direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C17×He3, C32⋊C51, C51.1C32, (C3×C51)⋊C3, C3.1(C3×C51), SmallGroup(459,3)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C17×He3
G = < a,b,c,d | a17=b3=c3=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >
Smallest permutation representation of C17×He3
►On 153 points
Generators in S153
(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 59 96)(2 60 97)(3 61 98)(4 62 99)(5 63 100)(6 64 101)(7 65 102)(8 66 86)(9 67 87)(10 68 88)(11 52 89)(12 53 90)(13 54 91)(14 55 92)(15 56 93)(16 57 94)(17 58 95)(18 115 42)(19 116 43)(20 117 44)(21 118 45)(22 119 46)(23 103 47)(24 104 48)(25 105 49)(26 106 50)(27 107 51)(28 108 35)(29 109 36)(30 110 37)(31 111 38)(32 112 39)(33 113 40)(34 114 41)(69 121 149)(70 122 150)(71 123 151)(72 124 152)(73 125 153)(74 126 137)(75 127 138)(76 128 139)(77 129 140)(78 130 141)(79 131 142)(80 132 143)(81 133 144)(82 134 145)(83 135 146)(84 136 147)(85 120 148)
(1 71 20)(2 72 21)(3 73 22)(4 74 23)(5 75 24)(6 76 25)(7 77 26)(8 78 27)(9 79 28)(10 80 29)(11 81 30)(12 82 31)(13 83 32)(14 84 33)(15 85 34)(16 69 18)(17 70 19)(35 87 142)(36 88 143)(37 89 144)(38 90 145)(39 91 146)(40 92 147)(41 93 148)(42 94 149)(43 95 150)(44 96 151)(45 97 152)(46 98 153)(47 99 137)(48 100 138)(49 101 139)(50 102 140)(51 86 141)(52 133 110)(53 134 111)(54 135 112)(55 136 113)(56 120 114)(57 121 115)(58 122 116)(59 123 117)(60 124 118)(61 125 119)(62 126 103)(63 127 104)(64 128 105)(65 129 106)(66 130 107)(67 131 108)(68 132 109)
(35 142 87)(36 143 88)(37 144 89)(38 145 90)(39 146 91)(40 147 92)(41 148 93)(42 149 94)(43 150 95)(44 151 96)(45 152 97)(46 153 98)(47 137 99)(48 138 100)(49 139 101)(50 140 102)(51 141 86)(52 133 110)(53 134 111)(54 135 112)(55 136 113)(56 120 114)(57 121 115)(58 122 116)(59 123 117)(60 124 118)(61 125 119)(62 126 103)(63 127 104)(64 128 105)(65 129 106)(66 130 107)(67 131 108)(68 132 109)
G:=sub<Sym(153)| (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), (1,59,96)(2,60,97)(3,61,98)(4,62,99)(5,63,100)(6,64,101)(7,65,102)(8,66,86)(9,67,87)(10,68,88)(11,52,89)(12,53,90)(13,54,91)(14,55,92)(15,56,93)(16,57,94)(17,58,95)(18,115,42)(19,116,43)(20,117,44)(21,118,45)(22,119,46)(23,103,47)(24,104,48)(25,105,49)(26,106,50)(27,107,51)(28,108,35)(29,109,36)(30,110,37)(31,111,38)(32,112,39)(33,113,40)(34,114,41)(69,121,149)(70,122,150)(71,123,151)(72,124,152)(73,125,153)(74,126,137)(75,127,138)(76,128,139)(77,129,140)(78,130,141)(79,131,142)(80,132,143)(81,133,144)(82,134,145)(83,135,146)(84,136,147)(85,120,148), (1,71,20)(2,72,21)(3,73,22)(4,74,23)(5,75,24)(6,76,25)(7,77,26)(8,78,27)(9,79,28)(10,80,29)(11,81,30)(12,82,31)(13,83,32)(14,84,33)(15,85,34)(16,69,18)(17,70,19)(35,87,142)(36,88,143)(37,89,144)(38,90,145)(39,91,146)(40,92,147)(41,93,148)(42,94,149)(43,95,150)(44,96,151)(45,97,152)(46,98,153)(47,99,137)(48,100,138)(49,101,139)(50,102,140)(51,86,141)(52,133,110)(53,134,111)(54,135,112)(55,136,113)(56,120,114)(57,121,115)(58,122,116)(59,123,117)(60,124,118)(61,125,119)(62,126,103)(63,127,104)(64,128,105)(65,129,106)(66,130,107)(67,131,108)(68,132,109), (35,142,87)(36,143,88)(37,144,89)(38,145,90)(39,146,91)(40,147,92)(41,148,93)(42,149,94)(43,150,95)(44,151,96)(45,152,97)(46,153,98)(47,137,99)(48,138,100)(49,139,101)(50,140,102)(51,141,86)(52,133,110)(53,134,111)(54,135,112)(55,136,113)(56,120,114)(57,121,115)(58,122,116)(59,123,117)(60,124,118)(61,125,119)(62,126,103)(63,127,104)(64,128,105)(65,129,106)(66,130,107)(67,131,108)(68,132,109)>;
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), (1,59,96)(2,60,97)(3,61,98)(4,62,99)(5,63,100)(6,64,101)(7,65,102)(8,66,86)(9,67,87)(10,68,88)(11,52,89)(12,53,90)(13,54,91)(14,55,92)(15,56,93)(16,57,94)(17,58,95)(18,115,42)(19,116,43)(20,117,44)(21,118,45)(22,119,46)(23,103,47)(24,104,48)(25,105,49)(26,106,50)(27,107,51)(28,108,35)(29,109,36)(30,110,37)(31,111,38)(32,112,39)(33,113,40)(34,114,41)(69,121,149)(70,122,150)(71,123,151)(72,124,152)(73,125,153)(74,126,137)(75,127,138)(76,128,139)(77,129,140)(78,130,141)(79,131,142)(80,132,143)(81,133,144)(82,134,145)(83,135,146)(84,136,147)(85,120,148), (1,71,20)(2,72,21)(3,73,22)(4,74,23)(5,75,24)(6,76,25)(7,77,26)(8,78,27)(9,79,28)(10,80,29)(11,81,30)(12,82,31)(13,83,32)(14,84,33)(15,85,34)(16,69,18)(17,70,19)(35,87,142)(36,88,143)(37,89,144)(38,90,145)(39,91,146)(40,92,147)(41,93,148)(42,94,149)(43,95,150)(44,96,151)(45,97,152)(46,98,153)(47,99,137)(48,100,138)(49,101,139)(50,102,140)(51,86,141)(52,133,110)(53,134,111)(54,135,112)(55,136,113)(56,120,114)(57,121,115)(58,122,116)(59,123,117)(60,124,118)(61,125,119)(62,126,103)(63,127,104)(64,128,105)(65,129,106)(66,130,107)(67,131,108)(68,132,109), (35,142,87)(36,143,88)(37,144,89)(38,145,90)(39,146,91)(40,147,92)(41,148,93)(42,149,94)(43,150,95)(44,151,96)(45,152,97)(46,153,98)(47,137,99)(48,138,100)(49,139,101)(50,140,102)(51,141,86)(52,133,110)(53,134,111)(54,135,112)(55,136,113)(56,120,114)(57,121,115)(58,122,116)(59,123,117)(60,124,118)(61,125,119)(62,126,103)(63,127,104)(64,128,105)(65,129,106)(66,130,107)(67,131,108)(68,132,109) );
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)], [(1,59,96),(2,60,97),(3,61,98),(4,62,99),(5,63,100),(6,64,101),(7,65,102),(8,66,86),(9,67,87),(10,68,88),(11,52,89),(12,53,90),(13,54,91),(14,55,92),(15,56,93),(16,57,94),(17,58,95),(18,115,42),(19,116,43),(20,117,44),(21,118,45),(22,119,46),(23,103,47),(24,104,48),(25,105,49),(26,106,50),(27,107,51),(28,108,35),(29,109,36),(30,110,37),(31,111,38),(32,112,39),(33,113,40),(34,114,41),(69,121,149),(70,122,150),(71,123,151),(72,124,152),(73,125,153),(74,126,137),(75,127,138),(76,128,139),(77,129,140),(78,130,141),(79,131,142),(80,132,143),(81,133,144),(82,134,145),(83,135,146),(84,136,147),(85,120,148)], [(1,71,20),(2,72,21),(3,73,22),(4,74,23),(5,75,24),(6,76,25),(7,77,26),(8,78,27),(9,79,28),(10,80,29),(11,81,30),(12,82,31),(13,83,32),(14,84,33),(15,85,34),(16,69,18),(17,70,19),(35,87,142),(36,88,143),(37,89,144),(38,90,145),(39,91,146),(40,92,147),(41,93,148),(42,94,149),(43,95,150),(44,96,151),(45,97,152),(46,98,153),(47,99,137),(48,100,138),(49,101,139),(50,102,140),(51,86,141),(52,133,110),(53,134,111),(54,135,112),(55,136,113),(56,120,114),(57,121,115),(58,122,116),(59,123,117),(60,124,118),(61,125,119),(62,126,103),(63,127,104),(64,128,105),(65,129,106),(66,130,107),(67,131,108),(68,132,109)], [(35,142,87),(36,143,88),(37,144,89),(38,145,90),(39,146,91),(40,147,92),(41,148,93),(42,149,94),(43,150,95),(44,151,96),(45,152,97),(46,153,98),(47,137,99),(48,138,100),(49,139,101),(50,140,102),(51,141,86),(52,133,110),(53,134,111),(54,135,112),(55,136,113),(56,120,114),(57,121,115),(58,122,116),(59,123,117),(60,124,118),(61,125,119),(62,126,103),(63,127,104),(64,128,105),(65,129,106),(66,130,107),(67,131,108),(68,132,109)]])
187 conjugacy classes
| class | 1 | 3A | 3B | 3C | ··· | 3J | 17A | ··· | 17P | 51A | ··· | 51AF | 51AG | ··· | 51FD |
| order | 1 | 3 | 3 | 3 | ··· | 3 | 17 | ··· | 17 | 51 | ··· | 51 | 51 | ··· | 51 |
| size | 1 | 1 | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 |
187 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C17 | C51 | He3 | C17×He3 |
| kernel | C17×He3 | C3×C51 | He3 | C32 | C17 | C1 |
| # reps | 1 | 8 | 16 | 128 | 2 | 32 |
Matrix representation of C17×He3 ►in GL3(𝔽103) generated by
| 93 | 0 | 0 |
| 0 | 93 | 0 |
| 0 | 0 | 93 |
| 38 | 45 | 0 |
| 7 | 65 | 1 |
| 7 | 95 | 0 |
| 46 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 0 | 46 |
| 1 | 0 | 0 |
| 35 | 56 | 0 |
| 8 | 0 | 46 |
G:=sub<GL(3,GF(103))| [93,0,0,0,93,0,0,0,93],[38,7,7,45,65,95,0,1,0],[46,0,0,0,46,0,0,0,46],[1,35,8,0,56,0,0,0,46] >;
C17×He3 in GAP, Magma, Sage, TeX
C_{17}\times {\rm He}_3 % in TeX
G:=Group("C17xHe3"); // GroupNames label
G:=SmallGroup(459,3);
// by ID
G=gap.SmallGroup(459,3);
# by ID
G:=PCGroup([4,-3,-3,-17,-3,1249]);
// Polycyclic
G:=Group<a,b,c,d|a^17=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations
Export