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## G = D8×D13order 416 = 25·13

### Direct product of D8 and D13

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D8×D13
 Chief series C1 — C13 — C26 — C52 — C4×D13 — D4×D13 — D8×D13
 Lower central C13 — C26 — C52 — D8×D13
 Upper central C1 — C2 — C4 — D8

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 2A 2B 2C 2D 2E 2F 2G 4A 4B 8A 8B 8C 8D 13A ··· 13F 26A ··· 26F 26G ··· 26R 52A ··· 52F 104A ··· 104L order 1 2 2 2 2 2 2 2 4 4 8 8 8 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 104 ··· 104 size 1 1 4 4 13 13 52 52 2 26 2 2 26 26 2 ··· 2 2 ··· 2 8 ··· 8 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D8 D13 D26 D26 D4×D13 D8×D13 kernel D8×D13 C8×D13 D104 D4⋊D13 C13×D8 D4×D13 Dic13 D26 D13 D8 C8 D4 C2 C1 # reps 1 1 1 2 1 2 1 1 4 6 6 12 6 12

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

 312 0 0 0 0 312 0 0 0 0 0 193 0 0 60 193
,
 312 0 0 0 0 312 0 0 0 0 0 120 0 0 60 0
,
 71 1 0 0 252 127 0 0 0 0 1 0 0 0 0 1
,
 54 222 0 0 290 259 0 0 0 0 1 0 0 0 0 1
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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