direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C11×D13, C13⋊C22, C143⋊2C2, SmallGroup(286,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C11×D13 |
Generators and relations for C11×D13
G = < a,b,c | a11=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C11×D13
►On 143 points
Generators in S143
(1 142 124 105 104 86 77 64 42 38 25)(2 143 125 106 92 87 78 65 43 39 26)(3 131 126 107 93 88 66 53 44 27 14)(4 132 127 108 94 89 67 54 45 28 15)(5 133 128 109 95 90 68 55 46 29 16)(6 134 129 110 96 91 69 56 47 30 17)(7 135 130 111 97 79 70 57 48 31 18)(8 136 118 112 98 80 71 58 49 32 19)(9 137 119 113 99 81 72 59 50 33 20)(10 138 120 114 100 82 73 60 51 34 21)(11 139 121 115 101 83 74 61 52 35 22)(12 140 122 116 102 84 75 62 40 36 23)(13 141 123 117 103 85 76 63 41 37 24)
(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)(118 119 120 121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140 141 142 143)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 22)(15 21)(16 20)(17 19)(23 26)(24 25)(27 35)(28 34)(29 33)(30 32)(36 39)(37 38)(40 43)(41 42)(44 52)(45 51)(46 50)(47 49)(53 61)(54 60)(55 59)(56 58)(62 65)(63 64)(66 74)(67 73)(68 72)(69 71)(75 78)(76 77)(80 91)(81 90)(82 89)(83 88)(84 87)(85 86)(92 102)(93 101)(94 100)(95 99)(96 98)(103 104)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(118 129)(119 128)(120 127)(121 126)(122 125)(123 124)(131 139)(132 138)(133 137)(134 136)(140 143)(141 142)
G:=sub<Sym(143)| (1,142,124,105,104,86,77,64,42,38,25)(2,143,125,106,92,87,78,65,43,39,26)(3,131,126,107,93,88,66,53,44,27,14)(4,132,127,108,94,89,67,54,45,28,15)(5,133,128,109,95,90,68,55,46,29,16)(6,134,129,110,96,91,69,56,47,30,17)(7,135,130,111,97,79,70,57,48,31,18)(8,136,118,112,98,80,71,58,49,32,19)(9,137,119,113,99,81,72,59,50,33,20)(10,138,120,114,100,82,73,60,51,34,21)(11,139,121,115,101,83,74,61,52,35,22)(12,140,122,116,102,84,75,62,40,36,23)(13,141,123,117,103,85,76,63,41,37,24), (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)(118,119,120,121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140,141,142,143), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,35)(28,34)(29,33)(30,32)(36,39)(37,38)(40,43)(41,42)(44,52)(45,51)(46,50)(47,49)(53,61)(54,60)(55,59)(56,58)(62,65)(63,64)(66,74)(67,73)(68,72)(69,71)(75,78)(76,77)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)(92,102)(93,101)(94,100)(95,99)(96,98)(103,104)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(118,129)(119,128)(120,127)(121,126)(122,125)(123,124)(131,139)(132,138)(133,137)(134,136)(140,143)(141,142)>;
G:=Group( (1,142,124,105,104,86,77,64,42,38,25)(2,143,125,106,92,87,78,65,43,39,26)(3,131,126,107,93,88,66,53,44,27,14)(4,132,127,108,94,89,67,54,45,28,15)(5,133,128,109,95,90,68,55,46,29,16)(6,134,129,110,96,91,69,56,47,30,17)(7,135,130,111,97,79,70,57,48,31,18)(8,136,118,112,98,80,71,58,49,32,19)(9,137,119,113,99,81,72,59,50,33,20)(10,138,120,114,100,82,73,60,51,34,21)(11,139,121,115,101,83,74,61,52,35,22)(12,140,122,116,102,84,75,62,40,36,23)(13,141,123,117,103,85,76,63,41,37,24), (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)(118,119,120,121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140,141,142,143), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,22)(15,21)(16,20)(17,19)(23,26)(24,25)(27,35)(28,34)(29,33)(30,32)(36,39)(37,38)(40,43)(41,42)(44,52)(45,51)(46,50)(47,49)(53,61)(54,60)(55,59)(56,58)(62,65)(63,64)(66,74)(67,73)(68,72)(69,71)(75,78)(76,77)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)(92,102)(93,101)(94,100)(95,99)(96,98)(103,104)(105,117)(106,116)(107,115)(108,114)(109,113)(110,112)(118,129)(119,128)(120,127)(121,126)(122,125)(123,124)(131,139)(132,138)(133,137)(134,136)(140,143)(141,142) );
G=PermutationGroup([[(1,142,124,105,104,86,77,64,42,38,25),(2,143,125,106,92,87,78,65,43,39,26),(3,131,126,107,93,88,66,53,44,27,14),(4,132,127,108,94,89,67,54,45,28,15),(5,133,128,109,95,90,68,55,46,29,16),(6,134,129,110,96,91,69,56,47,30,17),(7,135,130,111,97,79,70,57,48,31,18),(8,136,118,112,98,80,71,58,49,32,19),(9,137,119,113,99,81,72,59,50,33,20),(10,138,120,114,100,82,73,60,51,34,21),(11,139,121,115,101,83,74,61,52,35,22),(12,140,122,116,102,84,75,62,40,36,23),(13,141,123,117,103,85,76,63,41,37,24)], [(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),(118,119,120,121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140,141,142,143)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,22),(15,21),(16,20),(17,19),(23,26),(24,25),(27,35),(28,34),(29,33),(30,32),(36,39),(37,38),(40,43),(41,42),(44,52),(45,51),(46,50),(47,49),(53,61),(54,60),(55,59),(56,58),(62,65),(63,64),(66,74),(67,73),(68,72),(69,71),(75,78),(76,77),(80,91),(81,90),(82,89),(83,88),(84,87),(85,86),(92,102),(93,101),(94,100),(95,99),(96,98),(103,104),(105,117),(106,116),(107,115),(108,114),(109,113),(110,112),(118,129),(119,128),(120,127),(121,126),(122,125),(123,124),(131,139),(132,138),(133,137),(134,136),(140,143),(141,142)]])
88 conjugacy classes
| class | 1 | 2 | 11A | ··· | 11J | 13A | ··· | 13F | 22A | ··· | 22J | 143A | ··· | 143BH |
| order | 1 | 2 | 11 | ··· | 11 | 13 | ··· | 13 | 22 | ··· | 22 | 143 | ··· | 143 |
| size | 1 | 13 | 1 | ··· | 1 | 2 | ··· | 2 | 13 | ··· | 13 | 2 | ··· | 2 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C11 | C22 | D13 | C11×D13 |
| kernel | C11×D13 | C143 | D13 | C13 | C11 | C1 |
| # reps | 1 | 1 | 10 | 10 | 6 | 60 |
Matrix representation of C11×D13 ►in GL2(𝔽859) generated by
| 61 | 0 |
| 0 | 61 |
| 858 | 1 |
| 610 | 248 |
| 858 | 0 |
| 610 | 1 |
G:=sub<GL(2,GF(859))| [61,0,0,61],[858,610,1,248],[858,610,0,1] >;
C11×D13 in GAP, Magma, Sage, TeX
C_{11}\times D_{13} % in TeX
G:=Group("C11xD13"); // GroupNames label
G:=SmallGroup(286,2);
// by ID
G=gap.SmallGroup(286,2);
# by ID
G:=PCGroup([3,-2,-11,-13,2378]);
// Polycyclic
G:=Group<a,b,c|a^11=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export