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G = C15×D13order 390 = 2·3·5·13

Direct product of C15 and D13

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

Aliases: C15×D13, C656C6, C392C10, C1955C2, C133C30, SmallGroup(390,5)

Series: Derived Chief Lower central Upper central

C1C13 — C15×D13
C1C13C65C195 — C15×D13
C13 — C15×D13
C1C15

Generators and relations for C15×D13
 G = < a,b,c | a15=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >

13C2
13C6
13C10
13C30

Smallest permutation representation of C15×D13
On 195 points
Generators in S195
(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)
(1 150 37 171 119 87 71 122 56 23 185 96 160)(2 136 38 172 120 88 72 123 57 24 186 97 161)(3 137 39 173 106 89 73 124 58 25 187 98 162)(4 138 40 174 107 90 74 125 59 26 188 99 163)(5 139 41 175 108 76 75 126 60 27 189 100 164)(6 140 42 176 109 77 61 127 46 28 190 101 165)(7 141 43 177 110 78 62 128 47 29 191 102 151)(8 142 44 178 111 79 63 129 48 30 192 103 152)(9 143 45 179 112 80 64 130 49 16 193 104 153)(10 144 31 180 113 81 65 131 50 17 194 105 154)(11 145 32 166 114 82 66 132 51 18 195 91 155)(12 146 33 167 115 83 67 133 52 19 181 92 156)(13 147 34 168 116 84 68 134 53 20 182 93 157)(14 148 35 169 117 85 69 135 54 21 183 94 158)(15 149 36 170 118 86 70 121 55 22 184 95 159)
(1 160)(2 161)(3 162)(4 163)(5 164)(6 165)(7 151)(8 152)(9 153)(10 154)(11 155)(12 156)(13 157)(14 158)(15 159)(16 179)(17 180)(18 166)(19 167)(20 168)(21 169)(22 170)(23 171)(24 172)(25 173)(26 174)(27 175)(28 176)(29 177)(30 178)(31 194)(32 195)(33 181)(34 182)(35 183)(36 184)(37 185)(38 186)(39 187)(40 188)(41 189)(42 190)(43 191)(44 192)(45 193)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 120)(58 106)(59 107)(60 108)(76 126)(77 127)(78 128)(79 129)(80 130)(81 131)(82 132)(83 133)(84 134)(85 135)(86 121)(87 122)(88 123)(89 124)(90 125)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 136)(98 137)(99 138)(100 139)(101 140)(102 141)(103 142)(104 143)(105 144)

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

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

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

120 conjugacy classes

class 1  2 3A3B5A5B5C5D6A6B10A10B10C10D13A···13F15A···15H30A···30H39A···39L65A···65X195A···195AV
order12335555661010101013···1315···1530···3039···3965···65195···195
size1131111111313131313132···21···113···132···22···22···2

120 irreducible representations

dim111111112222
type+++
imageC1C2C3C5C6C10C15C30D13C3×D13C5×D13C15×D13
kernelC15×D13C195C5×D13C3×D13C65C39D13C13C15C5C3C1
# reps112424886122448

Matrix representation of C15×D13 in GL3(𝔽1171) generated by

75000
010680
001068
,
100
0821
011700
,
117000
001
010
G:=sub<GL(3,GF(1171))| [750,0,0,0,1068,0,0,0,1068],[1,0,0,0,82,1170,0,1,0],[1170,0,0,0,0,1,0,1,0] >;

C15×D13 in GAP, Magma, Sage, TeX

C_{15}\times D_{13}
% in TeX

G:=Group("C15xD13");
// GroupNames label

G:=SmallGroup(390,5);
// by ID

G=gap.SmallGroup(390,5);
# by ID

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

G:=Group<a,b,c|a^15=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C15×D13 in TeX

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