direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×D13, C4.6D26, Dic26⋊4C2, C52.6C22, C26.7C23, D26.9C22, Dic13.3C22, C13⋊2(C2×Q8), (Q8×C13)⋊2C2, (C4×D13).1C2, C2.8(C22×D13), SmallGroup(208,41)
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=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of Q8×D13
►On 104 points
Generators in S104
(1 45 23 32)(2 46 24 33)(3 47 25 34)(4 48 26 35)(5 49 14 36)(6 50 15 37)(7 51 16 38)(8 52 17 39)(9 40 18 27)(10 41 19 28)(11 42 20 29)(12 43 21 30)(13 44 22 31)(53 86 67 98)(54 87 68 99)(55 88 69 100)(56 89 70 101)(57 90 71 102)(58 91 72 103)(59 79 73 104)(60 80 74 92)(61 81 75 93)(62 82 76 94)(63 83 77 95)(64 84 78 96)(65 85 66 97)
(1 76 23 62)(2 77 24 63)(3 78 25 64)(4 66 26 65)(5 67 14 53)(6 68 15 54)(7 69 16 55)(8 70 17 56)(9 71 18 57)(10 72 19 58)(11 73 20 59)(12 74 21 60)(13 75 22 61)(27 102 40 90)(28 103 41 91)(29 104 42 79)(30 92 43 80)(31 93 44 81)(32 94 45 82)(33 95 46 83)(34 96 47 84)(35 97 48 85)(36 98 49 86)(37 99 50 87)(38 100 51 88)(39 101 52 89)
(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 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 52)(38 51)(39 50)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(79 96)(80 95)(81 94)(82 93)(83 92)(84 104)(85 103)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)
G:=sub<Sym(104)| (1,45,23,32)(2,46,24,33)(3,47,25,34)(4,48,26,35)(5,49,14,36)(6,50,15,37)(7,51,16,38)(8,52,17,39)(9,40,18,27)(10,41,19,28)(11,42,20,29)(12,43,21,30)(13,44,22,31)(53,86,67,98)(54,87,68,99)(55,88,69,100)(56,89,70,101)(57,90,71,102)(58,91,72,103)(59,79,73,104)(60,80,74,92)(61,81,75,93)(62,82,76,94)(63,83,77,95)(64,84,78,96)(65,85,66,97), (1,76,23,62)(2,77,24,63)(3,78,25,64)(4,66,26,65)(5,67,14,53)(6,68,15,54)(7,69,16,55)(8,70,17,56)(9,71,18,57)(10,72,19,58)(11,73,20,59)(12,74,21,60)(13,75,22,61)(27,102,40,90)(28,103,41,91)(29,104,42,79)(30,92,43,80)(31,93,44,81)(32,94,45,82)(33,95,46,83)(34,96,47,84)(35,97,48,85)(36,98,49,86)(37,99,50,87)(38,100,51,88)(39,101,52,89), (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)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,52)(38,51)(39,50)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(79,96)(80,95)(81,94)(82,93)(83,92)(84,104)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97)>;
G:=Group( (1,45,23,32)(2,46,24,33)(3,47,25,34)(4,48,26,35)(5,49,14,36)(6,50,15,37)(7,51,16,38)(8,52,17,39)(9,40,18,27)(10,41,19,28)(11,42,20,29)(12,43,21,30)(13,44,22,31)(53,86,67,98)(54,87,68,99)(55,88,69,100)(56,89,70,101)(57,90,71,102)(58,91,72,103)(59,79,73,104)(60,80,74,92)(61,81,75,93)(62,82,76,94)(63,83,77,95)(64,84,78,96)(65,85,66,97), (1,76,23,62)(2,77,24,63)(3,78,25,64)(4,66,26,65)(5,67,14,53)(6,68,15,54)(7,69,16,55)(8,70,17,56)(9,71,18,57)(10,72,19,58)(11,73,20,59)(12,74,21,60)(13,75,22,61)(27,102,40,90)(28,103,41,91)(29,104,42,79)(30,92,43,80)(31,93,44,81)(32,94,45,82)(33,95,46,83)(34,96,47,84)(35,97,48,85)(36,98,49,86)(37,99,50,87)(38,100,51,88)(39,101,52,89), (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)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,52)(38,51)(39,50)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(79,96)(80,95)(81,94)(82,93)(83,92)(84,104)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98)(91,97) );
G=PermutationGroup([[(1,45,23,32),(2,46,24,33),(3,47,25,34),(4,48,26,35),(5,49,14,36),(6,50,15,37),(7,51,16,38),(8,52,17,39),(9,40,18,27),(10,41,19,28),(11,42,20,29),(12,43,21,30),(13,44,22,31),(53,86,67,98),(54,87,68,99),(55,88,69,100),(56,89,70,101),(57,90,71,102),(58,91,72,103),(59,79,73,104),(60,80,74,92),(61,81,75,93),(62,82,76,94),(63,83,77,95),(64,84,78,96),(65,85,66,97)], [(1,76,23,62),(2,77,24,63),(3,78,25,64),(4,66,26,65),(5,67,14,53),(6,68,15,54),(7,69,16,55),(8,70,17,56),(9,71,18,57),(10,72,19,58),(11,73,20,59),(12,74,21,60),(13,75,22,61),(27,102,40,90),(28,103,41,91),(29,104,42,79),(30,92,43,80),(31,93,44,81),(32,94,45,82),(33,95,46,83),(34,96,47,84),(35,97,48,85),(36,98,49,86),(37,99,50,87),(38,100,51,88),(39,101,52,89)], [(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),(27,49),(28,48),(29,47),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(37,52),(38,51),(39,50),(53,71),(54,70),(55,69),(56,68),(57,67),(58,66),(59,78),(60,77),(61,76),(62,75),(63,74),(64,73),(65,72),(79,96),(80,95),(81,94),(82,93),(83,92),(84,104),(85,103),(86,102),(87,101),(88,100),(89,99),(90,98),(91,97)]])
Q8×D13 is a maximal subgroup of
D13.Q16 D4.D26 Q16⋊D13 Q8.10D26 D4.10D26
Q8×D13 is a maximal quotient of Dic13⋊3Q8 C52⋊Q8 Dic13.Q8 D26⋊Q8 D26⋊2Q8 Dic13⋊Q8 D26⋊3Q8
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 13A | ··· | 13F | 26A | ··· | 26F | 52A | ··· | 52R |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 13 | 13 | 2 | 2 | 2 | 26 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D13 | D26 | Q8×D13 |
| kernel | Q8×D13 | Dic26 | C4×D13 | Q8×C13 | D13 | Q8 | C4 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 18 | 6 |
Matrix representation of Q8×D13 ►in GL4(𝔽53) generated by
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 52 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 23 |
| 0 | 0 | 23 | 0 |
| 52 | 1 | 0 | 0 |
| 43 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 52 | 0 | 0 | 0 |
| 43 | 1 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,0,52,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,23,0,0,23,0],[52,43,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[52,43,0,0,0,1,0,0,0,0,52,0,0,0,0,52] >;
Q8×D13 in GAP, Magma, Sage, TeX
Q_8\times D_{13} % in TeX
G:=Group("Q8xD13"); // GroupNames label
G:=SmallGroup(208,41);
// by ID
G=gap.SmallGroup(208,41);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,46,97,42,4804]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^13=d^2=1,b^2=a^2,b*a*b^-1=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
Export