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G = D8×D13order 416 = 25·13

Direct product of D8 and D13

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D8×D13, C84D26, D41D26, D1044C2, C1042C22, D26.23D4, D521C22, C52.1C23, Dic13.7D4, C132(C2×D8), D4⋊D131C2, (C13×D8)⋊2C2, (C8×D13)⋊1C2, (D4×D13)⋊1C2, C26.27(C2×D4), C2.15(D4×D13), C132C85C22, (D4×C13)⋊1C22, C4.1(C22×D13), (C4×D13).15C22, SmallGroup(416,131)

Series: Derived Chief Lower central Upper central

C1C52 — D8×D13
C1C13C26C52C4×D13D4×D13 — D8×D13
C13C26C52 — D8×D13
C1C2C4D8

Generators and relations for D8×D13
 G = < a,b,c,d | a8=b2=c13=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 752 in 76 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2 [×6], C4, C4, C22 [×9], C8, C8, C2×C4, D4 [×2], D4 [×4], C23 [×2], C13, C2×C8, D8, D8 [×3], C2×D4 [×2], D13 [×2], D13 [×2], C26, C26 [×2], C2×D8, Dic13, C52, D26, D26 [×6], C2×C26 [×2], C132C8, C104, C4×D13, D52 [×2], C13⋊D4 [×2], D4×C13 [×2], C22×D13 [×2], C8×D13, D104, D4⋊D13 [×2], C13×D8, D4×D13 [×2], D8×D13
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D8 [×2], C2×D4, D13, C2×D8, D26 [×3], C22×D13, D4×D13, D8×D13

Smallest permutation representation of D8×D13
On 104 points
Generators in S104
(1 84 33 62 23 103 50 66)(2 85 34 63 24 104 51 67)(3 86 35 64 25 92 52 68)(4 87 36 65 26 93 40 69)(5 88 37 53 14 94 41 70)(6 89 38 54 15 95 42 71)(7 90 39 55 16 96 43 72)(8 91 27 56 17 97 44 73)(9 79 28 57 18 98 45 74)(10 80 29 58 19 99 46 75)(11 81 30 59 20 100 47 76)(12 82 31 60 21 101 48 77)(13 83 32 61 22 102 49 78)
(1 62)(2 63)(3 64)(4 65)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 66)(24 67)(25 68)(26 69)(27 91)(28 79)(29 80)(30 81)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 93)(41 94)(42 95)(43 96)(44 97)(45 98)(46 99)(47 100)(48 101)(49 102)(50 103)(51 104)(52 92)
(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 38)(28 37)(29 36)(30 35)(31 34)(32 33)(40 46)(41 45)(42 44)(47 52)(48 51)(49 50)(53 57)(54 56)(58 65)(59 64)(60 63)(61 62)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(79 88)(80 87)(81 86)(82 85)(83 84)(89 91)(92 100)(93 99)(94 98)(95 97)(101 104)(102 103)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B8A8B8C8D13A···13F26A···26F26G···26R52A···52F104A···104L
order1222222244888813···1326···2626···2652···52104···104
size1144131352522262226262···22···28···84···44···4

56 irreducible representations

dim11111122222244
type++++++++++++++
imageC1C2C2C2C2C2D4D4D8D13D26D26D4×D13D8×D13
kernelD8×D13C8×D13D104D4⋊D13C13×D8D4×D13Dic13D26D13D8C8D4C2C1
# reps1112121146612612

Matrix representation of D8×D13 in GL4(𝔽313) generated by

312000
031200
000193
0060193
,
312000
031200
000120
00600
,
71100
25212700
0010
0001
,
5422200
29025900
0010
0001
G:=sub<GL(4,GF(313))| [312,0,0,0,0,312,0,0,0,0,0,60,0,0,193,193],[312,0,0,0,0,312,0,0,0,0,0,60,0,0,120,0],[71,252,0,0,1,127,0,0,0,0,1,0,0,0,0,1],[54,290,0,0,222,259,0,0,0,0,1,0,0,0,0,1] >;

D8×D13 in GAP, Magma, Sage, TeX

D_8\times D_{13}
% in TeX

G:=Group("D8xD13");
// GroupNames label

G:=SmallGroup(416,131);
// by ID

G=gap.SmallGroup(416,131);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,116,297,159,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^13=d^2=1,b*a*b=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

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