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G = C5×D39order 390 = 2·3·5·13

Direct product of C5 and D39

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

Aliases: C5×D39, C652S3, C391C10, C1953C2, C153D13, C13⋊(C5×S3), C3⋊(C5×D13), SmallGroup(390,9)

Series: Derived Chief Lower central Upper central

C1C39 — C5×D39
C1C13C39C195 — C5×D39
C39 — C5×D39
C1C5

Generators and relations for C5×D39
 G = < a,b,c | a5=b39=c2=1, ab=ba, ac=ca, cbc=b-1 >

39C2
13S3
39C10
3D13
13C5×S3
3C5×D13

Smallest permutation representation of C5×D39
On 195 points
Generators in S195
(1 185 144 93 62)(2 186 145 94 63)(3 187 146 95 64)(4 188 147 96 65)(5 189 148 97 66)(6 190 149 98 67)(7 191 150 99 68)(8 192 151 100 69)(9 193 152 101 70)(10 194 153 102 71)(11 195 154 103 72)(12 157 155 104 73)(13 158 156 105 74)(14 159 118 106 75)(15 160 119 107 76)(16 161 120 108 77)(17 162 121 109 78)(18 163 122 110 40)(19 164 123 111 41)(20 165 124 112 42)(21 166 125 113 43)(22 167 126 114 44)(23 168 127 115 45)(24 169 128 116 46)(25 170 129 117 47)(26 171 130 79 48)(27 172 131 80 49)(28 173 132 81 50)(29 174 133 82 51)(30 175 134 83 52)(31 176 135 84 53)(32 177 136 85 54)(33 178 137 86 55)(34 179 138 87 56)(35 180 139 88 57)(36 181 140 89 58)(37 182 141 90 59)(38 183 142 91 60)(39 184 143 92 61)
(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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(40 44)(41 43)(45 78)(46 77)(47 76)(48 75)(49 74)(50 73)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(61 62)(79 106)(80 105)(81 104)(82 103)(83 102)(84 101)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(107 117)(108 116)(109 115)(110 114)(111 113)(118 130)(119 129)(120 128)(121 127)(122 126)(123 125)(131 156)(132 155)(133 154)(134 153)(135 152)(136 151)(137 150)(138 149)(139 148)(140 147)(141 146)(142 145)(143 144)(157 173)(158 172)(159 171)(160 170)(161 169)(162 168)(163 167)(164 166)(174 195)(175 194)(176 193)(177 192)(178 191)(179 190)(180 189)(181 188)(182 187)(183 186)(184 185)

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

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

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

105 conjugacy classes

class 1  2  3 5A5B5C5D10A10B10C10D13A···13F15A15B15C15D39A···39L65A···65X195A···195AV
order12355551010101013···131515151539···3965···65195···195
size13921111393939392···222222···22···22···2

105 irreducible representations

dim1111222222
type+++++
imageC1C2C5C10S3D13C5×S3D39C5×D13C5×D39
kernelC5×D39C195D39C39C65C15C13C5C3C1
# reps1144164122448

Matrix representation of C5×D39 in GL2(𝔽1171) generated by

2160
0216
,
10291037
134580
,
398834
682773
G:=sub<GL(2,GF(1171))| [216,0,0,216],[1029,134,1037,580],[398,682,834,773] >;

C5×D39 in GAP, Magma, Sage, TeX

C_5\times D_{39}
% in TeX

G:=Group("C5xD39");
// GroupNames label

G:=SmallGroup(390,9);
// by ID

G=gap.SmallGroup(390,9);
# by ID

G:=PCGroup([4,-2,-5,-3,-13,242,5763]);
// Polycyclic

G:=Group<a,b,c|a^5=b^39=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D39 in TeX

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