direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C15×D11, C55⋊2C6, C165⋊5C2, C11⋊3C30, C33⋊6C10, SmallGroup(330,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C15×D11 |
Generators and relations for C15×D11
G = < a,b,c | a15=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C15×D11
►On 165 points
Generators in S165
(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)(136 137 138 139 140 141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160 161 162 163 164 165)
(1 62 136 83 48 156 102 131 118 37 29)(2 63 137 84 49 157 103 132 119 38 30)(3 64 138 85 50 158 104 133 120 39 16)(4 65 139 86 51 159 105 134 106 40 17)(5 66 140 87 52 160 91 135 107 41 18)(6 67 141 88 53 161 92 121 108 42 19)(7 68 142 89 54 162 93 122 109 43 20)(8 69 143 90 55 163 94 123 110 44 21)(9 70 144 76 56 164 95 124 111 45 22)(10 71 145 77 57 165 96 125 112 31 23)(11 72 146 78 58 151 97 126 113 32 24)(12 73 147 79 59 152 98 127 114 33 25)(13 74 148 80 60 153 99 128 115 34 26)(14 75 149 81 46 154 100 129 116 35 27)(15 61 150 82 47 155 101 130 117 36 28)
(1 29)(2 30)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(31 71)(32 72)(33 73)(34 74)(35 75)(36 61)(37 62)(38 63)(39 64)(40 65)(41 66)(42 67)(43 68)(44 69)(45 70)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 97)(59 98)(60 99)(76 124)(77 125)(78 126)(79 127)(80 128)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 121)(89 122)(90 123)(106 139)(107 140)(108 141)(109 142)(110 143)(111 144)(112 145)(113 146)(114 147)(115 148)(116 149)(117 150)(118 136)(119 137)(120 138)
G:=sub<Sym(165)| (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)(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165), (1,62,136,83,48,156,102,131,118,37,29)(2,63,137,84,49,157,103,132,119,38,30)(3,64,138,85,50,158,104,133,120,39,16)(4,65,139,86,51,159,105,134,106,40,17)(5,66,140,87,52,160,91,135,107,41,18)(6,67,141,88,53,161,92,121,108,42,19)(7,68,142,89,54,162,93,122,109,43,20)(8,69,143,90,55,163,94,123,110,44,21)(9,70,144,76,56,164,95,124,111,45,22)(10,71,145,77,57,165,96,125,112,31,23)(11,72,146,78,58,151,97,126,113,32,24)(12,73,147,79,59,152,98,127,114,33,25)(13,74,148,80,60,153,99,128,115,34,26)(14,75,149,81,46,154,100,129,116,35,27)(15,61,150,82,47,155,101,130,117,36,28), (1,29)(2,30)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(31,71)(32,72)(33,73)(34,74)(35,75)(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,121)(89,122)(90,123)(106,139)(107,140)(108,141)(109,142)(110,143)(111,144)(112,145)(113,146)(114,147)(115,148)(116,149)(117,150)(118,136)(119,137)(120,138)>;
G:=Group( (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)(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165), (1,62,136,83,48,156,102,131,118,37,29)(2,63,137,84,49,157,103,132,119,38,30)(3,64,138,85,50,158,104,133,120,39,16)(4,65,139,86,51,159,105,134,106,40,17)(5,66,140,87,52,160,91,135,107,41,18)(6,67,141,88,53,161,92,121,108,42,19)(7,68,142,89,54,162,93,122,109,43,20)(8,69,143,90,55,163,94,123,110,44,21)(9,70,144,76,56,164,95,124,111,45,22)(10,71,145,77,57,165,96,125,112,31,23)(11,72,146,78,58,151,97,126,113,32,24)(12,73,147,79,59,152,98,127,114,33,25)(13,74,148,80,60,153,99,128,115,34,26)(14,75,149,81,46,154,100,129,116,35,27)(15,61,150,82,47,155,101,130,117,36,28), (1,29)(2,30)(3,16)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(31,71)(32,72)(33,73)(34,74)(35,75)(36,61)(37,62)(38,63)(39,64)(40,65)(41,66)(42,67)(43,68)(44,69)(45,70)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,97)(59,98)(60,99)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,121)(89,122)(90,123)(106,139)(107,140)(108,141)(109,142)(110,143)(111,144)(112,145)(113,146)(114,147)(115,148)(116,149)(117,150)(118,136)(119,137)(120,138) );
G=PermutationGroup([[(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),(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160,161,162,163,164,165)], [(1,62,136,83,48,156,102,131,118,37,29),(2,63,137,84,49,157,103,132,119,38,30),(3,64,138,85,50,158,104,133,120,39,16),(4,65,139,86,51,159,105,134,106,40,17),(5,66,140,87,52,160,91,135,107,41,18),(6,67,141,88,53,161,92,121,108,42,19),(7,68,142,89,54,162,93,122,109,43,20),(8,69,143,90,55,163,94,123,110,44,21),(9,70,144,76,56,164,95,124,111,45,22),(10,71,145,77,57,165,96,125,112,31,23),(11,72,146,78,58,151,97,126,113,32,24),(12,73,147,79,59,152,98,127,114,33,25),(13,74,148,80,60,153,99,128,115,34,26),(14,75,149,81,46,154,100,129,116,35,27),(15,61,150,82,47,155,101,130,117,36,28)], [(1,29),(2,30),(3,16),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,28),(31,71),(32,72),(33,73),(34,74),(35,75),(36,61),(37,62),(38,63),(39,64),(40,65),(41,66),(42,67),(43,68),(44,69),(45,70),(46,100),(47,101),(48,102),(49,103),(50,104),(51,105),(52,91),(53,92),(54,93),(55,94),(56,95),(57,96),(58,97),(59,98),(60,99),(76,124),(77,125),(78,126),(79,127),(80,128),(81,129),(82,130),(83,131),(84,132),(85,133),(86,134),(87,135),(88,121),(89,122),(90,123),(106,139),(107,140),(108,141),(109,142),(110,143),(111,144),(112,145),(113,146),(114,147),(115,148),(116,149),(117,150),(118,136),(119,137),(120,138)]])
105 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 11A | ··· | 11E | 15A | ··· | 15H | 30A | ··· | 30H | 33A | ··· | 33J | 55A | ··· | 55T | 165A | ··· | 165AN |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 11 | ··· | 11 | 15 | ··· | 15 | 30 | ··· | 30 | 33 | ··· | 33 | 55 | ··· | 55 | 165 | ··· | 165 |
| size | 1 | 11 | 1 | 1 | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 11 | 11 | 2 | ··· | 2 | 1 | ··· | 1 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | D11 | C3×D11 | C5×D11 | C15×D11 |
| kernel | C15×D11 | C165 | C5×D11 | C3×D11 | C55 | C33 | D11 | C11 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 5 | 10 | 20 | 40 |
Matrix representation of C15×D11 ►in GL3(𝔽331) generated by
| 323 | 0 | 0 |
| 0 | 299 | 0 |
| 0 | 0 | 299 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 330 | 123 |
| 330 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(331))| [323,0,0,0,299,0,0,0,299],[1,0,0,0,0,330,0,1,123],[330,0,0,0,0,1,0,1,0] >;
C15×D11 in GAP, Magma, Sage, TeX
C_{15}\times D_{11} % in TeX
G:=Group("C15xD11"); // GroupNames label
G:=SmallGroup(330,5);
// by ID
G=gap.SmallGroup(330,5);
# by ID
G:=PCGroup([4,-2,-3,-5,-11,4803]);
// Polycyclic
G:=Group<a,b,c|a^15=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export