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G = Q8×D13order 208 = 24·13

Direct product of Q8 and D13

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

Aliases: Q8×D13, C4.6D26, Dic264C2, C52.6C22, C26.7C23, D26.9C22, Dic13.3C22, C132(C2×Q8), (Q8×C13)⋊2C2, (C4×D13).1C2, C2.8(C22×D13), SmallGroup(208,41)

Series: Derived Chief Lower central Upper central

C1C26 — Q8×D13
C1C13C26D26C4×D13 — Q8×D13
C13C26 — Q8×D13
C1C2Q8

Generators and relations for Q8×D13
 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 >

13C2
13C2
13C4
13C4
13C22
13C4
13C2×C4
13C2×C4
13Q8
13Q8
13C2×C4
13Q8
13C2×Q8

Smallest permutation representation of Q8×D13
On 104 points
Generators in S104
(1 48 15 39)(2 49 16 27)(3 50 17 28)(4 51 18 29)(5 52 19 30)(6 40 20 31)(7 41 21 32)(8 42 22 33)(9 43 23 34)(10 44 24 35)(11 45 25 36)(12 46 26 37)(13 47 14 38)(53 88 69 94)(54 89 70 95)(55 90 71 96)(56 91 72 97)(57 79 73 98)(58 80 74 99)(59 81 75 100)(60 82 76 101)(61 83 77 102)(62 84 78 103)(63 85 66 104)(64 86 67 92)(65 87 68 93)
(1 72 15 56)(2 73 16 57)(3 74 17 58)(4 75 18 59)(5 76 19 60)(6 77 20 61)(7 78 21 62)(8 66 22 63)(9 67 23 64)(10 68 24 65)(11 69 25 53)(12 70 26 54)(13 71 14 55)(27 98 49 79)(28 99 50 80)(29 100 51 81)(30 101 52 82)(31 102 40 83)(32 103 41 84)(33 104 42 85)(34 92 43 86)(35 93 44 87)(36 94 45 88)(37 95 46 89)(38 96 47 90)(39 97 48 91)
(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 14)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 20)(9 19)(10 18)(11 17)(12 16)(13 15)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 78)(63 77)(64 76)(65 75)(79 95)(80 94)(81 93)(82 92)(83 104)(84 103)(85 102)(86 101)(87 100)(88 99)(89 98)(90 97)(91 96)

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

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

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

Q8×D13 is a maximal subgroup of   D13.Q16  D4.D26  Q16⋊D13  Q8.10D26  D4.10D26
Q8×D13 is a maximal quotient of   Dic133Q8  C52⋊Q8  Dic13.Q8  D26⋊Q8  D262Q8  Dic13⋊Q8  D263Q8

40 conjugacy classes

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

40 irreducible representations

dim11112224
type++++-++-
imageC1C2C2C2Q8D13D26Q8×D13
kernelQ8×D13Dic26C4×D13Q8×C13D13Q8C4C1
# reps133126186

Matrix representation of Q8×D13 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] >;

Q8×D13 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 Q8×D13 in TeX

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