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G = Q8⋊D13order 208 = 24·13

The semidirect product of Q8 and D13 acting via D13/C13=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8⋊D13, C4.3D26, C26.9D4, C133SD16, D52.2C2, C52.3C22, C132C83C2, (Q8×C13)⋊1C2, C2.6(C13⋊D4), SmallGroup(208,17)

Series: Derived Chief Lower central Upper central

C1C52 — Q8⋊D13
C1C13C26C52D52 — Q8⋊D13
C13C26C52 — Q8⋊D13
C1C2C4Q8

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 >

52C2
2C4
26C22
4D13
13C8
13D4
2D26
2C52
13SD16

Smallest permutation representation of Q8⋊D13
On 104 points
Generators in S104
(1 45 23 30)(2 46 24 31)(3 47 25 32)(4 48 26 33)(5 49 14 34)(6 50 15 35)(7 51 16 36)(8 52 17 37)(9 40 18 38)(10 41 19 39)(11 42 20 27)(12 43 21 28)(13 44 22 29)(53 90 76 93)(54 91 77 94)(55 79 78 95)(56 80 66 96)(57 81 67 97)(58 82 68 98)(59 83 69 99)(60 84 70 100)(61 85 71 101)(62 86 72 102)(63 87 73 103)(64 88 74 104)(65 89 75 92)
(1 72 23 62)(2 73 24 63)(3 74 25 64)(4 75 26 65)(5 76 14 53)(6 77 15 54)(7 78 16 55)(8 66 17 56)(9 67 18 57)(10 68 19 58)(11 69 20 59)(12 70 21 60)(13 71 22 61)(27 99 42 83)(28 100 43 84)(29 101 44 85)(30 102 45 86)(31 103 46 87)(32 104 47 88)(33 92 48 89)(34 93 49 90)(35 94 50 91)(36 95 51 79)(37 96 52 80)(38 97 40 81)(39 98 41 82)
(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 18)(15 17)(19 26)(20 25)(21 24)(22 23)(27 47)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 52)(36 51)(37 50)(38 49)(39 48)(53 81)(54 80)(55 79)(56 91)(57 90)(58 89)(59 88)(60 87)(61 86)(62 85)(63 84)(64 83)(65 82)(66 94)(67 93)(68 92)(69 104)(70 103)(71 102)(72 101)(73 100)(74 99)(75 98)(76 97)(77 96)(78 95)

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

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

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

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  D526C4  Q8⋊Dic13

37 conjugacy classes

class 1 2A2B4A4B8A8B13A···13F26A···26F52A···52R
order122448813···1326···2652···52
size11522426262···22···24···4

37 irreducible representations

dim1111222224
type++++++++
imageC1C2C2C2D4SD16D13D26C13⋊D4Q8⋊D13
kernelQ8⋊D13C132C8D52Q8×C13C26C13Q8C4C2C1
# reps11111266126

Matrix representation of Q8⋊D13 in GL4(𝔽313) generated by

312000
031200
001311
001312
,
628600
22125100
00130183
0065183
,
20100
79400
0010
0001
,
431200
1530900
0010
001312
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

Export

Subgroup lattice of Q8⋊D13 in TeX

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