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G = C3×D65order 390 = 2·3·5·13

Direct product of C3 and D65

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

Aliases: C3×D65, C655C6, C392D5, C1952C2, C152D13, C5⋊(C3×D13), C133(C3×D5), SmallGroup(390,7)

Series: Derived Chief Lower central Upper central

C1C65 — C3×D65
C1C13C65C195 — C3×D65
C65 — C3×D65
C1C3

Generators and relations for C3×D65
 G = < a,b,c | a3=b65=c2=1, ab=ba, ac=ca, cbc=b-1 >

65C2
65C6
13D5
5D13
13C3×D5
5C3×D13

Smallest permutation representation of C3×D65
On 195 points
Generators in S195
(1 156 80)(2 157 81)(3 158 82)(4 159 83)(5 160 84)(6 161 85)(7 162 86)(8 163 87)(9 164 88)(10 165 89)(11 166 90)(12 167 91)(13 168 92)(14 169 93)(15 170 94)(16 171 95)(17 172 96)(18 173 97)(19 174 98)(20 175 99)(21 176 100)(22 177 101)(23 178 102)(24 179 103)(25 180 104)(26 181 105)(27 182 106)(28 183 107)(29 184 108)(30 185 109)(31 186 110)(32 187 111)(33 188 112)(34 189 113)(35 190 114)(36 191 115)(37 192 116)(38 193 117)(39 194 118)(40 195 119)(41 131 120)(42 132 121)(43 133 122)(44 134 123)(45 135 124)(46 136 125)(47 137 126)(48 138 127)(49 139 128)(50 140 129)(51 141 130)(52 142 66)(53 143 67)(54 144 68)(55 145 69)(56 146 70)(57 147 71)(58 148 72)(59 149 73)(60 150 74)(61 151 75)(62 152 76)(63 153 77)(64 154 78)(65 155 79)
(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)
(1 65)(2 64)(3 63)(4 62)(5 61)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(66 93)(67 92)(68 91)(69 90)(70 89)(71 88)(72 87)(73 86)(74 85)(75 84)(76 83)(77 82)(78 81)(79 80)(94 130)(95 129)(96 128)(97 127)(98 126)(99 125)(100 124)(101 123)(102 122)(103 121)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)(131 180)(132 179)(133 178)(134 177)(135 176)(136 175)(137 174)(138 173)(139 172)(140 171)(141 170)(142 169)(143 168)(144 167)(145 166)(146 165)(147 164)(148 163)(149 162)(150 161)(151 160)(152 159)(153 158)(154 157)(155 156)(181 195)(182 194)(183 193)(184 192)(185 191)(186 190)(187 189)

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

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

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

102 conjugacy classes

class 1  2 3A3B5A5B6A6B13A···13F15A15B15C15D39A···39L65A···65X195A···195AV
order1233556613···131515151539···3965···65195···195
size165112265652···222222···22···22···2

102 irreducible representations

dim1111222222
type+++++
imageC1C2C3C6D5D13C3×D5C3×D13D65C3×D65
kernelC3×D65C195D65C65C39C15C13C5C3C1
# reps1122264122448

Matrix representation of C3×D65 in GL2(𝔽1171) generated by

4200
0420
,
866646
699228
,
618951
70553
G:=sub<GL(2,GF(1171))| [420,0,0,420],[866,699,646,228],[618,70,951,553] >;

C3×D65 in GAP, Magma, Sage, TeX

C_3\times D_{65}
% in TeX

G:=Group("C3xD65");
// GroupNames label

G:=SmallGroup(390,7);
// by ID

G=gap.SmallGroup(390,7);
# by ID

G:=PCGroup([4,-2,-3,-5,-13,290,5763]);
// Polycyclic

G:=Group<a,b,c|a^3=b^65=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D65 in TeX

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