direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C32×D17, C51⋊2C6, C17⋊(C3×C6), (C3×C51)⋊3C2, SmallGroup(306,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C32×D17 |
Generators and relations for C32×D17
G = < a,b,c,d | a3=b3=c17=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of C32×D17
►On 153 points
Generators in S153
(1 141 78)(2 142 79)(3 143 80)(4 144 81)(5 145 82)(6 146 83)(7 147 84)(8 148 85)(9 149 69)(10 150 70)(11 151 71)(12 152 72)(13 153 73)(14 137 74)(15 138 75)(16 139 76)(17 140 77)(18 104 92)(19 105 93)(20 106 94)(21 107 95)(22 108 96)(23 109 97)(24 110 98)(25 111 99)(26 112 100)(27 113 101)(28 114 102)(29 115 86)(30 116 87)(31 117 88)(32 118 89)(33 119 90)(34 103 91)(35 129 68)(36 130 52)(37 131 53)(38 132 54)(39 133 55)(40 134 56)(41 135 57)(42 136 58)(43 120 59)(44 121 60)(45 122 61)(46 123 62)(47 124 63)(48 125 64)(49 126 65)(50 127 66)(51 128 67)
(1 38 32)(2 39 33)(3 40 34)(4 41 18)(5 42 19)(6 43 20)(7 44 21)(8 45 22)(9 46 23)(10 47 24)(11 48 25)(12 49 26)(13 50 27)(14 51 28)(15 35 29)(16 36 30)(17 37 31)(52 87 76)(53 88 77)(54 89 78)(55 90 79)(56 91 80)(57 92 81)(58 93 82)(59 94 83)(60 95 84)(61 96 85)(62 97 69)(63 98 70)(64 99 71)(65 100 72)(66 101 73)(67 102 74)(68 86 75)(103 143 134)(104 144 135)(105 145 136)(106 146 120)(107 147 121)(108 148 122)(109 149 123)(110 150 124)(111 151 125)(112 152 126)(113 153 127)(114 137 128)(115 138 129)(116 139 130)(117 140 131)(118 141 132)(119 142 133)
(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)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 28)(19 27)(20 26)(21 25)(22 24)(29 34)(30 33)(31 32)(35 40)(36 39)(37 38)(41 51)(42 50)(43 49)(44 48)(45 47)(52 55)(53 54)(56 68)(57 67)(58 66)(59 65)(60 64)(61 63)(70 85)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(86 91)(87 90)(88 89)(92 102)(93 101)(94 100)(95 99)(96 98)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(116 119)(117 118)(120 126)(121 125)(122 124)(127 136)(128 135)(129 134)(130 133)(131 132)(137 144)(138 143)(139 142)(140 141)(145 153)(146 152)(147 151)(148 150)
G:=sub<Sym(153)| (1,141,78)(2,142,79)(3,143,80)(4,144,81)(5,145,82)(6,146,83)(7,147,84)(8,148,85)(9,149,69)(10,150,70)(11,151,71)(12,152,72)(13,153,73)(14,137,74)(15,138,75)(16,139,76)(17,140,77)(18,104,92)(19,105,93)(20,106,94)(21,107,95)(22,108,96)(23,109,97)(24,110,98)(25,111,99)(26,112,100)(27,113,101)(28,114,102)(29,115,86)(30,116,87)(31,117,88)(32,118,89)(33,119,90)(34,103,91)(35,129,68)(36,130,52)(37,131,53)(38,132,54)(39,133,55)(40,134,56)(41,135,57)(42,136,58)(43,120,59)(44,121,60)(45,122,61)(46,123,62)(47,124,63)(48,125,64)(49,126,65)(50,127,66)(51,128,67), (1,38,32)(2,39,33)(3,40,34)(4,41,18)(5,42,19)(6,43,20)(7,44,21)(8,45,22)(9,46,23)(10,47,24)(11,48,25)(12,49,26)(13,50,27)(14,51,28)(15,35,29)(16,36,30)(17,37,31)(52,87,76)(53,88,77)(54,89,78)(55,90,79)(56,91,80)(57,92,81)(58,93,82)(59,94,83)(60,95,84)(61,96,85)(62,97,69)(63,98,70)(64,99,71)(65,100,72)(66,101,73)(67,102,74)(68,86,75)(103,143,134)(104,144,135)(105,145,136)(106,146,120)(107,147,121)(108,148,122)(109,149,123)(110,150,124)(111,151,125)(112,152,126)(113,153,127)(114,137,128)(115,138,129)(116,139,130)(117,140,131)(118,141,132)(119,142,133), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,28)(19,27)(20,26)(21,25)(22,24)(29,34)(30,33)(31,32)(35,40)(36,39)(37,38)(41,51)(42,50)(43,49)(44,48)(45,47)(52,55)(53,54)(56,68)(57,67)(58,66)(59,65)(60,64)(61,63)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(86,91)(87,90)(88,89)(92,102)(93,101)(94,100)(95,99)(96,98)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(116,119)(117,118)(120,126)(121,125)(122,124)(127,136)(128,135)(129,134)(130,133)(131,132)(137,144)(138,143)(139,142)(140,141)(145,153)(146,152)(147,151)(148,150)>;
G:=Group( (1,141,78)(2,142,79)(3,143,80)(4,144,81)(5,145,82)(6,146,83)(7,147,84)(8,148,85)(9,149,69)(10,150,70)(11,151,71)(12,152,72)(13,153,73)(14,137,74)(15,138,75)(16,139,76)(17,140,77)(18,104,92)(19,105,93)(20,106,94)(21,107,95)(22,108,96)(23,109,97)(24,110,98)(25,111,99)(26,112,100)(27,113,101)(28,114,102)(29,115,86)(30,116,87)(31,117,88)(32,118,89)(33,119,90)(34,103,91)(35,129,68)(36,130,52)(37,131,53)(38,132,54)(39,133,55)(40,134,56)(41,135,57)(42,136,58)(43,120,59)(44,121,60)(45,122,61)(46,123,62)(47,124,63)(48,125,64)(49,126,65)(50,127,66)(51,128,67), (1,38,32)(2,39,33)(3,40,34)(4,41,18)(5,42,19)(6,43,20)(7,44,21)(8,45,22)(9,46,23)(10,47,24)(11,48,25)(12,49,26)(13,50,27)(14,51,28)(15,35,29)(16,36,30)(17,37,31)(52,87,76)(53,88,77)(54,89,78)(55,90,79)(56,91,80)(57,92,81)(58,93,82)(59,94,83)(60,95,84)(61,96,85)(62,97,69)(63,98,70)(64,99,71)(65,100,72)(66,101,73)(67,102,74)(68,86,75)(103,143,134)(104,144,135)(105,145,136)(106,146,120)(107,147,121)(108,148,122)(109,149,123)(110,150,124)(111,151,125)(112,152,126)(113,153,127)(114,137,128)(115,138,129)(116,139,130)(117,140,131)(118,141,132)(119,142,133), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,28)(19,27)(20,26)(21,25)(22,24)(29,34)(30,33)(31,32)(35,40)(36,39)(37,38)(41,51)(42,50)(43,49)(44,48)(45,47)(52,55)(53,54)(56,68)(57,67)(58,66)(59,65)(60,64)(61,63)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(86,91)(87,90)(88,89)(92,102)(93,101)(94,100)(95,99)(96,98)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(116,119)(117,118)(120,126)(121,125)(122,124)(127,136)(128,135)(129,134)(130,133)(131,132)(137,144)(138,143)(139,142)(140,141)(145,153)(146,152)(147,151)(148,150) );
G=PermutationGroup([[(1,141,78),(2,142,79),(3,143,80),(4,144,81),(5,145,82),(6,146,83),(7,147,84),(8,148,85),(9,149,69),(10,150,70),(11,151,71),(12,152,72),(13,153,73),(14,137,74),(15,138,75),(16,139,76),(17,140,77),(18,104,92),(19,105,93),(20,106,94),(21,107,95),(22,108,96),(23,109,97),(24,110,98),(25,111,99),(26,112,100),(27,113,101),(28,114,102),(29,115,86),(30,116,87),(31,117,88),(32,118,89),(33,119,90),(34,103,91),(35,129,68),(36,130,52),(37,131,53),(38,132,54),(39,133,55),(40,134,56),(41,135,57),(42,136,58),(43,120,59),(44,121,60),(45,122,61),(46,123,62),(47,124,63),(48,125,64),(49,126,65),(50,127,66),(51,128,67)], [(1,38,32),(2,39,33),(3,40,34),(4,41,18),(5,42,19),(6,43,20),(7,44,21),(8,45,22),(9,46,23),(10,47,24),(11,48,25),(12,49,26),(13,50,27),(14,51,28),(15,35,29),(16,36,30),(17,37,31),(52,87,76),(53,88,77),(54,89,78),(55,90,79),(56,91,80),(57,92,81),(58,93,82),(59,94,83),(60,95,84),(61,96,85),(62,97,69),(63,98,70),(64,99,71),(65,100,72),(66,101,73),(67,102,74),(68,86,75),(103,143,134),(104,144,135),(105,145,136),(106,146,120),(107,147,121),(108,148,122),(109,149,123),(110,150,124),(111,151,125),(112,152,126),(113,153,127),(114,137,128),(115,138,129),(116,139,130),(117,140,131),(118,141,132),(119,142,133)], [(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)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(18,28),(19,27),(20,26),(21,25),(22,24),(29,34),(30,33),(31,32),(35,40),(36,39),(37,38),(41,51),(42,50),(43,49),(44,48),(45,47),(52,55),(53,54),(56,68),(57,67),(58,66),(59,65),(60,64),(61,63),(70,85),(71,84),(72,83),(73,82),(74,81),(75,80),(76,79),(77,78),(86,91),(87,90),(88,89),(92,102),(93,101),(94,100),(95,99),(96,98),(103,115),(104,114),(105,113),(106,112),(107,111),(108,110),(116,119),(117,118),(120,126),(121,125),(122,124),(127,136),(128,135),(129,134),(130,133),(131,132),(137,144),(138,143),(139,142),(140,141),(145,153),(146,152),(147,151),(148,150)]])
90 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 17A | ··· | 17H | 51A | ··· | 51BL |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 17 | ··· | 17 | 51 | ··· | 51 |
| size | 1 | 17 | 1 | ··· | 1 | 17 | ··· | 17 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D17 | C3×D17 |
| kernel | C32×D17 | C3×C51 | C3×D17 | C51 | C32 | C3 |
| # reps | 1 | 1 | 8 | 8 | 8 | 64 |
Matrix representation of C32×D17 ►in GL3(𝔽103) generated by
| 56 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 56 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 0 | 46 |
| 1 | 0 | 0 |
| 0 | 88 | 15 |
| 0 | 102 | 49 |
| 102 | 0 | 0 |
| 0 | 48 | 100 |
| 0 | 81 | 55 |
G:=sub<GL(3,GF(103))| [56,0,0,0,1,0,0,0,1],[56,0,0,0,46,0,0,0,46],[1,0,0,0,88,102,0,15,49],[102,0,0,0,48,81,0,100,55] >;
C32×D17 in GAP, Magma, Sage, TeX
C_3^2\times D_{17} % in TeX
G:=Group("C3^2xD17"); // GroupNames label
G:=SmallGroup(306,5);
// by ID
G=gap.SmallGroup(306,5);
# by ID
G:=PCGroup([4,-2,-3,-3,-17,4611]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^17=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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