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G = Q8×C13order 104 = 23·13

Direct product of C13 and Q8

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

C1C2 — Q8×C13
C1C2C26C52 — Q8×C13
C1C2 — Q8×C13
C1C26 — Q8×C13

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 4A4B4C13A···13L26A···26L52A···52AJ
order1244413···1326···2652···52
size112221···11···12···2

65 irreducible representations

dim111122
type++-
imageC1C2C13C26Q8Q8×C13
kernelQ8×C13C52Q8C4C13C1
# reps131236112

Matrix representation of Q8×C13 in GL2(𝔽53) generated by

160
016
,
5251
11
,
4236
2911
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

Export

Subgroup lattice of Q8×C13 in TeX

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