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G = Q8xC13order 104 = 23·13

Direct product of C13 and Q8

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: Q8xC13, C4.C26, C52.3C2, C26.7C22, C2.2(C2xC26), SmallGroup(104,11)

Series: Derived Chief Lower central Upper central

C1C2 — Q8xC13
C1C2C26C52 — Q8xC13
C1C2 — Q8xC13
C1C26 — Q8xC13

Generators and relations for Q8xC13
 G = < a,b,c | a13=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 12, all normal (6 characteristic)
Quotients: C1, C2, C22, Q8, C13, C26, C2xC26, Q8xC13

Smallest permutation representation of Q8xC13
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)]])

Q8xC13 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++-
imageC1C2C13C26Q8Q8xC13
kernelQ8xC13C52Q8C4C13C1
# reps131236112

Matrix representation of Q8xC13 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] >;

Q8xC13 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 Q8xC13 in TeX

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