direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C13, C4.C26, C52.3C2, C26.7C22, C2.2(C2×C26), SmallGroup(104,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C13
G = < a,b,c | a13=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C13
►Regular action 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 89 26 27)(2 90 14 28)(3 91 15 29)(4 79 16 30)(5 80 17 31)(6 81 18 32)(7 82 19 33)(8 83 20 34)(9 84 21 35)(10 85 22 36)(11 86 23 37)(12 87 24 38)(13 88 25 39)(40 93 77 65)(41 94 78 53)(42 95 66 54)(43 96 67 55)(44 97 68 56)(45 98 69 57)(46 99 70 58)(47 100 71 59)(48 101 72 60)(49 102 73 61)(50 103 74 62)(51 104 75 63)(52 92 76 64)
(1 76 26 52)(2 77 14 40)(3 78 15 41)(4 66 16 42)(5 67 17 43)(6 68 18 44)(7 69 19 45)(8 70 20 46)(9 71 21 47)(10 72 22 48)(11 73 23 49)(12 74 24 50)(13 75 25 51)(27 64 89 92)(28 65 90 93)(29 53 91 94)(30 54 79 95)(31 55 80 96)(32 56 81 97)(33 57 82 98)(34 58 83 99)(35 59 84 100)(36 60 85 101)(37 61 86 102)(38 62 87 103)(39 63 88 104)
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), (1,89,26,27)(2,90,14,28)(3,91,15,29)(4,79,16,30)(5,80,17,31)(6,81,18,32)(7,82,19,33)(8,83,20,34)(9,84,21,35)(10,85,22,36)(11,86,23,37)(12,87,24,38)(13,88,25,39)(40,93,77,65)(41,94,78,53)(42,95,66,54)(43,96,67,55)(44,97,68,56)(45,98,69,57)(46,99,70,58)(47,100,71,59)(48,101,72,60)(49,102,73,61)(50,103,74,62)(51,104,75,63)(52,92,76,64), (1,76,26,52)(2,77,14,40)(3,78,15,41)(4,66,16,42)(5,67,17,43)(6,68,18,44)(7,69,19,45)(8,70,20,46)(9,71,21,47)(10,72,22,48)(11,73,23,49)(12,74,24,50)(13,75,25,51)(27,64,89,92)(28,65,90,93)(29,53,91,94)(30,54,79,95)(31,55,80,96)(32,56,81,97)(33,57,82,98)(34,58,83,99)(35,59,84,100)(36,60,85,101)(37,61,86,102)(38,62,87,103)(39,63,88,104)>;
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), (1,89,26,27)(2,90,14,28)(3,91,15,29)(4,79,16,30)(5,80,17,31)(6,81,18,32)(7,82,19,33)(8,83,20,34)(9,84,21,35)(10,85,22,36)(11,86,23,37)(12,87,24,38)(13,88,25,39)(40,93,77,65)(41,94,78,53)(42,95,66,54)(43,96,67,55)(44,97,68,56)(45,98,69,57)(46,99,70,58)(47,100,71,59)(48,101,72,60)(49,102,73,61)(50,103,74,62)(51,104,75,63)(52,92,76,64), (1,76,26,52)(2,77,14,40)(3,78,15,41)(4,66,16,42)(5,67,17,43)(6,68,18,44)(7,69,19,45)(8,70,20,46)(9,71,21,47)(10,72,22,48)(11,73,23,49)(12,74,24,50)(13,75,25,51)(27,64,89,92)(28,65,90,93)(29,53,91,94)(30,54,79,95)(31,55,80,96)(32,56,81,97)(33,57,82,98)(34,58,83,99)(35,59,84,100)(36,60,85,101)(37,61,86,102)(38,62,87,103)(39,63,88,104) );
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)], [(1,89,26,27),(2,90,14,28),(3,91,15,29),(4,79,16,30),(5,80,17,31),(6,81,18,32),(7,82,19,33),(8,83,20,34),(9,84,21,35),(10,85,22,36),(11,86,23,37),(12,87,24,38),(13,88,25,39),(40,93,77,65),(41,94,78,53),(42,95,66,54),(43,96,67,55),(44,97,68,56),(45,98,69,57),(46,99,70,58),(47,100,71,59),(48,101,72,60),(49,102,73,61),(50,103,74,62),(51,104,75,63),(52,92,76,64)], [(1,76,26,52),(2,77,14,40),(3,78,15,41),(4,66,16,42),(5,67,17,43),(6,68,18,44),(7,69,19,45),(8,70,20,46),(9,71,21,47),(10,72,22,48),(11,73,23,49),(12,74,24,50),(13,75,25,51),(27,64,89,92),(28,65,90,93),(29,53,91,94),(30,54,79,95),(31,55,80,96),(32,56,81,97),(33,57,82,98),(34,58,83,99),(35,59,84,100),(36,60,85,101),(37,61,86,102),(38,62,87,103),(39,63,88,104)]])
Q8×C13 is a maximal subgroup of
Q8⋊D13 C13⋊Q16 D52⋊C2 C26.A4
65 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 13A | ··· | 13L | 26A | ··· | 26L | 52A | ··· | 52AJ |
| order | 1 | 2 | 4 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | - | |||
| image | C1 | C2 | C13 | C26 | Q8 | Q8×C13 |
| kernel | Q8×C13 | C52 | Q8 | C4 | C13 | C1 |
| # reps | 1 | 3 | 12 | 36 | 1 | 12 |
Matrix representation of Q8×C13 ►in GL2(𝔽53) generated by
| 16 | 0 |
| 0 | 16 |
| 52 | 51 |
| 1 | 1 |
| 42 | 36 |
| 29 | 11 |
G:=sub<GL(2,GF(53))| [16,0,0,16],[52,1,51,1],[42,29,36,11] >;
Q8×C13 in GAP, Magma, Sage, TeX
Q_8\times C_{13} % in TeX
G:=Group("Q8xC13"); // GroupNames label
G:=SmallGroup(104,11);
// by ID
G=gap.SmallGroup(104,11);
# by ID
G:=PCGroup([4,-2,-2,-13,-2,208,433,213]);
// Polycyclic
G:=Group<a,b,c|a^13=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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