direct product, metabelian, supersoluble, monomial, A-group
Aliases: C17×C3⋊S3, C51⋊3S3, C32⋊2C34, C3⋊(S3×C17), (C3×C51)⋊5C2, SmallGroup(306,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C17×C3⋊S3 |
Generators and relations for C17×C3⋊S3
G = < a,b,c,d | a17=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Smallest permutation representation of C17×C3⋊S3
►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 88 74)(2 89 75)(3 90 76)(4 91 77)(5 92 78)(6 93 79)(7 94 80)(8 95 81)(9 96 82)(10 97 83)(11 98 84)(12 99 85)(13 100 69)(14 101 70)(15 102 71)(16 86 72)(17 87 73)(18 122 117)(19 123 118)(20 124 119)(21 125 103)(22 126 104)(23 127 105)(24 128 106)(25 129 107)(26 130 108)(27 131 109)(28 132 110)(29 133 111)(30 134 112)(31 135 113)(32 136 114)(33 120 115)(34 121 116)(35 52 142)(36 53 143)(37 54 144)(38 55 145)(39 56 146)(40 57 147)(41 58 148)(42 59 149)(43 60 150)(44 61 151)(45 62 152)(46 63 153)(47 64 137)(48 65 138)(49 66 139)(50 67 140)(51 68 141)
(1 147 18)(2 148 19)(3 149 20)(4 150 21)(5 151 22)(6 152 23)(7 153 24)(8 137 25)(9 138 26)(10 139 27)(11 140 28)(12 141 29)(13 142 30)(14 143 31)(15 144 32)(16 145 33)(17 146 34)(35 134 100)(36 135 101)(37 136 102)(38 120 86)(39 121 87)(40 122 88)(41 123 89)(42 124 90)(43 125 91)(44 126 92)(45 127 93)(46 128 94)(47 129 95)(48 130 96)(49 131 97)(50 132 98)(51 133 99)(52 112 69)(53 113 70)(54 114 71)(55 115 72)(56 116 73)(57 117 74)(58 118 75)(59 119 76)(60 103 77)(61 104 78)(62 105 79)(63 106 80)(64 107 81)(65 108 82)(66 109 83)(67 110 84)(68 111 85)
(18 147)(19 148)(20 149)(21 150)(22 151)(23 152)(24 153)(25 137)(26 138)(27 139)(28 140)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 118)(42 119)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 134)(53 135)(54 136)(55 120)(56 121)(57 122)(58 123)(59 124)(60 125)(61 126)(62 127)(63 128)(64 129)(65 130)(66 131)(67 132)(68 133)(69 100)(70 101)(71 102)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(85 99)
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), (1,88,74)(2,89,75)(3,90,76)(4,91,77)(5,92,78)(6,93,79)(7,94,80)(8,95,81)(9,96,82)(10,97,83)(11,98,84)(12,99,85)(13,100,69)(14,101,70)(15,102,71)(16,86,72)(17,87,73)(18,122,117)(19,123,118)(20,124,119)(21,125,103)(22,126,104)(23,127,105)(24,128,106)(25,129,107)(26,130,108)(27,131,109)(28,132,110)(29,133,111)(30,134,112)(31,135,113)(32,136,114)(33,120,115)(34,121,116)(35,52,142)(36,53,143)(37,54,144)(38,55,145)(39,56,146)(40,57,147)(41,58,148)(42,59,149)(43,60,150)(44,61,151)(45,62,152)(46,63,153)(47,64,137)(48,65,138)(49,66,139)(50,67,140)(51,68,141), (1,147,18)(2,148,19)(3,149,20)(4,150,21)(5,151,22)(6,152,23)(7,153,24)(8,137,25)(9,138,26)(10,139,27)(11,140,28)(12,141,29)(13,142,30)(14,143,31)(15,144,32)(16,145,33)(17,146,34)(35,134,100)(36,135,101)(37,136,102)(38,120,86)(39,121,87)(40,122,88)(41,123,89)(42,124,90)(43,125,91)(44,126,92)(45,127,93)(46,128,94)(47,129,95)(48,130,96)(49,131,97)(50,132,98)(51,133,99)(52,112,69)(53,113,70)(54,114,71)(55,115,72)(56,116,73)(57,117,74)(58,118,75)(59,119,76)(60,103,77)(61,104,78)(62,105,79)(63,106,80)(64,107,81)(65,108,82)(66,109,83)(67,110,84)(68,111,85), (18,147)(19,148)(20,149)(21,150)(22,151)(23,152)(24,153)(25,137)(26,138)(27,139)(28,140)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,118)(42,119)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,134)(53,135)(54,136)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,100)(70,101)(71,102)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(85,99)>;
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), (1,88,74)(2,89,75)(3,90,76)(4,91,77)(5,92,78)(6,93,79)(7,94,80)(8,95,81)(9,96,82)(10,97,83)(11,98,84)(12,99,85)(13,100,69)(14,101,70)(15,102,71)(16,86,72)(17,87,73)(18,122,117)(19,123,118)(20,124,119)(21,125,103)(22,126,104)(23,127,105)(24,128,106)(25,129,107)(26,130,108)(27,131,109)(28,132,110)(29,133,111)(30,134,112)(31,135,113)(32,136,114)(33,120,115)(34,121,116)(35,52,142)(36,53,143)(37,54,144)(38,55,145)(39,56,146)(40,57,147)(41,58,148)(42,59,149)(43,60,150)(44,61,151)(45,62,152)(46,63,153)(47,64,137)(48,65,138)(49,66,139)(50,67,140)(51,68,141), (1,147,18)(2,148,19)(3,149,20)(4,150,21)(5,151,22)(6,152,23)(7,153,24)(8,137,25)(9,138,26)(10,139,27)(11,140,28)(12,141,29)(13,142,30)(14,143,31)(15,144,32)(16,145,33)(17,146,34)(35,134,100)(36,135,101)(37,136,102)(38,120,86)(39,121,87)(40,122,88)(41,123,89)(42,124,90)(43,125,91)(44,126,92)(45,127,93)(46,128,94)(47,129,95)(48,130,96)(49,131,97)(50,132,98)(51,133,99)(52,112,69)(53,113,70)(54,114,71)(55,115,72)(56,116,73)(57,117,74)(58,118,75)(59,119,76)(60,103,77)(61,104,78)(62,105,79)(63,106,80)(64,107,81)(65,108,82)(66,109,83)(67,110,84)(68,111,85), (18,147)(19,148)(20,149)(21,150)(22,151)(23,152)(24,153)(25,137)(26,138)(27,139)(28,140)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,112)(36,113)(37,114)(38,115)(39,116)(40,117)(41,118)(42,119)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,134)(53,135)(54,136)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,100)(70,101)(71,102)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(85,99) );
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)], [(1,88,74),(2,89,75),(3,90,76),(4,91,77),(5,92,78),(6,93,79),(7,94,80),(8,95,81),(9,96,82),(10,97,83),(11,98,84),(12,99,85),(13,100,69),(14,101,70),(15,102,71),(16,86,72),(17,87,73),(18,122,117),(19,123,118),(20,124,119),(21,125,103),(22,126,104),(23,127,105),(24,128,106),(25,129,107),(26,130,108),(27,131,109),(28,132,110),(29,133,111),(30,134,112),(31,135,113),(32,136,114),(33,120,115),(34,121,116),(35,52,142),(36,53,143),(37,54,144),(38,55,145),(39,56,146),(40,57,147),(41,58,148),(42,59,149),(43,60,150),(44,61,151),(45,62,152),(46,63,153),(47,64,137),(48,65,138),(49,66,139),(50,67,140),(51,68,141)], [(1,147,18),(2,148,19),(3,149,20),(4,150,21),(5,151,22),(6,152,23),(7,153,24),(8,137,25),(9,138,26),(10,139,27),(11,140,28),(12,141,29),(13,142,30),(14,143,31),(15,144,32),(16,145,33),(17,146,34),(35,134,100),(36,135,101),(37,136,102),(38,120,86),(39,121,87),(40,122,88),(41,123,89),(42,124,90),(43,125,91),(44,126,92),(45,127,93),(46,128,94),(47,129,95),(48,130,96),(49,131,97),(50,132,98),(51,133,99),(52,112,69),(53,113,70),(54,114,71),(55,115,72),(56,116,73),(57,117,74),(58,118,75),(59,119,76),(60,103,77),(61,104,78),(62,105,79),(63,106,80),(64,107,81),(65,108,82),(66,109,83),(67,110,84),(68,111,85)], [(18,147),(19,148),(20,149),(21,150),(22,151),(23,152),(24,153),(25,137),(26,138),(27,139),(28,140),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,112),(36,113),(37,114),(38,115),(39,116),(40,117),(41,118),(42,119),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,134),(53,135),(54,136),(55,120),(56,121),(57,122),(58,123),(59,124),(60,125),(61,126),(62,127),(63,128),(64,129),(65,130),(66,131),(67,132),(68,133),(69,100),(70,101),(71,102),(72,86),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,97),(84,98),(85,99)]])
102 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 17A | ··· | 17P | 34A | ··· | 34P | 51A | ··· | 51BL |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 |
| size | 1 | 9 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C17 | C34 | S3 | S3×C17 |
| kernel | C17×C3⋊S3 | C3×C51 | C3⋊S3 | C32 | C51 | C3 |
| # reps | 1 | 1 | 16 | 16 | 4 | 64 |
Matrix representation of C17×C3⋊S3 ►in GL4(𝔽103) generated by
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 102 | 1 |
| 0 | 0 | 102 | 0 |
| 102 | 102 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 102 |
| 0 | 0 | 1 | 102 |
| 1 | 0 | 0 | 0 |
| 102 | 102 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(103))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,102,102,0,0,1,0],[102,1,0,0,102,0,0,0,0,0,0,1,0,0,102,102],[1,102,0,0,0,102,0,0,0,0,0,1,0,0,1,0] >;
C17×C3⋊S3 in GAP, Magma, Sage, TeX
C_{17}\times C_3\rtimes S_3 % in TeX
G:=Group("C17xC3:S3"); // GroupNames label
G:=SmallGroup(306,8);
// by ID
G=gap.SmallGroup(306,8);
# by ID
G:=PCGroup([4,-2,-17,-3,-3,818,3267]);
// Polycyclic
G:=Group<a,b,c,d|a^17=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
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