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G = D4⋊D13order 208 = 24·13

The semidirect product of D4 and D13 acting via D13/C13=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4⋊D13, C132D8, D522C2, C4.1D26, C26.7D4, C52.1C22, C132C81C2, (D4×C13)⋊1C2, C2.4(C13⋊D4), SmallGroup(208,15)

Series: Derived Chief Lower central Upper central

C1C52 — D4⋊D13
C1C13C26C52D52 — D4⋊D13
C13C26C52 — D4⋊D13
C1C2C4D4

Generators and relations for D4⋊D13
 G = < a,b,c,d | a4=b2=c13=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >

4C2
52C2
2C22
26C22
4D13
4C26
13C8
13D4
2D26
2C2×C26
13D8

Smallest permutation representation of D4⋊D13
On 104 points
Generators in S104
(1 42 15 33)(2 43 16 34)(3 44 17 35)(4 45 18 36)(5 46 19 37)(6 47 20 38)(7 48 21 39)(8 49 22 27)(9 50 23 28)(10 51 24 29)(11 52 25 30)(12 40 26 31)(13 41 14 32)(53 83 72 102)(54 84 73 103)(55 85 74 104)(56 86 75 92)(57 87 76 93)(58 88 77 94)(59 89 78 95)(60 90 66 96)(61 91 67 97)(62 79 68 98)(63 80 69 99)(64 81 70 100)(65 82 71 101)
(1 102)(2 103)(3 104)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 101)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 79)(25 80)(26 81)(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 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 66)(50 67)(51 68)(52 69)
(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 15)(16 26)(17 25)(18 24)(19 23)(20 22)(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 82)(54 81)(55 80)(56 79)(57 91)(58 90)(59 89)(60 88)(61 87)(62 86)(63 85)(64 84)(65 83)(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)| (1,42,15,33)(2,43,16,34)(3,44,17,35)(4,45,18,36)(5,46,19,37)(6,47,20,38)(7,48,21,39)(8,49,22,27)(9,50,23,28)(10,51,24,29)(11,52,25,30)(12,40,26,31)(13,41,14,32)(53,83,72,102)(54,84,73,103)(55,85,74,104)(56,86,75,92)(57,87,76,93)(58,88,77,94)(59,89,78,95)(60,90,66,96)(61,91,67,97)(62,79,68,98)(63,80,69,99)(64,81,70,100)(65,82,71,101), (1,102)(2,103)(3,104)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,79)(25,80)(26,81)(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)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,66)(50,67)(51,68)(52,69), (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)(14,15)(16,26)(17,25)(18,24)(19,23)(20,22)(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)(53,82)(54,81)(55,80)(56,79)(57,91)(58,90)(59,89)(60,88)(61,87)(62,86)(63,85)(64,84)(65,83)(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( (1,42,15,33)(2,43,16,34)(3,44,17,35)(4,45,18,36)(5,46,19,37)(6,47,20,38)(7,48,21,39)(8,49,22,27)(9,50,23,28)(10,51,24,29)(11,52,25,30)(12,40,26,31)(13,41,14,32)(53,83,72,102)(54,84,73,103)(55,85,74,104)(56,86,75,92)(57,87,76,93)(58,88,77,94)(59,89,78,95)(60,90,66,96)(61,91,67,97)(62,79,68,98)(63,80,69,99)(64,81,70,100)(65,82,71,101), (1,102)(2,103)(3,104)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,79)(25,80)(26,81)(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)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,66)(50,67)(51,68)(52,69), (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)(14,15)(16,26)(17,25)(18,24)(19,23)(20,22)(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)(53,82)(54,81)(55,80)(56,79)(57,91)(58,90)(59,89)(60,88)(61,87)(62,86)(63,85)(64,84)(65,83)(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([[(1,42,15,33),(2,43,16,34),(3,44,17,35),(4,45,18,36),(5,46,19,37),(6,47,20,38),(7,48,21,39),(8,49,22,27),(9,50,23,28),(10,51,24,29),(11,52,25,30),(12,40,26,31),(13,41,14,32),(53,83,72,102),(54,84,73,103),(55,85,74,104),(56,86,75,92),(57,87,76,93),(58,88,77,94),(59,89,78,95),(60,90,66,96),(61,91,67,97),(62,79,68,98),(63,80,69,99),(64,81,70,100),(65,82,71,101)], [(1,102),(2,103),(3,104),(4,92),(5,93),(6,94),(7,95),(8,96),(9,97),(10,98),(11,99),(12,100),(13,101),(14,82),(15,83),(16,84),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,79),(25,80),(26,81),(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),(40,70),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,66),(50,67),(51,68),(52,69)], [(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),(14,15),(16,26),(17,25),(18,24),(19,23),(20,22),(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),(53,82),(54,81),(55,80),(56,79),(57,91),(58,90),(59,89),(60,88),(61,87),(62,86),(63,85),(64,84),(65,83),(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)]])

D4⋊D13 is a maximal subgroup of   D8×D13  D8⋊D13  Q8⋊D26  D26.6D4  D526C22  D4⋊D26  C52.C23
D4⋊D13 is a maximal quotient of   C26.D8  D526C4  C13⋊D16  D8.D13  C8.6D26  C13⋊Q32  D4⋊Dic13

37 conjugacy classes

class 1 2A2B2C 4 8A8B13A···13F26A···26F26G···26R52A···52F
order122248813···1326···2626···2652···52
size11452226262···22···24···44···4

37 irreducible representations

dim1111222224
type+++++++++
imageC1C2C2C2D4D8D13D26C13⋊D4D4⋊D13
kernelD4⋊D13C132C8D52D4×C13C26C13D4C4C2C1
# reps11111266126

Matrix representation of D4⋊D13 in GL4(𝔽313) generated by

1000
0100
001264
00230312
,
312000
031200
000190
002850
,
289100
216400
0010
0001
,
431200
1530900
0010
00230312
G:=sub<GL(4,GF(313))| [1,0,0,0,0,1,0,0,0,0,1,230,0,0,264,312],[312,0,0,0,0,312,0,0,0,0,0,285,0,0,190,0],[289,216,0,0,1,4,0,0,0,0,1,0,0,0,0,1],[4,15,0,0,312,309,0,0,0,0,1,230,0,0,0,312] >;

D4⋊D13 in GAP, Magma, Sage, TeX

D_4\rtimes D_{13}
% in TeX

G:=Group("D4:D13");
// GroupNames label

G:=SmallGroup(208,15);
// by ID

G=gap.SmallGroup(208,15);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,61,182,97,42,4804]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^13=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of D4⋊D13 in TeX

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