direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9×D13, C117⋊2C2, C13⋊3C18, C39.3C6, C3.(C3×D13), (C3×D13).2C3, SmallGroup(234,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C9×D13 |
Generators and relations for C9×D13
G = < a,b,c | a9=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C9×D13
►On 117 points
Generators in S117
(1 111 76 39 97 59 14 80 50)(2 112 77 27 98 60 15 81 51)(3 113 78 28 99 61 16 82 52)(4 114 66 29 100 62 17 83 40)(5 115 67 30 101 63 18 84 41)(6 116 68 31 102 64 19 85 42)(7 117 69 32 103 65 20 86 43)(8 105 70 33 104 53 21 87 44)(9 106 71 34 92 54 22 88 45)(10 107 72 35 93 55 23 89 46)(11 108 73 36 94 56 24 90 47)(12 109 74 37 95 57 25 91 48)(13 110 75 38 96 58 26 79 49)
(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 26)(15 25)(16 24)(17 23)(18 22)(19 21)(27 37)(28 36)(29 35)(30 34)(31 33)(38 39)(40 46)(41 45)(42 44)(47 52)(48 51)(49 50)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(66 72)(67 71)(68 70)(73 78)(74 77)(75 76)(79 80)(81 91)(82 90)(83 89)(84 88)(85 87)(92 101)(93 100)(94 99)(95 98)(96 97)(102 104)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)
G:=sub<Sym(117)| (1,111,76,39,97,59,14,80,50)(2,112,77,27,98,60,15,81,51)(3,113,78,28,99,61,16,82,52)(4,114,66,29,100,62,17,83,40)(5,115,67,30,101,63,18,84,41)(6,116,68,31,102,64,19,85,42)(7,117,69,32,103,65,20,86,43)(8,105,70,33,104,53,21,87,44)(9,106,71,34,92,54,22,88,45)(10,107,72,35,93,55,23,89,46)(11,108,73,36,94,56,24,90,47)(12,109,74,37,95,57,25,91,48)(13,110,75,38,96,58,26,79,49), (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)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(27,37)(28,36)(29,35)(30,34)(31,33)(38,39)(40,46)(41,45)(42,44)(47,52)(48,51)(49,50)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(66,72)(67,71)(68,70)(73,78)(74,77)(75,76)(79,80)(81,91)(82,90)(83,89)(84,88)(85,87)(92,101)(93,100)(94,99)(95,98)(96,97)(102,104)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111)>;
G:=Group( (1,111,76,39,97,59,14,80,50)(2,112,77,27,98,60,15,81,51)(3,113,78,28,99,61,16,82,52)(4,114,66,29,100,62,17,83,40)(5,115,67,30,101,63,18,84,41)(6,116,68,31,102,64,19,85,42)(7,117,69,32,103,65,20,86,43)(8,105,70,33,104,53,21,87,44)(9,106,71,34,92,54,22,88,45)(10,107,72,35,93,55,23,89,46)(11,108,73,36,94,56,24,90,47)(12,109,74,37,95,57,25,91,48)(13,110,75,38,96,58,26,79,49), (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)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)(27,37)(28,36)(29,35)(30,34)(31,33)(38,39)(40,46)(41,45)(42,44)(47,52)(48,51)(49,50)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)(66,72)(67,71)(68,70)(73,78)(74,77)(75,76)(79,80)(81,91)(82,90)(83,89)(84,88)(85,87)(92,101)(93,100)(94,99)(95,98)(96,97)(102,104)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111) );
G=PermutationGroup([[(1,111,76,39,97,59,14,80,50),(2,112,77,27,98,60,15,81,51),(3,113,78,28,99,61,16,82,52),(4,114,66,29,100,62,17,83,40),(5,115,67,30,101,63,18,84,41),(6,116,68,31,102,64,19,85,42),(7,117,69,32,103,65,20,86,43),(8,105,70,33,104,53,21,87,44),(9,106,71,34,92,54,22,88,45),(10,107,72,35,93,55,23,89,46),(11,108,73,36,94,56,24,90,47),(12,109,74,37,95,57,25,91,48),(13,110,75,38,96,58,26,79,49)], [(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),(14,26),(15,25),(16,24),(17,23),(18,22),(19,21),(27,37),(28,36),(29,35),(30,34),(31,33),(38,39),(40,46),(41,45),(42,44),(47,52),(48,51),(49,50),(53,64),(54,63),(55,62),(56,61),(57,60),(58,59),(66,72),(67,71),(68,70),(73,78),(74,77),(75,76),(79,80),(81,91),(82,90),(83,89),(84,88),(85,87),(92,101),(93,100),(94,99),(95,98),(96,97),(102,104),(105,116),(106,115),(107,114),(108,113),(109,112),(110,111)]])
C9×D13 is a maximal subgroup of
C13⋊Dic9
72 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 9A | ··· | 9F | 13A | ··· | 13F | 18A | ··· | 18F | 39A | ··· | 39L | 117A | ··· | 117AJ |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 13 | ··· | 13 | 18 | ··· | 18 | 39 | ··· | 39 | 117 | ··· | 117 |
| size | 1 | 13 | 1 | 1 | 13 | 13 | 1 | ··· | 1 | 2 | ··· | 2 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | D13 | C3×D13 | C9×D13 |
| kernel | C9×D13 | C117 | C3×D13 | C39 | D13 | C13 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 12 | 36 |
Matrix representation of C9×D13 ►in GL2(𝔽937) generated by
| 451 | 0 |
| 0 | 451 |
| 0 | 1 |
| 936 | 232 |
| 0 | 1 |
| 1 | 0 |
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