direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C11×D15, C55⋊3S3, C33⋊3D5, C165⋊4C2, C15⋊1C22, C5⋊(S3×C11), C3⋊(D5×C11), SmallGroup(330,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C11×D15 |
Generators and relations for C11×D15
G = < a,b,c | a11=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C11×D15
►On 165 points
Generators in S165
(1 152 137 121 118 103 76 62 54 43 24)(2 153 138 122 119 104 77 63 55 44 25)(3 154 139 123 120 105 78 64 56 45 26)(4 155 140 124 106 91 79 65 57 31 27)(5 156 141 125 107 92 80 66 58 32 28)(6 157 142 126 108 93 81 67 59 33 29)(7 158 143 127 109 94 82 68 60 34 30)(8 159 144 128 110 95 83 69 46 35 16)(9 160 145 129 111 96 84 70 47 36 17)(10 161 146 130 112 97 85 71 48 37 18)(11 162 147 131 113 98 86 72 49 38 19)(12 163 148 132 114 99 87 73 50 39 20)(13 164 149 133 115 100 88 74 51 40 21)(14 165 150 134 116 101 89 75 52 41 22)(15 151 136 135 117 102 90 61 53 42 23)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(31 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 62)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(91 99)(92 98)(93 97)(94 96)(100 105)(101 104)(102 103)(106 114)(107 113)(108 112)(109 111)(115 120)(116 119)(117 118)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 137)(138 150)(139 149)(140 148)(141 147)(142 146)(143 145)(151 152)(153 165)(154 164)(155 163)(156 162)(157 161)(158 160)
G:=sub<Sym(165)| (1,152,137,121,118,103,76,62,54,43,24)(2,153,138,122,119,104,77,63,55,44,25)(3,154,139,123,120,105,78,64,56,45,26)(4,155,140,124,106,91,79,65,57,31,27)(5,156,141,125,107,92,80,66,58,32,28)(6,157,142,126,108,93,81,67,59,33,29)(7,158,143,127,109,94,82,68,60,34,30)(8,159,144,128,110,95,83,69,46,35,16)(9,160,145,129,111,96,84,70,47,36,17)(10,161,146,130,112,97,85,71,48,37,18)(11,162,147,131,113,98,86,72,49,38,19)(12,163,148,132,114,99,87,73,50,39,20)(13,164,149,133,115,100,88,74,51,40,21)(14,165,150,134,116,101,89,75,52,41,22)(15,151,136,135,117,102,90,61,53,42,23), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(31,39)(32,38)(33,37)(34,36)(40,45)(41,44)(42,43)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,62)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(91,99)(92,98)(93,97)(94,96)(100,105)(101,104)(102,103)(106,114)(107,113)(108,112)(109,111)(115,120)(116,119)(117,118)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,137)(138,150)(139,149)(140,148)(141,147)(142,146)(143,145)(151,152)(153,165)(154,164)(155,163)(156,162)(157,161)(158,160)>;
G:=Group( (1,152,137,121,118,103,76,62,54,43,24)(2,153,138,122,119,104,77,63,55,44,25)(3,154,139,123,120,105,78,64,56,45,26)(4,155,140,124,106,91,79,65,57,31,27)(5,156,141,125,107,92,80,66,58,32,28)(6,157,142,126,108,93,81,67,59,33,29)(7,158,143,127,109,94,82,68,60,34,30)(8,159,144,128,110,95,83,69,46,35,16)(9,160,145,129,111,96,84,70,47,36,17)(10,161,146,130,112,97,85,71,48,37,18)(11,162,147,131,113,98,86,72,49,38,19)(12,163,148,132,114,99,87,73,50,39,20)(13,164,149,133,115,100,88,74,51,40,21)(14,165,150,134,116,101,89,75,52,41,22)(15,151,136,135,117,102,90,61,53,42,23), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(31,39)(32,38)(33,37)(34,36)(40,45)(41,44)(42,43)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,62)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(76,90)(77,89)(78,88)(79,87)(80,86)(81,85)(82,84)(91,99)(92,98)(93,97)(94,96)(100,105)(101,104)(102,103)(106,114)(107,113)(108,112)(109,111)(115,120)(116,119)(117,118)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,137)(138,150)(139,149)(140,148)(141,147)(142,146)(143,145)(151,152)(153,165)(154,164)(155,163)(156,162)(157,161)(158,160) );
G=PermutationGroup([[(1,152,137,121,118,103,76,62,54,43,24),(2,153,138,122,119,104,77,63,55,44,25),(3,154,139,123,120,105,78,64,56,45,26),(4,155,140,124,106,91,79,65,57,31,27),(5,156,141,125,107,92,80,66,58,32,28),(6,157,142,126,108,93,81,67,59,33,29),(7,158,143,127,109,94,82,68,60,34,30),(8,159,144,128,110,95,83,69,46,35,16),(9,160,145,129,111,96,84,70,47,36,17),(10,161,146,130,112,97,85,71,48,37,18),(11,162,147,131,113,98,86,72,49,38,19),(12,163,148,132,114,99,87,73,50,39,20),(13,164,149,133,115,100,88,74,51,40,21),(14,165,150,134,116,101,89,75,52,41,22),(15,151,136,135,117,102,90,61,53,42,23)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,24),(31,39),(32,38),(33,37),(34,36),(40,45),(41,44),(42,43),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(61,62),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(76,90),(77,89),(78,88),(79,87),(80,86),(81,85),(82,84),(91,99),(92,98),(93,97),(94,96),(100,105),(101,104),(102,103),(106,114),(107,113),(108,112),(109,111),(115,120),(116,119),(117,118),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129),(136,137),(138,150),(139,149),(140,148),(141,147),(142,146),(143,145),(151,152),(153,165),(154,164),(155,163),(156,162),(157,161),(158,160)]])
99 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 11A | ··· | 11J | 15A | 15B | 15C | 15D | 22A | ··· | 22J | 33A | ··· | 33J | 55A | ··· | 55T | 165A | ··· | 165AN |
| order | 1 | 2 | 3 | 5 | 5 | 11 | ··· | 11 | 15 | 15 | 15 | 15 | 22 | ··· | 22 | 33 | ··· | 33 | 55 | ··· | 55 | 165 | ··· | 165 |
| size | 1 | 15 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 15 | ··· | 15 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C11 | C22 | S3 | D5 | D15 | S3×C11 | D5×C11 | C11×D15 |
| kernel | C11×D15 | C165 | D15 | C15 | C55 | C33 | C11 | C5 | C3 | C1 |
| # reps | 1 | 1 | 10 | 10 | 1 | 2 | 4 | 10 | 20 | 40 |
Matrix representation of C11×D15 ►in GL2(𝔽331) generated by
| 180 | 0 |
| 0 | 180 |
| 274 | 54 |
| 252 | 220 |
| 330 | 0 |
| 1 | 1 |
G:=sub<GL(2,GF(331))| [180,0,0,180],[274,252,54,220],[330,1,0,1] >;
C11×D15 in GAP, Magma, Sage, TeX
C_{11}\times D_{15} % in TeX
G:=Group("C11xD15"); // GroupNames label
G:=SmallGroup(330,10);
// by ID
G=gap.SmallGroup(330,10);
# by ID
G:=PCGroup([4,-2,-11,-3,-5,530,4227]);
// Polycyclic
G:=Group<a,b,c|a^11=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export