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G = C3×Dic17order 204 = 22·3·17

Direct product of C3 and Dic17

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

Aliases: C3×Dic17, C514C4, C34.C6, C172C12, C6.2D17, C102.2C2, C2.(C3×D17), SmallGroup(204,2)

Series: Derived Chief Lower central Upper central

C1C17 — C3×Dic17
C1C17C34C102 — C3×Dic17
C17 — C3×Dic17
C1C6

Generators and relations for C3×Dic17
 G = < a,b,c | a3=b34=1, c2=b17, ab=ba, ac=ca, cbc-1=b-1 >

17C4
17C12

Smallest permutation representation of C3×Dic17
Regular action on 204 points
Generators in S204
(1 100 66)(2 101 67)(3 102 68)(4 69 35)(5 70 36)(6 71 37)(7 72 38)(8 73 39)(9 74 40)(10 75 41)(11 76 42)(12 77 43)(13 78 44)(14 79 45)(15 80 46)(16 81 47)(17 82 48)(18 83 49)(19 84 50)(20 85 51)(21 86 52)(22 87 53)(23 88 54)(24 89 55)(25 90 56)(26 91 57)(27 92 58)(28 93 59)(29 94 60)(30 95 61)(31 96 62)(32 97 63)(33 98 64)(34 99 65)(103 171 154)(104 172 155)(105 173 156)(106 174 157)(107 175 158)(108 176 159)(109 177 160)(110 178 161)(111 179 162)(112 180 163)(113 181 164)(114 182 165)(115 183 166)(116 184 167)(117 185 168)(118 186 169)(119 187 170)(120 188 137)(121 189 138)(122 190 139)(123 191 140)(124 192 141)(125 193 142)(126 194 143)(127 195 144)(128 196 145)(129 197 146)(130 198 147)(131 199 148)(132 200 149)(133 201 150)(134 202 151)(135 203 152)(136 204 153)
(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)
(1 120 18 103)(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)(35 168 52 151)(36 167 53 150)(37 166 54 149)(38 165 55 148)(39 164 56 147)(40 163 57 146)(41 162 58 145)(42 161 59 144)(43 160 60 143)(44 159 61 142)(45 158 62 141)(46 157 63 140)(47 156 64 139)(48 155 65 138)(49 154 66 137)(50 153 67 170)(51 152 68 169)(69 185 86 202)(70 184 87 201)(71 183 88 200)(72 182 89 199)(73 181 90 198)(74 180 91 197)(75 179 92 196)(76 178 93 195)(77 177 94 194)(78 176 95 193)(79 175 96 192)(80 174 97 191)(81 173 98 190)(82 172 99 189)(83 171 100 188)(84 204 101 187)(85 203 102 186)

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

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

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

C3×Dic17 is a maximal subgroup of   C513C8  D512C4  C17⋊D12  C51⋊Q8  C12×D17

60 conjugacy classes

class 1  2 3A3B4A4B6A6B12A12B12C12D17A···17H34A···34H51A···51P102A···102P
order123344661212121217···1734···3451···51102···102
size1111171711171717172···22···22···22···2

60 irreducible representations

dim1111112222
type+++-
imageC1C2C3C4C6C12D17Dic17C3×D17C3×Dic17
kernelC3×Dic17C102Dic17C51C34C17C6C3C2C1
# reps112224881616

Matrix representation of C3×Dic17 in GL3(𝔽409) generated by

5300
010
001
,
100
04040
00327
,
40800
001
04080
G:=sub<GL(3,GF(409))| [53,0,0,0,1,0,0,0,1],[1,0,0,0,404,0,0,0,327],[408,0,0,0,0,408,0,1,0] >;

C3×Dic17 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{17}
% in TeX

G:=Group("C3xDic17");
// GroupNames label

G:=SmallGroup(204,2);
// by ID

G=gap.SmallGroup(204,2);
# by ID

G:=PCGroup([4,-2,-3,-2,-17,24,3075]);
// Polycyclic

G:=Group<a,b,c|a^3=b^34=1,c^2=b^17,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×Dic17 in TeX

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