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## G = SD16×D13order 416 = 25·13

### Direct product of SD16 and D13

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — SD16×D13
 Chief series C1 — C13 — C26 — C52 — C4×D13 — D4×D13 — SD16×D13
 Lower central C13 — C26 — C52 — SD16×D13
 Upper central C1 — C2 — C4 — SD16

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 SD16 D13 D26 D26 D26 D4×D13 SD16×D13 kernel SD16×D13 C8×D13 C104⋊C2 D4.D13 Q8⋊D13 C13×SD16 D4×D13 Q8×D13 Dic13 D26 D13 SD16 C8 D4 Q8 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 4 6 6 6 6 6 12

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

 1 0 0 0 0 1 0 0 0 0 183 130 0 0 248 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 312
,
 115 1 0 0 305 283 0 0 0 0 1 0 0 0 0 1
,
 283 312 0 0 273 30 0 0 0 0 1 0 0 0 0 1
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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