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 48 15 39)(2 49 16 27)(3 50 17 28)(4 51 18 29)(5 52 19 30)(6 40 20 31)(7 41 21 32)(8 42 22 33)(9 43 23 34)(10 44 24 35)(11 45 25 36)(12 46 26 37)(13 47 14 38)(53 88 69 94)(54 89 70 95)(55 90 71 96)(56 91 72 97)(57 79 73 98)(58 80 74 99)(59 81 75 100)(60 82 76 101)(61 83 77 102)(62 84 78 103)(63 85 66 104)(64 86 67 92)(65 87 68 93)
(1 72 15 56)(2 73 16 57)(3 74 17 58)(4 75 18 59)(5 76 19 60)(6 77 20 61)(7 78 21 62)(8 66 22 63)(9 67 23 64)(10 68 24 65)(11 69 25 53)(12 70 26 54)(13 71 14 55)(27 98 49 79)(28 99 50 80)(29 100 51 81)(30 101 52 82)(31 102 40 83)(32 103 41 84)(33 104 42 85)(34 92 43 86)(35 93 44 87)(36 94 45 88)(37 95 46 89)(38 96 47 90)(39 97 48 91)
(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 14)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 15)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 78)(63 77)(64 76)(65 75)(79 95)(80 94)(81 93)(82 92)(83 104)(84 103)(85 102)(86 101)(87 100)(88 99)(89 98)(90 97)(91 96)
G:=sub<Sym(104)| (1,48,15,39)(2,49,16,27)(3,50,17,28)(4,51,18,29)(5,52,19,30)(6,40,20,31)(7,41,21,32)(8,42,22,33)(9,43,23,34)(10,44,24,35)(11,45,25,36)(12,46,26,37)(13,47,14,38)(53,88,69,94)(54,89,70,95)(55,90,71,96)(56,91,72,97)(57,79,73,98)(58,80,74,99)(59,81,75,100)(60,82,76,101)(61,83,77,102)(62,84,78,103)(63,85,66,104)(64,86,67,92)(65,87,68,93), (1,72,15,56)(2,73,16,57)(3,74,17,58)(4,75,18,59)(5,76,19,60)(6,77,20,61)(7,78,21,62)(8,66,22,63)(9,67,23,64)(10,68,24,65)(11,69,25,53)(12,70,26,54)(13,71,14,55)(27,98,49,79)(28,99,50,80)(29,100,51,81)(30,101,52,82)(31,102,40,83)(32,103,41,84)(33,104,42,85)(34,92,43,86)(35,93,44,87)(36,94,45,88)(37,95,46,89)(38,96,47,90)(39,97,48,91), (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), (1,14)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,78)(63,77)(64,76)(65,75)(79,95)(80,94)(81,93)(82,92)(83,104)(84,103)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96)>;
G:=Group( (1,48,15,39)(2,49,16,27)(3,50,17,28)(4,51,18,29)(5,52,19,30)(6,40,20,31)(7,41,21,32)(8,42,22,33)(9,43,23,34)(10,44,24,35)(11,45,25,36)(12,46,26,37)(13,47,14,38)(53,88,69,94)(54,89,70,95)(55,90,71,96)(56,91,72,97)(57,79,73,98)(58,80,74,99)(59,81,75,100)(60,82,76,101)(61,83,77,102)(62,84,78,103)(63,85,66,104)(64,86,67,92)(65,87,68,93), (1,72,15,56)(2,73,16,57)(3,74,17,58)(4,75,18,59)(5,76,19,60)(6,77,20,61)(7,78,21,62)(8,66,22,63)(9,67,23,64)(10,68,24,65)(11,69,25,53)(12,70,26,54)(13,71,14,55)(27,98,49,79)(28,99,50,80)(29,100,51,81)(30,101,52,82)(31,102,40,83)(32,103,41,84)(33,104,42,85)(34,92,43,86)(35,93,44,87)(36,94,45,88)(37,95,46,89)(38,96,47,90)(39,97,48,91), (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), (1,14)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,78)(63,77)(64,76)(65,75)(79,95)(80,94)(81,93)(82,92)(83,104)(84,103)(85,102)(86,101)(87,100)(88,99)(89,98)(90,97)(91,96) );
G=PermutationGroup([(1,48,15,39),(2,49,16,27),(3,50,17,28),(4,51,18,29),(5,52,19,30),(6,40,20,31),(7,41,21,32),(8,42,22,33),(9,43,23,34),(10,44,24,35),(11,45,25,36),(12,46,26,37),(13,47,14,38),(53,88,69,94),(54,89,70,95),(55,90,71,96),(56,91,72,97),(57,79,73,98),(58,80,74,99),(59,81,75,100),(60,82,76,101),(61,83,77,102),(62,84,78,103),(63,85,66,104),(64,86,67,92),(65,87,68,93)], [(1,72,15,56),(2,73,16,57),(3,74,17,58),(4,75,18,59),(5,76,19,60),(6,77,20,61),(7,78,21,62),(8,66,22,63),(9,67,23,64),(10,68,24,65),(11,69,25,53),(12,70,26,54),(13,71,14,55),(27,98,49,79),(28,99,50,80),(29,100,51,81),(30,101,52,82),(31,102,40,83),(32,103,41,84),(33,104,42,85),(34,92,43,86),(35,93,44,87),(36,94,45,88),(37,95,46,89),(38,96,47,90),(39,97,48,91)], [(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)], [(1,14),(2,26),(3,25),(4,24),(5,23),(6,22),(7,21),(8,20),(9,19),(10,18),(11,17),(12,16),(13,15),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,40),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(53,74),(54,73),(55,72),(56,71),(57,70),(58,69),(59,68),(60,67),(61,66),(62,78),(63,77),(64,76),(65,75),(79,95),(80,94),(81,93),(82,92),(83,104),(84,103),(85,102),(86,101),(87,100),(88,99),(89,98),(90,97),(91,96)])
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