direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×D33, C55⋊2S3, C165⋊3C2, C33⋊5C10, C15⋊3D11, C3⋊(C5×D11), C11⋊3(C5×S3), SmallGroup(330,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C5×D33 |
Generators and relations for C5×D33
G = < a,b,c | a5=b33=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C5×D33
►On 165 points
Generators in S165
(1 145 108 74 42)(2 146 109 75 43)(3 147 110 76 44)(4 148 111 77 45)(5 149 112 78 46)(6 150 113 79 47)(7 151 114 80 48)(8 152 115 81 49)(9 153 116 82 50)(10 154 117 83 51)(11 155 118 84 52)(12 156 119 85 53)(13 157 120 86 54)(14 158 121 87 55)(15 159 122 88 56)(16 160 123 89 57)(17 161 124 90 58)(18 162 125 91 59)(19 163 126 92 60)(20 164 127 93 61)(21 165 128 94 62)(22 133 129 95 63)(23 134 130 96 64)(24 135 131 97 65)(25 136 132 98 66)(26 137 100 99 34)(27 138 101 67 35)(28 139 102 68 36)(29 140 103 69 37)(30 141 104 70 38)(31 142 105 71 39)(32 143 106 72 40)(33 144 107 73 41)
(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(50 66)(51 65)(52 64)(53 63)(54 62)(55 61)(56 60)(57 59)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(81 99)(82 98)(83 97)(84 96)(85 95)(86 94)(87 93)(88 92)(89 91)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(116 132)(117 131)(118 130)(119 129)(120 128)(121 127)(122 126)(123 125)(133 156)(134 155)(135 154)(136 153)(137 152)(138 151)(139 150)(140 149)(141 148)(142 147)(143 146)(144 145)(157 165)(158 164)(159 163)(160 162)
G:=sub<Sym(165)| (1,145,108,74,42)(2,146,109,75,43)(3,147,110,76,44)(4,148,111,77,45)(5,149,112,78,46)(6,150,113,79,47)(7,151,114,80,48)(8,152,115,81,49)(9,153,116,82,50)(10,154,117,83,51)(11,155,118,84,52)(12,156,119,85,53)(13,157,120,86,54)(14,158,121,87,55)(15,159,122,88,56)(16,160,123,89,57)(17,161,124,90,58)(18,162,125,91,59)(19,163,126,92,60)(20,164,127,93,61)(21,165,128,94,62)(22,133,129,95,63)(23,134,130,96,64)(24,135,131,97,65)(25,136,132,98,66)(26,137,100,99,34)(27,138,101,67,35)(28,139,102,68,36)(29,140,103,69,37)(30,141,104,70,38)(31,142,105,71,39)(32,143,106,72,40)(33,144,107,73,41), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(133,156)(134,155)(135,154)(136,153)(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)(157,165)(158,164)(159,163)(160,162)>;
G:=Group( (1,145,108,74,42)(2,146,109,75,43)(3,147,110,76,44)(4,148,111,77,45)(5,149,112,78,46)(6,150,113,79,47)(7,151,114,80,48)(8,152,115,81,49)(9,153,116,82,50)(10,154,117,83,51)(11,155,118,84,52)(12,156,119,85,53)(13,157,120,86,54)(14,158,121,87,55)(15,159,122,88,56)(16,160,123,89,57)(17,161,124,90,58)(18,162,125,91,59)(19,163,126,92,60)(20,164,127,93,61)(21,165,128,94,62)(22,133,129,95,63)(23,134,130,96,64)(24,135,131,97,65)(25,136,132,98,66)(26,137,100,99,34)(27,138,101,67,35)(28,139,102,68,36)(29,140,103,69,37)(30,141,104,70,38)(31,142,105,71,39)(32,143,106,72,40)(33,144,107,73,41), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)(50,66)(51,65)(52,64)(53,63)(54,62)(55,61)(56,60)(57,59)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,99)(82,98)(83,97)(84,96)(85,95)(86,94)(87,93)(88,92)(89,91)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,126)(123,125)(133,156)(134,155)(135,154)(136,153)(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)(157,165)(158,164)(159,163)(160,162) );
G=PermutationGroup([[(1,145,108,74,42),(2,146,109,75,43),(3,147,110,76,44),(4,148,111,77,45),(5,149,112,78,46),(6,150,113,79,47),(7,151,114,80,48),(8,152,115,81,49),(9,153,116,82,50),(10,154,117,83,51),(11,155,118,84,52),(12,156,119,85,53),(13,157,120,86,54),(14,158,121,87,55),(15,159,122,88,56),(16,160,123,89,57),(17,161,124,90,58),(18,162,125,91,59),(19,163,126,92,60),(20,164,127,93,61),(21,165,128,94,62),(22,133,129,95,63),(23,134,130,96,64),(24,135,131,97,65),(25,136,132,98,66),(26,137,100,99,34),(27,138,101,67,35),(28,139,102,68,36),(29,140,103,69,37),(30,141,104,70,38),(31,142,105,71,39),(32,143,106,72,40),(33,144,107,73,41)], [(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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(41,42),(50,66),(51,65),(52,64),(53,63),(54,62),(55,61),(56,60),(57,59),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(81,99),(82,98),(83,97),(84,96),(85,95),(86,94),(87,93),(88,92),(89,91),(100,115),(101,114),(102,113),(103,112),(104,111),(105,110),(106,109),(107,108),(116,132),(117,131),(118,130),(119,129),(120,128),(121,127),(122,126),(123,125),(133,156),(134,155),(135,154),(136,153),(137,152),(138,151),(139,150),(140,149),(141,148),(142,147),(143,146),(144,145),(157,165),(158,164),(159,163),(160,162)]])
90 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 11A | ··· | 11E | 15A | 15B | 15C | 15D | 33A | ··· | 33J | 55A | ··· | 55T | 165A | ··· | 165AN |
| order | 1 | 2 | 3 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 11 | ··· | 11 | 15 | 15 | 15 | 15 | 33 | ··· | 33 | 55 | ··· | 55 | 165 | ··· | 165 |
| size | 1 | 33 | 2 | 1 | 1 | 1 | 1 | 33 | 33 | 33 | 33 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C5 | C10 | S3 | D11 | C5×S3 | D33 | C5×D11 | C5×D33 |
| kernel | C5×D33 | C165 | D33 | C33 | C55 | C15 | C11 | C5 | C3 | C1 |
| # reps | 1 | 1 | 4 | 4 | 1 | 5 | 4 | 10 | 20 | 40 |
Matrix representation of C5×D33 ►in GL2(𝔽331) generated by
| 323 | 0 |
| 0 | 323 |
| 188 | 250 |
| 81 | 113 |
| 201 | 232 |
| 271 | 130 |
G:=sub<GL(2,GF(331))| [323,0,0,323],[188,81,250,113],[201,271,232,130] >;
C5×D33 in GAP, Magma, Sage, TeX
C_5\times D_{33} % in TeX
G:=Group("C5xD33"); // GroupNames label
G:=SmallGroup(330,9);
// by ID
G=gap.SmallGroup(330,9);
# by ID
G:=PCGroup([4,-2,-5,-3,-11,242,4803]);
// Polycyclic
G:=Group<a,b,c|a^5=b^33=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export