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G = Dic3×C17order 204 = 22·3·17

Direct product of C17 and Dic3

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

Aliases: Dic3×C17, C3⋊C68, C515C4, C6.C34, C34.2S3, C102.3C2, C2.(S3×C17), SmallGroup(204,1)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3×C17
C1C3C6C102 — Dic3×C17
C3 — Dic3×C17
C1C34

Generators and relations for Dic3×C17
 G = < a,b,c | a17=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C68

Smallest permutation representation of Dic3×C17
Regular action 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)
(1 158 187 204 75 31)(2 159 171 188 76 32)(3 160 172 189 77 33)(4 161 173 190 78 34)(5 162 174 191 79 18)(6 163 175 192 80 19)(7 164 176 193 81 20)(8 165 177 194 82 21)(9 166 178 195 83 22)(10 167 179 196 84 23)(11 168 180 197 85 24)(12 169 181 198 69 25)(13 170 182 199 70 26)(14 154 183 200 71 27)(15 155 184 201 72 28)(16 156 185 202 73 29)(17 157 186 203 74 30)(35 62 149 94 130 115)(36 63 150 95 131 116)(37 64 151 96 132 117)(38 65 152 97 133 118)(39 66 153 98 134 119)(40 67 137 99 135 103)(41 68 138 100 136 104)(42 52 139 101 120 105)(43 53 140 102 121 106)(44 54 141 86 122 107)(45 55 142 87 123 108)(46 56 143 88 124 109)(47 57 144 89 125 110)(48 58 145 90 126 111)(49 59 146 91 127 112)(50 60 147 92 128 113)(51 61 148 93 129 114)
(1 108 204 142)(2 109 188 143)(3 110 189 144)(4 111 190 145)(5 112 191 146)(6 113 192 147)(7 114 193 148)(8 115 194 149)(9 116 195 150)(10 117 196 151)(11 118 197 152)(12 119 198 153)(13 103 199 137)(14 104 200 138)(15 105 201 139)(16 106 202 140)(17 107 203 141)(18 49 174 91)(19 50 175 92)(20 51 176 93)(21 35 177 94)(22 36 178 95)(23 37 179 96)(24 38 180 97)(25 39 181 98)(26 40 182 99)(27 41 183 100)(28 42 184 101)(29 43 185 102)(30 44 186 86)(31 45 187 87)(32 46 171 88)(33 47 172 89)(34 48 173 90)(52 155 120 72)(53 156 121 73)(54 157 122 74)(55 158 123 75)(56 159 124 76)(57 160 125 77)(58 161 126 78)(59 162 127 79)(60 163 128 80)(61 164 129 81)(62 165 130 82)(63 166 131 83)(64 167 132 84)(65 168 133 85)(66 169 134 69)(67 170 135 70)(68 154 136 71)

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

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

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

Dic3×C17 is a maximal subgroup of   D512C4  C3⋊D68  C51⋊Q8  S3×C68

102 conjugacy classes

class 1  2  3 4A4B 6 17A···17P34A···34P51A···51P68A···68AF102A···102P
order12344617···1734···3451···5168···68102···102
size1123321···11···12···23···32···2

102 irreducible representations

dim1111112222
type+++-
imageC1C2C4C17C34C68S3Dic3S3×C17Dic3×C17
kernelDic3×C17C102C51Dic3C6C3C34C17C2C1
# reps112161632111616

Matrix representation of Dic3×C17 in GL2(𝔽409) generated by

300
030
,
1408
10
,
50371
12359
G:=sub<GL(2,GF(409))| [30,0,0,30],[1,1,408,0],[50,12,371,359] >;

Dic3×C17 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{17}
% in TeX

G:=Group("Dic3xC17");
// GroupNames label

G:=SmallGroup(204,1);
// by ID

G=gap.SmallGroup(204,1);
# by ID

G:=PCGroup([4,-2,-17,-2,-3,136,2179]);
// Polycyclic

G:=Group<a,b,c|a^17=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Dic3×C17 in TeX

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