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## G = C13×SD16order 208 = 24·13

### Direct product of C13 and SD16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C13×SD16, Q8⋊C26, C82C26, D4.C26, C1046C2, C26.15D4, C52.18C22, C4.2(C2×C26), (Q8×C13)⋊4C2, C2.4(D4×C13), (D4×C13).2C2, SmallGroup(208,26)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C13×SD16
 Chief series C1 — C2 — C4 — C52 — Q8×C13 — C13×SD16
 Lower central C1 — C2 — C4 — C13×SD16
 Upper central C1 — C26 — C52 — C13×SD16

Generators and relations for C13×SD16
G = < a,b,c | a13=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Smallest permutation representation of C13×SD16
On 104 points
Generators in S104
(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 74 40 56 86 102 20 29)(2 75 41 57 87 103 21 30)(3 76 42 58 88 104 22 31)(4 77 43 59 89 92 23 32)(5 78 44 60 90 93 24 33)(6 66 45 61 91 94 25 34)(7 67 46 62 79 95 26 35)(8 68 47 63 80 96 14 36)(9 69 48 64 81 97 15 37)(10 70 49 65 82 98 16 38)(11 71 50 53 83 99 17 39)(12 72 51 54 84 100 18 27)(13 73 52 55 85 101 19 28)
(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 100)(28 101)(29 102)(30 103)(31 104)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 66)(62 67)(63 68)(64 69)(65 70)

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

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

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

C13×SD16 is a maximal subgroup of   Q8⋊D26  D4.D26  D26.6D4

91 conjugacy classes

 class 1 2A 2B 4A 4B 8A 8B 13A ··· 13L 26A ··· 26L 26M ··· 26X 52A ··· 52L 52M ··· 52X 104A ··· 104X order 1 2 2 4 4 8 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 52 ··· 52 104 ··· 104 size 1 1 4 2 4 2 2 1 ··· 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2

91 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C2 C13 C26 C26 C26 D4 SD16 D4×C13 C13×SD16 kernel C13×SD16 C104 D4×C13 Q8×C13 SD16 C8 D4 Q8 C26 C13 C2 C1 # reps 1 1 1 1 12 12 12 12 1 2 12 24

Matrix representation of C13×SD16 in GL2(𝔽313) generated by

 280 0 0 280
,
 183 183 65 0
,
 1 0 312 312
G:=sub<GL(2,GF(313))| [280,0,0,280],[183,65,183,0],[1,312,0,312] >;

C13×SD16 in GAP, Magma, Sage, TeX

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

G:=Group("C13xSD16");
// GroupNames label

G:=SmallGroup(208,26);
// by ID

G=gap.SmallGroup(208,26);
# by ID

G:=PCGroup([5,-2,-2,-13,-2,-2,520,541,3123,1568,58]);
// Polycyclic

G:=Group<a,b,c|a^13=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations

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