direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C17⋊D4, C51⋊8D4, D34⋊2C6, Dic17⋊C6, C6.17D34, C102.17C22, C17⋊2(C3×D4), (C2×C34)⋊4C6, (C2×C6)⋊1D17, (C2×C102)⋊4C2, (C6×D17)⋊5C2, C34.5(C2×C6), C2.5(C6×D17), C22⋊2(C3×D17), (C3×Dic17)⋊4C2, SmallGroup(408,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C17⋊D4
G = < a,b,c,d | a3=b17=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Smallest permutation representation of C3×C17⋊D4
►On 204 points
Generators in S204
(1 69 35)(2 70 36)(3 71 37)(4 72 38)(5 73 39)(6 74 40)(7 75 41)(8 76 42)(9 77 43)(10 78 44)(11 79 45)(12 80 46)(13 81 47)(14 82 48)(15 83 49)(16 84 50)(17 85 51)(18 86 52)(19 87 53)(20 88 54)(21 89 55)(22 90 56)(23 91 57)(24 92 58)(25 93 59)(26 94 60)(27 95 61)(28 96 62)(29 97 63)(30 98 64)(31 99 65)(32 100 66)(33 101 67)(34 102 68)(103 171 137)(104 172 138)(105 173 139)(106 174 140)(107 175 141)(108 176 142)(109 177 143)(110 178 144)(111 179 145)(112 180 146)(113 181 147)(114 182 148)(115 183 149)(116 184 150)(117 185 151)(118 186 152)(119 187 153)(120 188 154)(121 189 155)(122 190 156)(123 191 157)(124 192 158)(125 193 159)(126 194 160)(127 195 161)(128 196 162)(129 197 163)(130 198 164)(131 199 165)(132 200 166)(133 201 167)(134 202 168)(135 203 169)(136 204 170)
(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)(154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170)(171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187)(188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204)
(1 103 18 120)(2 119 19 136)(3 118 20 135)(4 117 21 134)(5 116 22 133)(6 115 23 132)(7 114 24 131)(8 113 25 130)(9 112 26 129)(10 111 27 128)(11 110 28 127)(12 109 29 126)(13 108 30 125)(14 107 31 124)(15 106 32 123)(16 105 33 122)(17 104 34 121)(35 137 52 154)(36 153 53 170)(37 152 54 169)(38 151 55 168)(39 150 56 167)(40 149 57 166)(41 148 58 165)(42 147 59 164)(43 146 60 163)(44 145 61 162)(45 144 62 161)(46 143 63 160)(47 142 64 159)(48 141 65 158)(49 140 66 157)(50 139 67 156)(51 138 68 155)(69 171 86 188)(70 187 87 204)(71 186 88 203)(72 185 89 202)(73 184 90 201)(74 183 91 200)(75 182 92 199)(76 181 93 198)(77 180 94 197)(78 179 95 196)(79 178 96 195)(80 177 97 194)(81 176 98 193)(82 175 99 192)(83 174 100 191)(84 173 101 190)(85 172 102 189)
(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(36 51)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)(70 85)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(87 102)(88 101)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)(103 120)(104 136)(105 135)(106 134)(107 133)(108 132)(109 131)(110 130)(111 129)(112 128)(113 127)(114 126)(115 125)(116 124)(117 123)(118 122)(119 121)(137 154)(138 170)(139 169)(140 168)(141 167)(142 166)(143 165)(144 164)(145 163)(146 162)(147 161)(148 160)(149 159)(150 158)(151 157)(152 156)(153 155)(171 188)(172 204)(173 203)(174 202)(175 201)(176 200)(177 199)(178 198)(179 197)(180 196)(181 195)(182 194)(183 193)(184 192)(185 191)(186 190)(187 189)
G:=sub<Sym(204)| (1,69,35)(2,70,36)(3,71,37)(4,72,38)(5,73,39)(6,74,40)(7,75,41)(8,76,42)(9,77,43)(10,78,44)(11,79,45)(12,80,46)(13,81,47)(14,82,48)(15,83,49)(16,84,50)(17,85,51)(18,86,52)(19,87,53)(20,88,54)(21,89,55)(22,90,56)(23,91,57)(24,92,58)(25,93,59)(26,94,60)(27,95,61)(28,96,62)(29,97,63)(30,98,64)(31,99,65)(32,100,66)(33,101,67)(34,102,68)(103,171,137)(104,172,138)(105,173,139)(106,174,140)(107,175,141)(108,176,142)(109,177,143)(110,178,144)(111,179,145)(112,180,146)(113,181,147)(114,182,148)(115,183,149)(116,184,150)(117,185,151)(118,186,152)(119,187,153)(120,188,154)(121,189,155)(122,190,156)(123,191,157)(124,192,158)(125,193,159)(126,194,160)(127,195,161)(128,196,162)(129,197,163)(130,198,164)(131,199,165)(132,200,166)(133,201,167)(134,202,168)(135,203,169)(136,204,170), (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)(154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170)(171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187)(188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204), (1,103,18,120)(2,119,19,136)(3,118,20,135)(4,117,21,134)(5,116,22,133)(6,115,23,132)(7,114,24,131)(8,113,25,130)(9,112,26,129)(10,111,27,128)(11,110,28,127)(12,109,29,126)(13,108,30,125)(14,107,31,124)(15,106,32,123)(16,105,33,122)(17,104,34,121)(35,137,52,154)(36,153,53,170)(37,152,54,169)(38,151,55,168)(39,150,56,167)(40,149,57,166)(41,148,58,165)(42,147,59,164)(43,146,60,163)(44,145,61,162)(45,144,62,161)(46,143,63,160)(47,142,64,159)(48,141,65,158)(49,140,66,157)(50,139,67,156)(51,138,68,155)(69,171,86,188)(70,187,87,204)(71,186,88,203)(72,185,89,202)(73,184,90,201)(74,183,91,200)(75,182,92,199)(76,181,93,198)(77,180,94,197)(78,179,95,196)(79,178,96,195)(80,177,97,194)(81,176,98,193)(82,175,99,192)(83,174,100,191)(84,173,101,190)(85,172,102,189), (2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)(103,120)(104,136)(105,135)(106,134)(107,133)(108,132)(109,131)(110,130)(111,129)(112,128)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(137,154)(138,170)(139,169)(140,168)(141,167)(142,166)(143,165)(144,164)(145,163)(146,162)(147,161)(148,160)(149,159)(150,158)(151,157)(152,156)(153,155)(171,188)(172,204)(173,203)(174,202)(175,201)(176,200)(177,199)(178,198)(179,197)(180,196)(181,195)(182,194)(183,193)(184,192)(185,191)(186,190)(187,189)>;
G:=Group( (1,69,35)(2,70,36)(3,71,37)(4,72,38)(5,73,39)(6,74,40)(7,75,41)(8,76,42)(9,77,43)(10,78,44)(11,79,45)(12,80,46)(13,81,47)(14,82,48)(15,83,49)(16,84,50)(17,85,51)(18,86,52)(19,87,53)(20,88,54)(21,89,55)(22,90,56)(23,91,57)(24,92,58)(25,93,59)(26,94,60)(27,95,61)(28,96,62)(29,97,63)(30,98,64)(31,99,65)(32,100,66)(33,101,67)(34,102,68)(103,171,137)(104,172,138)(105,173,139)(106,174,140)(107,175,141)(108,176,142)(109,177,143)(110,178,144)(111,179,145)(112,180,146)(113,181,147)(114,182,148)(115,183,149)(116,184,150)(117,185,151)(118,186,152)(119,187,153)(120,188,154)(121,189,155)(122,190,156)(123,191,157)(124,192,158)(125,193,159)(126,194,160)(127,195,161)(128,196,162)(129,197,163)(130,198,164)(131,199,165)(132,200,166)(133,201,167)(134,202,168)(135,203,169)(136,204,170), (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)(154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170)(171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187)(188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204), (1,103,18,120)(2,119,19,136)(3,118,20,135)(4,117,21,134)(5,116,22,133)(6,115,23,132)(7,114,24,131)(8,113,25,130)(9,112,26,129)(10,111,27,128)(11,110,28,127)(12,109,29,126)(13,108,30,125)(14,107,31,124)(15,106,32,123)(16,105,33,122)(17,104,34,121)(35,137,52,154)(36,153,53,170)(37,152,54,169)(38,151,55,168)(39,150,56,167)(40,149,57,166)(41,148,58,165)(42,147,59,164)(43,146,60,163)(44,145,61,162)(45,144,62,161)(46,143,63,160)(47,142,64,159)(48,141,65,158)(49,140,66,157)(50,139,67,156)(51,138,68,155)(69,171,86,188)(70,187,87,204)(71,186,88,203)(72,185,89,202)(73,184,90,201)(74,183,91,200)(75,182,92,199)(76,181,93,198)(77,180,94,197)(78,179,95,196)(79,178,96,195)(80,177,97,194)(81,176,98,193)(82,175,99,192)(83,174,100,191)(84,173,101,190)(85,172,102,189), (2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)(103,120)(104,136)(105,135)(106,134)(107,133)(108,132)(109,131)(110,130)(111,129)(112,128)(113,127)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(137,154)(138,170)(139,169)(140,168)(141,167)(142,166)(143,165)(144,164)(145,163)(146,162)(147,161)(148,160)(149,159)(150,158)(151,157)(152,156)(153,155)(171,188)(172,204)(173,203)(174,202)(175,201)(176,200)(177,199)(178,198)(179,197)(180,196)(181,195)(182,194)(183,193)(184,192)(185,191)(186,190)(187,189) );
G=PermutationGroup([[(1,69,35),(2,70,36),(3,71,37),(4,72,38),(5,73,39),(6,74,40),(7,75,41),(8,76,42),(9,77,43),(10,78,44),(11,79,45),(12,80,46),(13,81,47),(14,82,48),(15,83,49),(16,84,50),(17,85,51),(18,86,52),(19,87,53),(20,88,54),(21,89,55),(22,90,56),(23,91,57),(24,92,58),(25,93,59),(26,94,60),(27,95,61),(28,96,62),(29,97,63),(30,98,64),(31,99,65),(32,100,66),(33,101,67),(34,102,68),(103,171,137),(104,172,138),(105,173,139),(106,174,140),(107,175,141),(108,176,142),(109,177,143),(110,178,144),(111,179,145),(112,180,146),(113,181,147),(114,182,148),(115,183,149),(116,184,150),(117,185,151),(118,186,152),(119,187,153),(120,188,154),(121,189,155),(122,190,156),(123,191,157),(124,192,158),(125,193,159),(126,194,160),(127,195,161),(128,196,162),(129,197,163),(130,198,164),(131,199,165),(132,200,166),(133,201,167),(134,202,168),(135,203,169),(136,204,170)], [(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),(154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170),(171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187),(188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204)], [(1,103,18,120),(2,119,19,136),(3,118,20,135),(4,117,21,134),(5,116,22,133),(6,115,23,132),(7,114,24,131),(8,113,25,130),(9,112,26,129),(10,111,27,128),(11,110,28,127),(12,109,29,126),(13,108,30,125),(14,107,31,124),(15,106,32,123),(16,105,33,122),(17,104,34,121),(35,137,52,154),(36,153,53,170),(37,152,54,169),(38,151,55,168),(39,150,56,167),(40,149,57,166),(41,148,58,165),(42,147,59,164),(43,146,60,163),(44,145,61,162),(45,144,62,161),(46,143,63,160),(47,142,64,159),(48,141,65,158),(49,140,66,157),(50,139,67,156),(51,138,68,155),(69,171,86,188),(70,187,87,204),(71,186,88,203),(72,185,89,202),(73,184,90,201),(74,183,91,200),(75,182,92,199),(76,181,93,198),(77,180,94,197),(78,179,95,196),(79,178,96,195),(80,177,97,194),(81,176,98,193),(82,175,99,192),(83,174,100,191),(84,173,101,190),(85,172,102,189)], [(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27),(36,51),(37,50),(38,49),(39,48),(40,47),(41,46),(42,45),(43,44),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61),(70,85),(71,84),(72,83),(73,82),(74,81),(75,80),(76,79),(77,78),(87,102),(88,101),(89,100),(90,99),(91,98),(92,97),(93,96),(94,95),(103,120),(104,136),(105,135),(106,134),(107,133),(108,132),(109,131),(110,130),(111,129),(112,128),(113,127),(114,126),(115,125),(116,124),(117,123),(118,122),(119,121),(137,154),(138,170),(139,169),(140,168),(141,167),(142,166),(143,165),(144,164),(145,163),(146,162),(147,161),(148,160),(149,159),(150,158),(151,157),(152,156),(153,155),(171,188),(172,204),(173,203),(174,202),(175,201),(176,200),(177,199),(178,198),(179,197),(180,196),(181,195),(182,194),(183,193),(184,192),(185,191),(186,190),(187,189)]])
111 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 17A | ··· | 17H | 34A | ··· | 34X | 51A | ··· | 51P | 102A | ··· | 102AV |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 102 | ··· | 102 |
| size | 1 | 1 | 2 | 34 | 1 | 1 | 34 | 1 | 1 | 2 | 2 | 34 | 34 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
111 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | C3×D4 | D17 | D34 | C3×D17 | C17⋊D4 | C6×D17 | C3×C17⋊D4 |
| kernel | C3×C17⋊D4 | C3×Dic17 | C6×D17 | C2×C102 | C17⋊D4 | Dic17 | D34 | C2×C34 | C51 | C17 | C2×C6 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 8 | 8 | 16 | 16 | 16 | 32 |
Matrix representation of C3×C17⋊D4 ►in GL4(𝔽409) generated by
| 53 | 0 | 0 | 0 |
| 0 | 53 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 292 | 408 | 0 | 0 |
| 47 | 322 | 0 | 0 |
| 0 | 0 | 117 | 4 |
| 0 | 0 | 1 | 35 |
| 264 | 205 | 0 | 0 |
| 79 | 145 | 0 | 0 |
| 0 | 0 | 277 | 406 |
| 0 | 0 | 355 | 132 |
| 145 | 204 | 0 | 0 |
| 330 | 264 | 0 | 0 |
| 0 | 0 | 395 | 59 |
| 0 | 0 | 170 | 14 |
G:=sub<GL(4,GF(409))| [53,0,0,0,0,53,0,0,0,0,1,0,0,0,0,1],[292,47,0,0,408,322,0,0,0,0,117,1,0,0,4,35],[264,79,0,0,205,145,0,0,0,0,277,355,0,0,406,132],[145,330,0,0,204,264,0,0,0,0,395,170,0,0,59,14] >;
C3×C17⋊D4 in GAP, Magma, Sage, TeX
C_3\times C_{17}\rtimes D_4 % in TeX
G:=Group("C3xC17:D4"); // GroupNames label
G:=SmallGroup(408,19);
// by ID
G=gap.SmallGroup(408,19);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-17,141,9604]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^17=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export