direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D5×C33, C5⋊C66, C55⋊3C6, C165⋊6C2, C15⋊2C22, SmallGroup(330,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C33 |
Generators and relations for D5×C33
G = < a,b,c | a33=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C33
►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 48 139 121 69)(2 49 140 122 70)(3 50 141 123 71)(4 51 142 124 72)(5 52 143 125 73)(6 53 144 126 74)(7 54 145 127 75)(8 55 146 128 76)(9 56 147 129 77)(10 57 148 130 78)(11 58 149 131 79)(12 59 150 132 80)(13 60 151 100 81)(14 61 152 101 82)(15 62 153 102 83)(16 63 154 103 84)(17 64 155 104 85)(18 65 156 105 86)(19 66 157 106 87)(20 34 158 107 88)(21 35 159 108 89)(22 36 160 109 90)(23 37 161 110 91)(24 38 162 111 92)(25 39 163 112 93)(26 40 164 113 94)(27 41 165 114 95)(28 42 133 115 96)(29 43 134 116 97)(30 44 135 117 98)(31 45 136 118 99)(32 46 137 119 67)(33 47 138 120 68)
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 81)(14 82)(15 83)(16 84)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 99)(32 67)(33 68)(34 107)(35 108)(36 109)(37 110)(38 111)(39 112)(40 113)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 121)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 129)(57 130)(58 131)(59 132)(60 100)(61 101)(62 102)(63 103)(64 104)(65 105)(66 106)
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), (1,48,139,121,69)(2,49,140,122,70)(3,50,141,123,71)(4,51,142,124,72)(5,52,143,125,73)(6,53,144,126,74)(7,54,145,127,75)(8,55,146,128,76)(9,56,147,129,77)(10,57,148,130,78)(11,58,149,131,79)(12,59,150,132,80)(13,60,151,100,81)(14,61,152,101,82)(15,62,153,102,83)(16,63,154,103,84)(17,64,155,104,85)(18,65,156,105,86)(19,66,157,106,87)(20,34,158,107,88)(21,35,159,108,89)(22,36,160,109,90)(23,37,161,110,91)(24,38,162,111,92)(25,39,163,112,93)(26,40,164,113,94)(27,41,165,114,95)(28,42,133,115,96)(29,43,134,116,97)(30,44,135,117,98)(31,45,136,118,99)(32,46,137,119,67)(33,47,138,120,68), (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,67)(33,68)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,113)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,100)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106)>;
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), (1,48,139,121,69)(2,49,140,122,70)(3,50,141,123,71)(4,51,142,124,72)(5,52,143,125,73)(6,53,144,126,74)(7,54,145,127,75)(8,55,146,128,76)(9,56,147,129,77)(10,57,148,130,78)(11,58,149,131,79)(12,59,150,132,80)(13,60,151,100,81)(14,61,152,101,82)(15,62,153,102,83)(16,63,154,103,84)(17,64,155,104,85)(18,65,156,105,86)(19,66,157,106,87)(20,34,158,107,88)(21,35,159,108,89)(22,36,160,109,90)(23,37,161,110,91)(24,38,162,111,92)(25,39,163,112,93)(26,40,164,113,94)(27,41,165,114,95)(28,42,133,115,96)(29,43,134,116,97)(30,44,135,117,98)(31,45,136,118,99)(32,46,137,119,67)(33,47,138,120,68), (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,83)(16,84)(17,85)(18,86)(19,87)(20,88)(21,89)(22,90)(23,91)(24,92)(25,93)(26,94)(27,95)(28,96)(29,97)(30,98)(31,99)(32,67)(33,68)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,113)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,100)(61,101)(62,102)(63,103)(64,104)(65,105)(66,106) );
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)], [(1,48,139,121,69),(2,49,140,122,70),(3,50,141,123,71),(4,51,142,124,72),(5,52,143,125,73),(6,53,144,126,74),(7,54,145,127,75),(8,55,146,128,76),(9,56,147,129,77),(10,57,148,130,78),(11,58,149,131,79),(12,59,150,132,80),(13,60,151,100,81),(14,61,152,101,82),(15,62,153,102,83),(16,63,154,103,84),(17,64,155,104,85),(18,65,156,105,86),(19,66,157,106,87),(20,34,158,107,88),(21,35,159,108,89),(22,36,160,109,90),(23,37,161,110,91),(24,38,162,111,92),(25,39,163,112,93),(26,40,164,113,94),(27,41,165,114,95),(28,42,133,115,96),(29,43,134,116,97),(30,44,135,117,98),(31,45,136,118,99),(32,46,137,119,67),(33,47,138,120,68)], [(1,69),(2,70),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,79),(12,80),(13,81),(14,82),(15,83),(16,84),(17,85),(18,86),(19,87),(20,88),(21,89),(22,90),(23,91),(24,92),(25,93),(26,94),(27,95),(28,96),(29,97),(30,98),(31,99),(32,67),(33,68),(34,107),(35,108),(36,109),(37,110),(38,111),(39,112),(40,113),(41,114),(42,115),(43,116),(44,117),(45,118),(46,119),(47,120),(48,121),(49,122),(50,123),(51,124),(52,125),(53,126),(54,127),(55,128),(56,129),(57,130),(58,131),(59,132),(60,100),(61,101),(62,102),(63,103),(64,104),(65,105),(66,106)]])
132 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 6A | 6B | 11A | ··· | 11J | 15A | 15B | 15C | 15D | 22A | ··· | 22J | 33A | ··· | 33T | 55A | ··· | 55T | 66A | ··· | 66T | 165A | ··· | 165AN |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 6 | 6 | 11 | ··· | 11 | 15 | 15 | 15 | 15 | 22 | ··· | 22 | 33 | ··· | 33 | 55 | ··· | 55 | 66 | ··· | 66 | 165 | ··· | 165 |
| size | 1 | 5 | 1 | 1 | 2 | 2 | 5 | 5 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C6 | C11 | C22 | C33 | C66 | D5 | C3×D5 | D5×C11 | D5×C33 |
| kernel | D5×C33 | C165 | D5×C11 | C55 | C3×D5 | C15 | D5 | C5 | C33 | C11 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 10 | 10 | 20 | 20 | 2 | 4 | 20 | 40 |
Matrix representation of D5×C33 ►in GL2(𝔽331) generated by
| 219 | 0 |
| 0 | 219 |
| 116 | 1 |
| 330 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(331))| [219,0,0,219],[116,330,1,0],[0,1,1,0] >;
D5×C33 in GAP, Magma, Sage, TeX
D_5\times C_{33} % in TeX
G:=Group("D5xC33"); // GroupNames label
G:=SmallGroup(330,6);
// by ID
G=gap.SmallGroup(330,6);
# by ID
G:=PCGroup([4,-2,-3,-11,-5,4227]);
// Polycyclic
G:=Group<a,b,c|a^33=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export