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G = SD16×D13order 416 = 25·13

Direct product of SD16 and D13

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

Aliases: SD16×D13, C85D26, Q81D26, D4.2D26, C1045C22, D26.24D4, C52.4C23, Dic13.8D4, D52.2C22, Dic262C22, (C8×D13)⋊4C2, Q8⋊D131C2, (Q8×D13)⋊1C2, C132(C2×SD16), C104⋊C25C2, D4.D133C2, (D4×D13).1C2, C2.18(D4×D13), C26.30(C2×D4), C132C86C22, (C13×SD16)⋊3C2, (Q8×C13)⋊1C22, C4.4(C22×D13), (D4×C13).2C22, (C4×D13).17C22, SmallGroup(416,134)

Series: Derived Chief Lower central Upper central

C1C52 — SD16×D13
C1C13C26C52C4×D13D4×D13 — SD16×D13
C13C26C52 — SD16×D13
C1C2C4SD16

Generators and relations for SD16×D13
 G = < a,b,c,d | a8=b2=c13=d2=1, bab=a3, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 576 in 68 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2 [×4], C4, C4 [×3], C22 [×5], C8, C8, C2×C4 [×2], D4, D4 [×2], Q8, Q8 [×2], C23, C13, C2×C8, SD16, SD16 [×3], C2×D4, C2×Q8, D13 [×2], D13, C26, C26, C2×SD16, Dic13, Dic13, C52, C52, D26, D26 [×3], C2×C26, C132C8, C104, Dic26, Dic26, C4×D13, C4×D13, D52, C13⋊D4, D4×C13, Q8×C13, C22×D13, C8×D13, C104⋊C2, D4.D13, Q8⋊D13, C13×SD16, D4×D13, Q8×D13, SD16×D13
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, SD16 [×2], C2×D4, D13, C2×SD16, D26 [×3], C22×D13, D4×D13, SD16×D13

Smallest permutation representation of SD16×D13
On 104 points
Generators in S104
(1 97 29 70 23 80 47 65)(2 98 30 71 24 81 48 53)(3 99 31 72 25 82 49 54)(4 100 32 73 26 83 50 55)(5 101 33 74 14 84 51 56)(6 102 34 75 15 85 52 57)(7 103 35 76 16 86 40 58)(8 104 36 77 17 87 41 59)(9 92 37 78 18 88 42 60)(10 93 38 66 19 89 43 61)(11 94 39 67 20 90 44 62)(12 95 27 68 21 91 45 63)(13 96 28 69 22 79 46 64)
(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 40)(36 41)(37 42)(38 43)(39 44)(53 81)(54 82)(55 83)(56 84)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 79)(65 80)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 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 30)(28 29)(31 39)(32 38)(33 37)(34 36)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 63)(54 62)(55 61)(56 60)(57 59)(64 65)(66 73)(67 72)(68 71)(69 70)(74 78)(75 77)(79 80)(81 91)(82 90)(83 89)(84 88)(85 87)(92 101)(93 100)(94 99)(95 98)(96 97)(102 104)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D8A8B8C8D13A···13F26A···26F26G···26L52A···52F52G···52L104A···104L
order1222224444888813···1326···2626···2652···5252···52104···104
size1141313522426522226262···22···28···84···48···84···4

56 irreducible representations

dim11111111222222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4SD16D13D26D26D26D4×D13SD16×D13
kernelSD16×D13C8×D13C104⋊C2D4.D13Q8⋊D13C13×SD16D4×D13Q8×D13Dic13D26D13SD16C8D4Q8C2C1
# reps111111111146666612

Matrix representation of SD16×D13 in GL4(𝔽313) generated by

1000
0100
00183130
002480
,
1000
0100
0010
001312
,
115100
30528300
0010
0001
,
28331200
2733000
0010
0001
G:=sub<GL(4,GF(313))| [1,0,0,0,0,1,0,0,0,0,183,248,0,0,130,0],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,312],[115,305,0,0,1,283,0,0,0,0,1,0,0,0,0,1],[283,273,0,0,312,30,0,0,0,0,1,0,0,0,0,1] >;

SD16×D13 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times D_{13}
% in TeX

G:=Group("SD16xD13");
// GroupNames label

G:=SmallGroup(416,134);
// by ID

G=gap.SmallGroup(416,134);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,116,86,297,159,69,13829]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^13=d^2=1,b*a*b=a^3,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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