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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

C1C4 — C13×SD16
C1C2C4C52Q8×C13 — C13×SD16
C1C2C4 — C13×SD16
C1C26C52 — C13×SD16

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

4C2
2C4
2C22
4C26
2C52
2C2×C26

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

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

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

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

C13×SD16 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+++++
imageC1C2C2C2C13C26C26C26D4SD16D4×C13C13×SD16
kernelC13×SD16C104D4×C13Q8×C13SD16C8D4Q8C26C13C2C1
# reps111112121212121224

Matrix representation of C13×SD16 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] >;

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

Export

Subgroup lattice of C13×SD16 in TeX

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