direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D55, C55⋊1C6, C33⋊2D5, C165⋊2C2, C15⋊2D11, C11⋊(C3×D5), C5⋊(C3×D11), SmallGroup(330,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C55 — C3×D55 |
Generators and relations for C3×D55
G = < a,b,c | a3=b55=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D55
►On 165 points
Generators in S165
(1 147 109)(2 148 110)(3 149 56)(4 150 57)(5 151 58)(6 152 59)(7 153 60)(8 154 61)(9 155 62)(10 156 63)(11 157 64)(12 158 65)(13 159 66)(14 160 67)(15 161 68)(16 162 69)(17 163 70)(18 164 71)(19 165 72)(20 111 73)(21 112 74)(22 113 75)(23 114 76)(24 115 77)(25 116 78)(26 117 79)(27 118 80)(28 119 81)(29 120 82)(30 121 83)(31 122 84)(32 123 85)(33 124 86)(34 125 87)(35 126 88)(36 127 89)(37 128 90)(38 129 91)(39 130 92)(40 131 93)(41 132 94)(42 133 95)(43 134 96)(44 135 97)(45 136 98)(46 137 99)(47 138 100)(48 139 101)(49 140 102)(50 141 103)(51 142 104)(52 143 105)(53 144 106)(54 145 107)(55 146 108)
(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 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(56 106)(57 105)(58 104)(59 103)(60 102)(61 101)(62 100)(63 99)(64 98)(65 97)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(107 110)(108 109)(111 127)(112 126)(113 125)(114 124)(115 123)(116 122)(117 121)(118 120)(128 165)(129 164)(130 163)(131 162)(132 161)(133 160)(134 159)(135 158)(136 157)(137 156)(138 155)(139 154)(140 153)(141 152)(142 151)(143 150)(144 149)(145 148)(146 147)
G:=sub<Sym(165)| (1,147,109)(2,148,110)(3,149,56)(4,150,57)(5,151,58)(6,152,59)(7,153,60)(8,154,61)(9,155,62)(10,156,63)(11,157,64)(12,158,65)(13,159,66)(14,160,67)(15,161,68)(16,162,69)(17,163,70)(18,164,71)(19,165,72)(20,111,73)(21,112,74)(22,113,75)(23,114,76)(24,115,77)(25,116,78)(26,117,79)(27,118,80)(28,119,81)(29,120,82)(30,121,83)(31,122,84)(32,123,85)(33,124,86)(34,125,87)(35,126,88)(36,127,89)(37,128,90)(38,129,91)(39,130,92)(40,131,93)(41,132,94)(42,133,95)(43,134,96)(44,135,97)(45,136,98)(46,137,99)(47,138,100)(48,139,101)(49,140,102)(50,141,103)(51,142,104)(52,143,105)(53,144,106)(54,145,107)(55,146,108), (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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(56,106)(57,105)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(107,110)(108,109)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(128,165)(129,164)(130,163)(131,162)(132,161)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)(139,154)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)>;
G:=Group( (1,147,109)(2,148,110)(3,149,56)(4,150,57)(5,151,58)(6,152,59)(7,153,60)(8,154,61)(9,155,62)(10,156,63)(11,157,64)(12,158,65)(13,159,66)(14,160,67)(15,161,68)(16,162,69)(17,163,70)(18,164,71)(19,165,72)(20,111,73)(21,112,74)(22,113,75)(23,114,76)(24,115,77)(25,116,78)(26,117,79)(27,118,80)(28,119,81)(29,120,82)(30,121,83)(31,122,84)(32,123,85)(33,124,86)(34,125,87)(35,126,88)(36,127,89)(37,128,90)(38,129,91)(39,130,92)(40,131,93)(41,132,94)(42,133,95)(43,134,96)(44,135,97)(45,136,98)(46,137,99)(47,138,100)(48,139,101)(49,140,102)(50,141,103)(51,142,104)(52,143,105)(53,144,106)(54,145,107)(55,146,108), (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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(56,106)(57,105)(58,104)(59,103)(60,102)(61,101)(62,100)(63,99)(64,98)(65,97)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(107,110)(108,109)(111,127)(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(128,165)(129,164)(130,163)(131,162)(132,161)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)(139,154)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147) );
G=PermutationGroup([[(1,147,109),(2,148,110),(3,149,56),(4,150,57),(5,151,58),(6,152,59),(7,153,60),(8,154,61),(9,155,62),(10,156,63),(11,157,64),(12,158,65),(13,159,66),(14,160,67),(15,161,68),(16,162,69),(17,163,70),(18,164,71),(19,165,72),(20,111,73),(21,112,74),(22,113,75),(23,114,76),(24,115,77),(25,116,78),(26,117,79),(27,118,80),(28,119,81),(29,120,82),(30,121,83),(31,122,84),(32,123,85),(33,124,86),(34,125,87),(35,126,88),(36,127,89),(37,128,90),(38,129,91),(39,130,92),(40,131,93),(41,132,94),(42,133,95),(43,134,96),(44,135,97),(45,136,98),(46,137,99),(47,138,100),(48,139,101),(49,140,102),(50,141,103),(51,142,104),(52,143,105),(53,144,106),(54,145,107),(55,146,108)], [(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)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,48),(9,47),(10,46),(11,45),(12,44),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(56,106),(57,105),(58,104),(59,103),(60,102),(61,101),(62,100),(63,99),(64,98),(65,97),(66,96),(67,95),(68,94),(69,93),(70,92),(71,91),(72,90),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84),(79,83),(80,82),(107,110),(108,109),(111,127),(112,126),(113,125),(114,124),(115,123),(116,122),(117,121),(118,120),(128,165),(129,164),(130,163),(131,162),(132,161),(133,160),(134,159),(135,158),(136,157),(137,156),(138,155),(139,154),(140,153),(141,152),(142,151),(143,150),(144,149),(145,148),(146,147)]])
87 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 6A | 6B | 11A | ··· | 11E | 15A | 15B | 15C | 15D | 33A | ··· | 33J | 55A | ··· | 55T | 165A | ··· | 165AN |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 6 | 6 | 11 | ··· | 11 | 15 | 15 | 15 | 15 | 33 | ··· | 33 | 55 | ··· | 55 | 165 | ··· | 165 |
| size | 1 | 55 | 1 | 1 | 2 | 2 | 55 | 55 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
87 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | D5 | D11 | C3×D5 | C3×D11 | D55 | C3×D55 |
| kernel | C3×D55 | C165 | D55 | C55 | C33 | C15 | C11 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 5 | 4 | 10 | 20 | 40 |
Matrix representation of C3×D55 ►in GL2(𝔽331) generated by
| 31 | 0 |
| 0 | 31 |
| 205 | 325 |
| 6 | 129 |
| 205 | 325 |
| 53 | 126 |
G:=sub<GL(2,GF(331))| [31,0,0,31],[205,6,325,129],[205,53,325,126] >;
C3×D55 in GAP, Magma, Sage, TeX
C_3\times D_{55} % in TeX
G:=Group("C3xD55"); // GroupNames label
G:=SmallGroup(330,7);
// by ID
G=gap.SmallGroup(330,7);
# by ID
G:=PCGroup([4,-2,-3,-5,-11,290,4803]);
// Polycyclic
G:=Group<a,b,c|a^3=b^55=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export