direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C17×D12, C51⋊6D4, C68⋊3S3, C12⋊1C34, C204⋊5C2, D6⋊1C34, C34.15D6, C102.20C22, C4⋊(S3×C17), C3⋊1(D4×C17), (S3×C34)⋊4C2, C2.4(S3×C34), C6.3(C2×C34), SmallGroup(408,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C17×D12
G = < a,b,c | a17=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C17×D12
►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)
(1 201 32 51 181 80 64 169 145 89 113 132)(2 202 33 35 182 81 65 170 146 90 114 133)(3 203 34 36 183 82 66 154 147 91 115 134)(4 204 18 37 184 83 67 155 148 92 116 135)(5 188 19 38 185 84 68 156 149 93 117 136)(6 189 20 39 186 85 52 157 150 94 118 120)(7 190 21 40 187 69 53 158 151 95 119 121)(8 191 22 41 171 70 54 159 152 96 103 122)(9 192 23 42 172 71 55 160 153 97 104 123)(10 193 24 43 173 72 56 161 137 98 105 124)(11 194 25 44 174 73 57 162 138 99 106 125)(12 195 26 45 175 74 58 163 139 100 107 126)(13 196 27 46 176 75 59 164 140 101 108 127)(14 197 28 47 177 76 60 165 141 102 109 128)(15 198 29 48 178 77 61 166 142 86 110 129)(16 199 30 49 179 78 62 167 143 87 111 130)(17 200 31 50 180 79 63 168 144 88 112 131)
(1 32)(2 33)(3 34)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 31)(35 133)(36 134)(37 135)(38 136)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 150)(53 151)(54 152)(55 153)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 145)(65 146)(66 147)(67 148)(68 149)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 102)(77 86)(78 87)(79 88)(80 89)(81 90)(82 91)(83 92)(84 93)(85 94)(103 171)(104 172)(105 173)(106 174)(107 175)(108 176)(109 177)(110 178)(111 179)(112 180)(113 181)(114 182)(115 183)(116 184)(117 185)(118 186)(119 187)
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), (1,201,32,51,181,80,64,169,145,89,113,132)(2,202,33,35,182,81,65,170,146,90,114,133)(3,203,34,36,183,82,66,154,147,91,115,134)(4,204,18,37,184,83,67,155,148,92,116,135)(5,188,19,38,185,84,68,156,149,93,117,136)(6,189,20,39,186,85,52,157,150,94,118,120)(7,190,21,40,187,69,53,158,151,95,119,121)(8,191,22,41,171,70,54,159,152,96,103,122)(9,192,23,42,172,71,55,160,153,97,104,123)(10,193,24,43,173,72,56,161,137,98,105,124)(11,194,25,44,174,73,57,162,138,99,106,125)(12,195,26,45,175,74,58,163,139,100,107,126)(13,196,27,46,176,75,59,164,140,101,108,127)(14,197,28,47,177,76,60,165,141,102,109,128)(15,198,29,48,178,77,61,166,142,86,110,129)(16,199,30,49,179,78,62,167,143,87,111,130)(17,200,31,50,180,79,63,168,144,88,112,131), (1,32)(2,33)(3,34)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(35,133)(36,134)(37,135)(38,136)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,150)(53,151)(54,152)(55,153)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,148)(68,149)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93)(85,94)(103,171)(104,172)(105,173)(106,174)(107,175)(108,176)(109,177)(110,178)(111,179)(112,180)(113,181)(114,182)(115,183)(116,184)(117,185)(118,186)(119,187)>;
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), (1,201,32,51,181,80,64,169,145,89,113,132)(2,202,33,35,182,81,65,170,146,90,114,133)(3,203,34,36,183,82,66,154,147,91,115,134)(4,204,18,37,184,83,67,155,148,92,116,135)(5,188,19,38,185,84,68,156,149,93,117,136)(6,189,20,39,186,85,52,157,150,94,118,120)(7,190,21,40,187,69,53,158,151,95,119,121)(8,191,22,41,171,70,54,159,152,96,103,122)(9,192,23,42,172,71,55,160,153,97,104,123)(10,193,24,43,173,72,56,161,137,98,105,124)(11,194,25,44,174,73,57,162,138,99,106,125)(12,195,26,45,175,74,58,163,139,100,107,126)(13,196,27,46,176,75,59,164,140,101,108,127)(14,197,28,47,177,76,60,165,141,102,109,128)(15,198,29,48,178,77,61,166,142,86,110,129)(16,199,30,49,179,78,62,167,143,87,111,130)(17,200,31,50,180,79,63,168,144,88,112,131), (1,32)(2,33)(3,34)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(35,133)(36,134)(37,135)(38,136)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,150)(53,151)(54,152)(55,153)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,148)(68,149)(69,95)(70,96)(71,97)(72,98)(73,99)(74,100)(75,101)(76,102)(77,86)(78,87)(79,88)(80,89)(81,90)(82,91)(83,92)(84,93)(85,94)(103,171)(104,172)(105,173)(106,174)(107,175)(108,176)(109,177)(110,178)(111,179)(112,180)(113,181)(114,182)(115,183)(116,184)(117,185)(118,186)(119,187) );
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)], [(1,201,32,51,181,80,64,169,145,89,113,132),(2,202,33,35,182,81,65,170,146,90,114,133),(3,203,34,36,183,82,66,154,147,91,115,134),(4,204,18,37,184,83,67,155,148,92,116,135),(5,188,19,38,185,84,68,156,149,93,117,136),(6,189,20,39,186,85,52,157,150,94,118,120),(7,190,21,40,187,69,53,158,151,95,119,121),(8,191,22,41,171,70,54,159,152,96,103,122),(9,192,23,42,172,71,55,160,153,97,104,123),(10,193,24,43,173,72,56,161,137,98,105,124),(11,194,25,44,174,73,57,162,138,99,106,125),(12,195,26,45,175,74,58,163,139,100,107,126),(13,196,27,46,176,75,59,164,140,101,108,127),(14,197,28,47,177,76,60,165,141,102,109,128),(15,198,29,48,178,77,61,166,142,86,110,129),(16,199,30,49,179,78,62,167,143,87,111,130),(17,200,31,50,180,79,63,168,144,88,112,131)], [(1,32),(2,33),(3,34),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,31),(35,133),(36,134),(37,135),(38,136),(39,120),(40,121),(41,122),(42,123),(43,124),(44,125),(45,126),(46,127),(47,128),(48,129),(49,130),(50,131),(51,132),(52,150),(53,151),(54,152),(55,153),(56,137),(57,138),(58,139),(59,140),(60,141),(61,142),(62,143),(63,144),(64,145),(65,146),(66,147),(67,148),(68,149),(69,95),(70,96),(71,97),(72,98),(73,99),(74,100),(75,101),(76,102),(77,86),(78,87),(79,88),(80,89),(81,90),(82,91),(83,92),(84,93),(85,94),(103,171),(104,172),(105,173),(106,174),(107,175),(108,176),(109,177),(110,178),(111,179),(112,180),(113,181),(114,182),(115,183),(116,184),(117,185),(118,186),(119,187)]])
153 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4 | 6 | 12A | 12B | 17A | ··· | 17P | 34A | ··· | 34P | 34Q | ··· | 34AV | 51A | ··· | 51P | 68A | ··· | 68P | 102A | ··· | 102P | 204A | ··· | 204AF |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 12 | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 | 204 | ··· | 204 |
| size | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
153 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C17 | C34 | C34 | S3 | D4 | D6 | D12 | S3×C17 | D4×C17 | S3×C34 | C17×D12 |
| kernel | C17×D12 | C204 | S3×C34 | D12 | C12 | D6 | C68 | C51 | C34 | C17 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 16 | 16 | 32 | 1 | 1 | 1 | 2 | 16 | 16 | 16 | 32 |
Matrix representation of C17×D12 ►in GL2(𝔽409) generated by
| 36 | 0 |
| 0 | 36 |
| 353 | 56 |
| 353 | 297 |
| 1 | 1 |
| 0 | 408 |
G:=sub<GL(2,GF(409))| [36,0,0,36],[353,353,56,297],[1,0,1,408] >;
C17×D12 in GAP, Magma, Sage, TeX
C_{17}\times D_{12} % in TeX
G:=Group("C17xD12"); // GroupNames label
G:=SmallGroup(408,22);
// by ID
G=gap.SmallGroup(408,22);
# by ID
G:=PCGroup([5,-2,-2,-17,-2,-3,701,346,6804]);
// Polycyclic
G:=Group<a,b,c|a^17=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export