direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C13×SD16, Q8⋊C26, C8⋊2C26, D4.C26, C104⋊6C2, 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
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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