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G = C17×D12order 408 = 23·3·17

Direct product of C17 and D12

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C17×D12, C516D4, C683S3, C121C34, C2045C2, D61C34, C34.15D6, C102.20C22, C4⋊(S3×C17), C31(D4×C17), (S3×C34)⋊4C2, C2.4(S3×C34), C6.3(C2×C34), SmallGroup(408,22)

Series: Derived Chief Lower central Upper central

C1C6 — C17×D12
C1C3C6C102S3×C34 — C17×D12
C3C6 — C17×D12
C1C34C68

Generators and relations for C17×D12
 G = < a,b,c | a17=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

6C2
6C2
3C22
3C22
2S3
2S3
6C34
6C34
3D4
3C2×C34
3C2×C34
2S3×C17
2S3×C17
3D4×C17

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)
(1 196 141 29 93 40 115 176 155 129 67 71)(2 197 142 30 94 41 116 177 156 130 68 72)(3 198 143 31 95 42 117 178 157 131 52 73)(4 199 144 32 96 43 118 179 158 132 53 74)(5 200 145 33 97 44 119 180 159 133 54 75)(6 201 146 34 98 45 103 181 160 134 55 76)(7 202 147 18 99 46 104 182 161 135 56 77)(8 203 148 19 100 47 105 183 162 136 57 78)(9 204 149 20 101 48 106 184 163 120 58 79)(10 188 150 21 102 49 107 185 164 121 59 80)(11 189 151 22 86 50 108 186 165 122 60 81)(12 190 152 23 87 51 109 187 166 123 61 82)(13 191 153 24 88 35 110 171 167 124 62 83)(14 192 137 25 89 36 111 172 168 125 63 84)(15 193 138 26 90 37 112 173 169 126 64 85)(16 194 139 27 91 38 113 174 170 127 65 69)(17 195 140 28 92 39 114 175 154 128 66 70)
(1 141)(2 142)(3 143)(4 144)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 153)(14 137)(15 138)(16 139)(17 140)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 84)(26 85)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 124)(36 125)(37 126)(38 127)(39 128)(40 129)(41 130)(42 131)(43 132)(44 133)(45 134)(46 135)(47 136)(48 120)(49 121)(50 122)(51 123)(52 95)(53 96)(54 97)(55 98)(56 99)(57 100)(58 101)(59 102)(60 86)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(103 160)(104 161)(105 162)(106 163)(107 164)(108 165)(109 166)(110 167)(111 168)(112 169)(113 170)(114 154)(115 155)(116 156)(117 157)(118 158)(119 159)

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), (1,196,141,29,93,40,115,176,155,129,67,71)(2,197,142,30,94,41,116,177,156,130,68,72)(3,198,143,31,95,42,117,178,157,131,52,73)(4,199,144,32,96,43,118,179,158,132,53,74)(5,200,145,33,97,44,119,180,159,133,54,75)(6,201,146,34,98,45,103,181,160,134,55,76)(7,202,147,18,99,46,104,182,161,135,56,77)(8,203,148,19,100,47,105,183,162,136,57,78)(9,204,149,20,101,48,106,184,163,120,58,79)(10,188,150,21,102,49,107,185,164,121,59,80)(11,189,151,22,86,50,108,186,165,122,60,81)(12,190,152,23,87,51,109,187,166,123,61,82)(13,191,153,24,88,35,110,171,167,124,62,83)(14,192,137,25,89,36,111,172,168,125,63,84)(15,193,138,26,90,37,112,173,169,126,64,85)(16,194,139,27,91,38,113,174,170,127,65,69)(17,195,140,28,92,39,114,175,154,128,66,70), (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,137)(15,138)(16,139)(17,140)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,85)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,120)(49,121)(50,122)(51,123)(52,95)(53,96)(54,97)(55,98)(56,99)(57,100)(58,101)(59,102)(60,86)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(103,160)(104,161)(105,162)(106,163)(107,164)(108,165)(109,166)(110,167)(111,168)(112,169)(113,170)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159)>;

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), (1,196,141,29,93,40,115,176,155,129,67,71)(2,197,142,30,94,41,116,177,156,130,68,72)(3,198,143,31,95,42,117,178,157,131,52,73)(4,199,144,32,96,43,118,179,158,132,53,74)(5,200,145,33,97,44,119,180,159,133,54,75)(6,201,146,34,98,45,103,181,160,134,55,76)(7,202,147,18,99,46,104,182,161,135,56,77)(8,203,148,19,100,47,105,183,162,136,57,78)(9,204,149,20,101,48,106,184,163,120,58,79)(10,188,150,21,102,49,107,185,164,121,59,80)(11,189,151,22,86,50,108,186,165,122,60,81)(12,190,152,23,87,51,109,187,166,123,61,82)(13,191,153,24,88,35,110,171,167,124,62,83)(14,192,137,25,89,36,111,172,168,125,63,84)(15,193,138,26,90,37,112,173,169,126,64,85)(16,194,139,27,91,38,113,174,170,127,65,69)(17,195,140,28,92,39,114,175,154,128,66,70), (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,137)(15,138)(16,139)(17,140)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,85)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(41,130)(42,131)(43,132)(44,133)(45,134)(46,135)(47,136)(48,120)(49,121)(50,122)(51,123)(52,95)(53,96)(54,97)(55,98)(56,99)(57,100)(58,101)(59,102)(60,86)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(103,160)(104,161)(105,162)(106,163)(107,164)(108,165)(109,166)(110,167)(111,168)(112,169)(113,170)(114,154)(115,155)(116,156)(117,157)(118,158)(119,159) );

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)], [(1,196,141,29,93,40,115,176,155,129,67,71),(2,197,142,30,94,41,116,177,156,130,68,72),(3,198,143,31,95,42,117,178,157,131,52,73),(4,199,144,32,96,43,118,179,158,132,53,74),(5,200,145,33,97,44,119,180,159,133,54,75),(6,201,146,34,98,45,103,181,160,134,55,76),(7,202,147,18,99,46,104,182,161,135,56,77),(8,203,148,19,100,47,105,183,162,136,57,78),(9,204,149,20,101,48,106,184,163,120,58,79),(10,188,150,21,102,49,107,185,164,121,59,80),(11,189,151,22,86,50,108,186,165,122,60,81),(12,190,152,23,87,51,109,187,166,123,61,82),(13,191,153,24,88,35,110,171,167,124,62,83),(14,192,137,25,89,36,111,172,168,125,63,84),(15,193,138,26,90,37,112,173,169,126,64,85),(16,194,139,27,91,38,113,174,170,127,65,69),(17,195,140,28,92,39,114,175,154,128,66,70)], [(1,141),(2,142),(3,143),(4,144),(5,145),(6,146),(7,147),(8,148),(9,149),(10,150),(11,151),(12,152),(13,153),(14,137),(15,138),(16,139),(17,140),(18,77),(19,78),(20,79),(21,80),(22,81),(23,82),(24,83),(25,84),(26,85),(27,69),(28,70),(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,124),(36,125),(37,126),(38,127),(39,128),(40,129),(41,130),(42,131),(43,132),(44,133),(45,134),(46,135),(47,136),(48,120),(49,121),(50,122),(51,123),(52,95),(53,96),(54,97),(55,98),(56,99),(57,100),(58,101),(59,102),(60,86),(61,87),(62,88),(63,89),(64,90),(65,91),(66,92),(67,93),(68,94),(103,160),(104,161),(105,162),(106,163),(107,164),(108,165),(109,166),(110,167),(111,168),(112,169),(113,170),(114,154),(115,155),(116,156),(117,157),(118,158),(119,159)])

153 conjugacy classes

class 1 2A2B2C 3  4  6 12A12B17A···17P34A···34P34Q···34AV51A···51P68A···68P102A···102P204A···204AF
order1222346121217···1734···3434···3451···5168···68102···102204···204
size1166222221···11···16···62···22···22···22···2

153 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C17C34C34S3D4D6D12S3×C17D4×C17S3×C34C17×D12
kernelC17×D12C204S3×C34D12C12D6C68C51C34C17C4C3C2C1
# reps112161632111216161632

Matrix representation of C17×D12 in GL2(𝔽409) generated by

360
036
,
35356
353297
,
11
0408
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

Subgroup lattice of C17×D12 in TeX

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