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G = D8⋊D13order 416 = 25·13

2nd semidirect product of D8 and D13 acting via D13/C13=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C82D26, D82D13, D42D26, C1044C22, D26.14D4, C52.2C23, D52.1C22, Dic13.16D4, Dic261C22, D4⋊D132C2, (C13×D8)⋊4C2, (D4×D13)⋊2C2, C104⋊C23C2, C8⋊D133C2, C132(C8⋊C22), D4.D131C2, C2.16(D4×D13), C26.28(C2×D4), D42D131C2, C132C81C22, (D4×C13)⋊2C22, C4.2(C22×D13), (C4×D13).1C22, SmallGroup(416,132)

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 | a4=b2=c26=d2=1, bab=cac-1=dad=a-1, cbc-1=ab, dbd=a-1b, dcd=c-1 >

Subgroups: 584 in 68 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2 [×4], C4, C4 [×2], C22 [×6], C8, C8, C2×C4 [×2], D4 [×2], D4 [×3], Q8, C23, C13, M4(2), D8, D8, SD16 [×2], C2×D4, C4○D4, D13 [×2], C26, C26 [×2], C8⋊C22, Dic13, Dic13, C52, D26, D26 [×3], C2×C26 [×2], C132C8, C104, Dic26, C4×D13, D52, C2×Dic13, C13⋊D4 [×2], D4×C13 [×2], C22×D13, C8⋊D13, C104⋊C2, D4⋊D13, D4.D13, C13×D8, D4×D13, D42D13, D8⋊D13
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, C8⋊C22, D26 [×3], C22×D13, D4×D13, D8⋊D13

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

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E4A4B4C8A8B13A···13F26A···26F26G···26R52A···52F104A···104L
order1222224448813···1326···2626···2652···52104···104
size11442652226524522···22···28···84···44···4

53 irreducible representations

dim1111111122222444
type+++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D13D26D26C8⋊C22D4×D13D8⋊D13
kernelD8⋊D13C8⋊D13C104⋊C2D4⋊D13D4.D13C13×D8D4×D13D42D13Dic13D26D8C8D4C13C2C1
# reps111111111166121612

Matrix representation of D8⋊D13 in GL6(𝔽313)

100000
010000
00128900
0028731200
0029278312259
00980581
,
31200000
03120000
0023707024
00980581
00226547635
0028731200
,
14130000
2722750000
00128900
00031200
00292783120
00980581
,
84370000
1902290000
00128900
00031200
00027810
002150255312

G:=sub<GL(6,GF(313))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,287,29,98,0,0,289,312,278,0,0,0,0,0,312,58,0,0,0,0,259,1],[312,0,0,0,0,0,0,312,0,0,0,0,0,0,237,98,226,287,0,0,0,0,54,312,0,0,70,58,76,0,0,0,24,1,35,0],[14,272,0,0,0,0,13,275,0,0,0,0,0,0,1,0,29,98,0,0,289,312,278,0,0,0,0,0,312,58,0,0,0,0,0,1],[84,190,0,0,0,0,37,229,0,0,0,0,0,0,1,0,0,215,0,0,289,312,278,0,0,0,0,0,1,255,0,0,0,0,0,312] >;

D8⋊D13 in GAP, Magma, Sage, TeX

D_8\rtimes D_{13}
% in TeX

G:=Group("D8:D13");
// GroupNames label

G:=SmallGroup(416,132);
// by ID

G=gap.SmallGroup(416,132);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,362,116,297,159,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^26=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations

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