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G = C13xSD16order 208 = 24·13

Direct product of C13 and SD16

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

Aliases: C13xSD16, Q8:C26, C8:2C26, D4.C26, C104:6C2, C26.15D4, C52.18C22, C4.2(C2xC26), (Q8xC13):4C2, C2.4(D4xC13), (D4xC13).2C2, SmallGroup(208,26)

Series: Derived Chief Lower central Upper central

C1C4 — C13xSD16
C1C2C4C52Q8xC13 — C13xSD16
C1C2C4 — C13xSD16
C1C26C52 — C13xSD16

Generators and relations for C13xSD16
 G = < a,b,c | a13=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Subgroups: 30 in 20 conjugacy classes, 14 normal (all characteristic)
Quotients: C1, C2, C22, D4, C13, SD16, C26, C2xC26, D4xC13, C13xSD16
4C2
2C4
2C22
4C26
2C52
2C2xC26

Smallest permutation representation of C13xSD16
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)]])

C13xSD16 is a maximal subgroup of   Q8:D26  D4.D26  D26.6D4

91 conjugacy classes

class 1 2A2B4A4B8A8B13A···13L26A···26L26M···26X52A···52L52M···52X104A···104X
order122448813···1326···2626···2652···5252···52104···104
size11424221···11···14···42···24···42···2

91 irreducible representations

dim111111112222
type+++++
imageC1C2C2C2C13C26C26C26D4SD16D4xC13C13xSD16
kernelC13xSD16C104D4xC13Q8xC13SD16C8D4Q8C26C13C2C1
# reps111112121212121224

Matrix representation of C13xSD16 in GL2(F313) generated by

2800
0280
,
183183
650
,
10
312312
G:=sub<GL(2,GF(313))| [280,0,0,280],[183,65,183,0],[1,312,0,312] >;

C13xSD16 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

Export

Subgroup lattice of C13xSD16 in TeX

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