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