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G = C9×D13order 234 = 2·32·13

Direct product of C9 and D13

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C9×D13, C1172C2, C133C18, C39.3C6, C3.(C3×D13), (C3×D13).2C3, SmallGroup(234,4)

Series: Derived Chief Lower central Upper central

C1C13 — C9×D13
C1C13C39C117 — C9×D13
C13 — C9×D13
C1C9

Generators and relations for C9×D13
 G = < a,b,c | a9=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >

13C2
13C6
13C18

Smallest permutation representation of C9×D13
On 117 points
Generators in S117
(1 109 77 33 93 59 20 89 47)(2 110 78 34 94 60 21 90 48)(3 111 66 35 95 61 22 91 49)(4 112 67 36 96 62 23 79 50)(5 113 68 37 97 63 24 80 51)(6 114 69 38 98 64 25 81 52)(7 115 70 39 99 65 26 82 40)(8 116 71 27 100 53 14 83 41)(9 117 72 28 101 54 15 84 42)(10 105 73 29 102 55 16 85 43)(11 106 74 30 103 56 17 86 44)(12 107 75 31 104 57 18 87 45)(13 108 76 32 92 58 19 88 46)
(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)(105 106 107 108 109 110 111 112 113 114 115 116 117)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 25)(15 24)(16 23)(17 22)(18 21)(19 20)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(66 74)(67 73)(68 72)(69 71)(75 78)(76 77)(79 85)(80 84)(81 83)(86 91)(87 90)(88 89)(92 93)(94 104)(95 103)(96 102)(97 101)(98 100)(105 112)(106 111)(107 110)(108 109)(113 117)(114 116)

G:=sub<Sym(117)| (1,109,77,33,93,59,20,89,47)(2,110,78,34,94,60,21,90,48)(3,111,66,35,95,61,22,91,49)(4,112,67,36,96,62,23,79,50)(5,113,68,37,97,63,24,80,51)(6,114,69,38,98,64,25,81,52)(7,115,70,39,99,65,26,82,40)(8,116,71,27,100,53,14,83,41)(9,117,72,28,101,54,15,84,42)(10,105,73,29,102,55,16,85,43)(11,106,74,30,103,56,17,86,44)(12,107,75,31,104,57,18,87,45)(13,108,76,32,92,58,19,88,46), (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)(105,106,107,108,109,110,111,112,113,114,115,116,117), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(66,74)(67,73)(68,72)(69,71)(75,78)(76,77)(79,85)(80,84)(81,83)(86,91)(87,90)(88,89)(92,93)(94,104)(95,103)(96,102)(97,101)(98,100)(105,112)(106,111)(107,110)(108,109)(113,117)(114,116)>;

G:=Group( (1,109,77,33,93,59,20,89,47)(2,110,78,34,94,60,21,90,48)(3,111,66,35,95,61,22,91,49)(4,112,67,36,96,62,23,79,50)(5,113,68,37,97,63,24,80,51)(6,114,69,38,98,64,25,81,52)(7,115,70,39,99,65,26,82,40)(8,116,71,27,100,53,14,83,41)(9,117,72,28,101,54,15,84,42)(10,105,73,29,102,55,16,85,43)(11,106,74,30,103,56,17,86,44)(12,107,75,31,104,57,18,87,45)(13,108,76,32,92,58,19,88,46), (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)(105,106,107,108,109,110,111,112,113,114,115,116,117), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(66,74)(67,73)(68,72)(69,71)(75,78)(76,77)(79,85)(80,84)(81,83)(86,91)(87,90)(88,89)(92,93)(94,104)(95,103)(96,102)(97,101)(98,100)(105,112)(106,111)(107,110)(108,109)(113,117)(114,116) );

G=PermutationGroup([(1,109,77,33,93,59,20,89,47),(2,110,78,34,94,60,21,90,48),(3,111,66,35,95,61,22,91,49),(4,112,67,36,96,62,23,79,50),(5,113,68,37,97,63,24,80,51),(6,114,69,38,98,64,25,81,52),(7,115,70,39,99,65,26,82,40),(8,116,71,27,100,53,14,83,41),(9,117,72,28,101,54,15,84,42),(10,105,73,29,102,55,16,85,43),(11,106,74,30,103,56,17,86,44),(12,107,75,31,104,57,18,87,45),(13,108,76,32,92,58,19,88,46)], [(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),(105,106,107,108,109,110,111,112,113,114,115,116,117)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,25),(15,24),(16,23),(17,22),(18,21),(19,20),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(53,64),(54,63),(55,62),(56,61),(57,60),(58,59),(66,74),(67,73),(68,72),(69,71),(75,78),(76,77),(79,85),(80,84),(81,83),(86,91),(87,90),(88,89),(92,93),(94,104),(95,103),(96,102),(97,101),(98,100),(105,112),(106,111),(107,110),(108,109),(113,117),(114,116)])

C9×D13 is a maximal subgroup of   C13⋊Dic9

72 conjugacy classes

class 1  2 3A3B6A6B9A···9F13A···13F18A···18F39A···39L117A···117AJ
order1233669···913···1318···1839···39117···117
size1131113131···12···213···132···22···2

72 irreducible representations

dim111111222
type+++
imageC1C2C3C6C9C18D13C3×D13C9×D13
kernelC9×D13C117C3×D13C39D13C13C9C3C1
# reps11226661236

Matrix representation of C9×D13 in GL2(𝔽937) generated by

4510
0451
,
01
936232
,
01
10
G:=sub<GL(2,GF(937))| [451,0,0,451],[0,936,1,232],[0,1,1,0] >;

C9×D13 in GAP, Magma, Sage, TeX

C_9\times D_{13}
% in TeX

G:=Group("C9xD13");
// GroupNames label

G:=SmallGroup(234,4);
// by ID

G=gap.SmallGroup(234,4);
# by ID

G:=PCGroup([4,-2,-3,-3,-13,29,3459]);
// Polycyclic

G:=Group<a,b,c|a^9=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C9×D13 in TeX

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