,0,311,312],[62,221,0,0,86,251,0,0,0,0,130,65,0,0,183,183],[20,79,0,0,1,4,0,0,0,0,1,0,0,0,0,1],[4,15,0,0,312,309,0,0,0,0,1,1,0,0,0,312] >;
Q8⋊D13 in GAP, Magma, Sage, TeX
Q_8\rtimes D_{13} % in TeX
G:=Group("Q8:D13"); // GroupNames label
G:=SmallGroup(208,17);
// by ID
G=gap.SmallGroup(208,17);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,61,46,182,97,42,4804]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^13=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations
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