direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C17×C3⋊D4, C51⋊9D4, D6⋊2C34, Dic3⋊C34, C34.17D6, C102.22C22, C3⋊2(D4×C17), (C2×C6)⋊2C34, (C2×C34)⋊3S3, (S3×C34)⋊5C2, (C2×C102)⋊6C2, C2.5(S3×C34), C6.5(C2×C34), C22⋊2(S3×C17), (Dic3×C17)⋊4C2, SmallGroup(408,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C17×C3⋊D4
G = < a,b,c,d | a17=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Smallest permutation representation of C17×C3⋊D4
►On 204 points
Generators in S204
(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)
(1 114 70)(2 115 71)(3 116 72)(4 117 73)(5 118 74)(6 119 75)(7 103 76)(8 104 77)(9 105 78)(10 106 79)(11 107 80)(12 108 81)(13 109 82)(14 110 83)(15 111 84)(16 112 85)(17 113 69)(18 170 189)(19 154 190)(20 155 191)(21 156 192)(22 157 193)(23 158 194)(24 159 195)(25 160 196)(26 161 197)(27 162 198)(28 163 199)(29 164 200)(30 165 201)(31 166 202)(32 167 203)(33 168 204)(34 169 188)(35 171 128)(36 172 129)(37 173 130)(38 174 131)(39 175 132)(40 176 133)(41 177 134)(42 178 135)(43 179 136)(44 180 120)(45 181 121)(46 182 122)(47 183 123)(48 184 124)(49 185 125)(50 186 126)(51 187 127)(52 149 88)(53 150 89)(54 151 90)(55 152 91)(56 153 92)(57 137 93)(58 138 94)(59 139 95)(60 140 96)(61 141 97)(62 142 98)(63 143 99)(64 144 100)(65 145 101)(66 146 102)(67 147 86)(68 148 87)
(1 180 194 66)(2 181 195 67)(3 182 196 68)(4 183 197 52)(5 184 198 53)(6 185 199 54)(7 186 200 55)(8 187 201 56)(9 171 202 57)(10 172 203 58)(11 173 204 59)(12 174 188 60)(13 175 189 61)(14 176 190 62)(15 177 191 63)(16 178 192 64)(17 179 193 65)(18 97 109 39)(19 98 110 40)(20 99 111 41)(21 100 112 42)(22 101 113 43)(23 102 114 44)(24 86 115 45)(25 87 116 46)(26 88 117 47)(27 89 118 48)(28 90 119 49)(29 91 103 50)(30 92 104 51)(31 93 105 35)(32 94 106 36)(33 95 107 37)(34 96 108 38)(69 136 157 145)(70 120 158 146)(71 121 159 147)(72 122 160 148)(73 123 161 149)(74 124 162 150)(75 125 163 151)(76 126 164 152)(77 127 165 153)(78 128 166 137)(79 129 167 138)(80 130 168 139)(81 131 169 140)(82 132 170 141)(83 133 154 142)(84 134 155 143)(85 135 156 144)
(18 170)(19 154)(20 155)(21 156)(22 157)(23 158)(24 159)(25 160)(26 161)(27 162)(28 163)(29 164)(30 165)(31 166)(32 167)(33 168)(34 169)(35 137)(36 138)(37 139)(38 140)(39 141)(40 142)(41 143)(42 144)(43 145)(44 146)(45 147)(46 148)(47 149)(48 150)(49 151)(50 152)(51 153)(52 183)(53 184)(54 185)(55 186)(56 187)(57 171)(58 172)(59 173)(60 174)(61 175)(62 176)(63 177)(64 178)(65 179)(66 180)(67 181)(68 182)(69 113)(70 114)(71 115)(72 116)(73 117)(74 118)(75 119)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)(82 109)(83 110)(84 111)(85 112)(86 121)(87 122)(88 123)(89 124)(90 125)(91 126)(92 127)(93 128)(94 129)(95 130)(96 131)(97 132)(98 133)(99 134)(100 135)(101 136)(102 120)
G:=sub<Sym(204)| (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), (1,114,70)(2,115,71)(3,116,72)(4,117,73)(5,118,74)(6,119,75)(7,103,76)(8,104,77)(9,105,78)(10,106,79)(11,107,80)(12,108,81)(13,109,82)(14,110,83)(15,111,84)(16,112,85)(17,113,69)(18,170,189)(19,154,190)(20,155,191)(21,156,192)(22,157,193)(23,158,194)(24,159,195)(25,160,196)(26,161,197)(27,162,198)(28,163,199)(29,164,200)(30,165,201)(31,166,202)(32,167,203)(33,168,204)(34,169,188)(35,171,128)(36,172,129)(37,173,130)(38,174,131)(39,175,132)(40,176,133)(41,177,134)(42,178,135)(43,179,136)(44,180,120)(45,181,121)(46,182,122)(47,183,123)(48,184,124)(49,185,125)(50,186,126)(51,187,127)(52,149,88)(53,150,89)(54,151,90)(55,152,91)(56,153,92)(57,137,93)(58,138,94)(59,139,95)(60,140,96)(61,141,97)(62,142,98)(63,143,99)(64,144,100)(65,145,101)(66,146,102)(67,147,86)(68,148,87), (1,180,194,66)(2,181,195,67)(3,182,196,68)(4,183,197,52)(5,184,198,53)(6,185,199,54)(7,186,200,55)(8,187,201,56)(9,171,202,57)(10,172,203,58)(11,173,204,59)(12,174,188,60)(13,175,189,61)(14,176,190,62)(15,177,191,63)(16,178,192,64)(17,179,193,65)(18,97,109,39)(19,98,110,40)(20,99,111,41)(21,100,112,42)(22,101,113,43)(23,102,114,44)(24,86,115,45)(25,87,116,46)(26,88,117,47)(27,89,118,48)(28,90,119,49)(29,91,103,50)(30,92,104,51)(31,93,105,35)(32,94,106,36)(33,95,107,37)(34,96,108,38)(69,136,157,145)(70,120,158,146)(71,121,159,147)(72,122,160,148)(73,123,161,149)(74,124,162,150)(75,125,163,151)(76,126,164,152)(77,127,165,153)(78,128,166,137)(79,129,167,138)(80,130,168,139)(81,131,169,140)(82,132,170,141)(83,133,154,142)(84,134,155,143)(85,135,156,144), (18,170)(19,154)(20,155)(21,156)(22,157)(23,158)(24,159)(25,160)(26,161)(27,162)(28,163)(29,164)(30,165)(31,166)(32,167)(33,168)(34,169)(35,137)(36,138)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,183)(53,184)(54,185)(55,186)(56,187)(57,171)(58,172)(59,173)(60,174)(61,175)(62,176)(63,177)(64,178)(65,179)(66,180)(67,181)(68,182)(69,113)(70,114)(71,115)(72,116)(73,117)(74,118)(75,119)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108)(82,109)(83,110)(84,111)(85,112)(86,121)(87,122)(88,123)(89,124)(90,125)(91,126)(92,127)(93,128)(94,129)(95,130)(96,131)(97,132)(98,133)(99,134)(100,135)(101,136)(102,120)>;
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)(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), (1,114,70)(2,115,71)(3,116,72)(4,117,73)(5,118,74)(6,119,75)(7,103,76)(8,104,77)(9,105,78)(10,106,79)(11,107,80)(12,108,81)(13,109,82)(14,110,83)(15,111,84)(16,112,85)(17,113,69)(18,170,189)(19,154,190)(20,155,191)(21,156,192)(22,157,193)(23,158,194)(24,159,195)(25,160,196)(26,161,197)(27,162,198)(28,163,199)(29,164,200)(30,165,201)(31,166,202)(32,167,203)(33,168,204)(34,169,188)(35,171,128)(36,172,129)(37,173,130)(38,174,131)(39,175,132)(40,176,133)(41,177,134)(42,178,135)(43,179,136)(44,180,120)(45,181,121)(46,182,122)(47,183,123)(48,184,124)(49,185,125)(50,186,126)(51,187,127)(52,149,88)(53,150,89)(54,151,90)(55,152,91)(56,153,92)(57,137,93)(58,138,94)(59,139,95)(60,140,96)(61,141,97)(62,142,98)(63,143,99)(64,144,100)(65,145,101)(66,146,102)(67,147,86)(68,148,87), (1,180,194,66)(2,181,195,67)(3,182,196,68)(4,183,197,52)(5,184,198,53)(6,185,199,54)(7,186,200,55)(8,187,201,56)(9,171,202,57)(10,172,203,58)(11,173,204,59)(12,174,188,60)(13,175,189,61)(14,176,190,62)(15,177,191,63)(16,178,192,64)(17,179,193,65)(18,97,109,39)(19,98,110,40)(20,99,111,41)(21,100,112,42)(22,101,113,43)(23,102,114,44)(24,86,115,45)(25,87,116,46)(26,88,117,47)(27,89,118,48)(28,90,119,49)(29,91,103,50)(30,92,104,51)(31,93,105,35)(32,94,106,36)(33,95,107,37)(34,96,108,38)(69,136,157,145)(70,120,158,146)(71,121,159,147)(72,122,160,148)(73,123,161,149)(74,124,162,150)(75,125,163,151)(76,126,164,152)(77,127,165,153)(78,128,166,137)(79,129,167,138)(80,130,168,139)(81,131,169,140)(82,132,170,141)(83,133,154,142)(84,134,155,143)(85,135,156,144), (18,170)(19,154)(20,155)(21,156)(22,157)(23,158)(24,159)(25,160)(26,161)(27,162)(28,163)(29,164)(30,165)(31,166)(32,167)(33,168)(34,169)(35,137)(36,138)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,183)(53,184)(54,185)(55,186)(56,187)(57,171)(58,172)(59,173)(60,174)(61,175)(62,176)(63,177)(64,178)(65,179)(66,180)(67,181)(68,182)(69,113)(70,114)(71,115)(72,116)(73,117)(74,118)(75,119)(76,103)(77,104)(78,105)(79,106)(80,107)(81,108)(82,109)(83,110)(84,111)(85,112)(86,121)(87,122)(88,123)(89,124)(90,125)(91,126)(92,127)(93,128)(94,129)(95,130)(96,131)(97,132)(98,133)(99,134)(100,135)(101,136)(102,120) );
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),(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)], [(1,114,70),(2,115,71),(3,116,72),(4,117,73),(5,118,74),(6,119,75),(7,103,76),(8,104,77),(9,105,78),(10,106,79),(11,107,80),(12,108,81),(13,109,82),(14,110,83),(15,111,84),(16,112,85),(17,113,69),(18,170,189),(19,154,190),(20,155,191),(21,156,192),(22,157,193),(23,158,194),(24,159,195),(25,160,196),(26,161,197),(27,162,198),(28,163,199),(29,164,200),(30,165,201),(31,166,202),(32,167,203),(33,168,204),(34,169,188),(35,171,128),(36,172,129),(37,173,130),(38,174,131),(39,175,132),(40,176,133),(41,177,134),(42,178,135),(43,179,136),(44,180,120),(45,181,121),(46,182,122),(47,183,123),(48,184,124),(49,185,125),(50,186,126),(51,187,127),(52,149,88),(53,150,89),(54,151,90),(55,152,91),(56,153,92),(57,137,93),(58,138,94),(59,139,95),(60,140,96),(61,141,97),(62,142,98),(63,143,99),(64,144,100),(65,145,101),(66,146,102),(67,147,86),(68,148,87)], [(1,180,194,66),(2,181,195,67),(3,182,196,68),(4,183,197,52),(5,184,198,53),(6,185,199,54),(7,186,200,55),(8,187,201,56),(9,171,202,57),(10,172,203,58),(11,173,204,59),(12,174,188,60),(13,175,189,61),(14,176,190,62),(15,177,191,63),(16,178,192,64),(17,179,193,65),(18,97,109,39),(19,98,110,40),(20,99,111,41),(21,100,112,42),(22,101,113,43),(23,102,114,44),(24,86,115,45),(25,87,116,46),(26,88,117,47),(27,89,118,48),(28,90,119,49),(29,91,103,50),(30,92,104,51),(31,93,105,35),(32,94,106,36),(33,95,107,37),(34,96,108,38),(69,136,157,145),(70,120,158,146),(71,121,159,147),(72,122,160,148),(73,123,161,149),(74,124,162,150),(75,125,163,151),(76,126,164,152),(77,127,165,153),(78,128,166,137),(79,129,167,138),(80,130,168,139),(81,131,169,140),(82,132,170,141),(83,133,154,142),(84,134,155,143),(85,135,156,144)], [(18,170),(19,154),(20,155),(21,156),(22,157),(23,158),(24,159),(25,160),(26,161),(27,162),(28,163),(29,164),(30,165),(31,166),(32,167),(33,168),(34,169),(35,137),(36,138),(37,139),(38,140),(39,141),(40,142),(41,143),(42,144),(43,145),(44,146),(45,147),(46,148),(47,149),(48,150),(49,151),(50,152),(51,153),(52,183),(53,184),(54,185),(55,186),(56,187),(57,171),(58,172),(59,173),(60,174),(61,175),(62,176),(63,177),(64,178),(65,179),(66,180),(67,181),(68,182),(69,113),(70,114),(71,115),(72,116),(73,117),(74,118),(75,119),(76,103),(77,104),(78,105),(79,106),(80,107),(81,108),(82,109),(83,110),(84,111),(85,112),(86,121),(87,122),(88,123),(89,124),(90,125),(91,126),(92,127),(93,128),(94,129),(95,130),(96,131),(97,132),(98,133),(99,134),(100,135),(101,136),(102,120)]])
153 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6A | 6B | 6C | 17A | ··· | 17P | 34A | ··· | 34P | 34Q | ··· | 34AF | 34AG | ··· | 34AV | 51A | ··· | 51P | 68A | ··· | 68P | 102A | ··· | 102AV |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 6 | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 |
| size | 1 | 1 | 2 | 6 | 2 | 6 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
153 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 | C17 | C34 | C34 | C34 | S3 | D4 | D6 | C3⋊D4 | S3×C17 | D4×C17 | S3×C34 | C17×C3⋊D4 |
| kernel | C17×C3⋊D4 | Dic3×C17 | S3×C34 | C2×C102 | C3⋊D4 | Dic3 | D6 | C2×C6 | C2×C34 | C51 | C34 | C17 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 16 | 16 | 16 | 16 | 1 | 1 | 1 | 2 | 16 | 16 | 16 | 32 |
Matrix representation of C17×C3⋊D4 ►in GL2(𝔽409) generated by
| 150 | 0 |
| 0 | 150 |
| 408 | 408 |
| 1 | 0 |
| 237 | 65 |
| 237 | 172 |
| 1 | 0 |
| 408 | 408 |
G:=sub<GL(2,GF(409))| [150,0,0,150],[408,1,408,0],[237,237,65,172],[1,408,0,408] >;
C17×C3⋊D4 in GAP, Magma, Sage, TeX
C_{17}\times C_3\rtimes D_4 % in TeX
G:=Group("C17xC3:D4"); // GroupNames label
G:=SmallGroup(408,24);
// by ID
G=gap.SmallGroup(408,24);
# by ID
G:=PCGroup([5,-2,-2,-17,-2,-3,701,6804]);
// Polycyclic
G:=Group<a,b,c,d|a^17=b^3=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