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G = Q8xD13order 208 = 24·13

Direct product of Q8 and D13

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8xD13, C4.6D26, Dic26:4C2, C52.6C22, C26.7C23, D26.9C22, Dic13.3C22, C13:2(C2xQ8), (Q8xC13):2C2, (C4xD13).1C2, C2.8(C22xD13), SmallGroup(208,41)

Series: Derived Chief Lower central Upper central

C1C26 — Q8xD13
C1C13C26D26C4xD13 — Q8xD13
C13C26 — Q8xD13
C1C2Q8

Generators and relations for Q8xD13
 G = < a,b,c,d | a4=c13=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 194 in 38 conjugacy classes, 25 normal (8 characteristic)
Quotients: C1, C2, C22, Q8, C23, C2xQ8, D13, D26, C22xD13, Q8xD13
13C2
13C2
13C4
13C4
13C22
13C4
13C2xC4
13C2xC4
13Q8
13Q8
13C2xC4
13Q8
13C2xQ8

Smallest permutation representation of Q8xD13
On 104 points
Generators in S104
(1 45 23 32)(2 46 24 33)(3 47 25 34)(4 48 26 35)(5 49 14 36)(6 50 15 37)(7 51 16 38)(8 52 17 39)(9 40 18 27)(10 41 19 28)(11 42 20 29)(12 43 21 30)(13 44 22 31)(53 86 67 98)(54 87 68 99)(55 88 69 100)(56 89 70 101)(57 90 71 102)(58 91 72 103)(59 79 73 104)(60 80 74 92)(61 81 75 93)(62 82 76 94)(63 83 77 95)(64 84 78 96)(65 85 66 97)
(1 76 23 62)(2 77 24 63)(3 78 25 64)(4 66 26 65)(5 67 14 53)(6 68 15 54)(7 69 16 55)(8 70 17 56)(9 71 18 57)(10 72 19 58)(11 73 20 59)(12 74 21 60)(13 75 22 61)(27 102 40 90)(28 103 41 91)(29 104 42 79)(30 92 43 80)(31 93 44 81)(32 94 45 82)(33 95 46 83)(34 96 47 84)(35 97 48 85)(36 98 49 86)(37 99 50 87)(38 100 51 88)(39 101 52 89)
(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 22)(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 26)(11 25)(12 24)(13 23)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 52)(38 51)(39 50)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(79 96)(80 95)(81 94)(82 93)(83 92)(84 104)(85 103)(86 102)(87 101)(88 100)(89 99)(90 98)(91 97)

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

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

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

Q8xD13 is a maximal subgroup of   D13.Q16  D4.D26  Q16:D13  Q8.10D26  D4.10D26
Q8xD13 is a maximal quotient of   Dic13:3Q8  C52:Q8  Dic13.Q8  D26:Q8  D26:2Q8  Dic13:Q8  D26:3Q8

40 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F13A···13F26A···26F52A···52R
order122244444413···1326···2652···52
size1113132222626262···22···24···4

40 irreducible representations

dim11112224
type++++-++-
imageC1C2C2C2Q8D13D26Q8xD13
kernelQ8xD13Dic26C4xD13Q8xC13D13Q8C4C1
# reps133126186

Matrix representation of Q8xD13 in GL4(F53) generated by

52000
05200
0001
00520
,
1000
0100
00023
00230
,
52100
43900
0010
0001
,
52000
43100
00520
00052
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,0,52,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,23,0,0,23,0],[52,43,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[52,43,0,0,0,1,0,0,0,0,52,0,0,0,0,52] >;

Q8xD13 in GAP, Magma, Sage, TeX

Q_8\times D_{13}
% in TeX

G:=Group("Q8xD13");
// GroupNames label

G:=SmallGroup(208,41);
// by ID

G=gap.SmallGroup(208,41);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,46,97,42,4804]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^13=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of Q8xD13 in TeX

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