Copied to
clipboard

G = C13×D15order 390 = 2·3·5·13

Direct product of C13 and D15

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

Aliases: C13×D15, C653S3, C393D5, C1954C2, C151C26, C5⋊(S3×C13), C3⋊(D5×C13), SmallGroup(390,10)

Series: Derived Chief Lower central Upper central

C1C15 — C13×D15
C1C5C15C195 — C13×D15
C15 — C13×D15
C1C13

Generators and relations for C13×D15
 G = < a,b,c | a13=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

15C2
5S3
3D5
15C26
5S3×C13
3D5×C13

Smallest permutation representation of C13×D15
On 195 points
Generators in S195
(1 193 170 163 137 133 118 92 84 69 46 32 17)(2 194 171 164 138 134 119 93 85 70 47 33 18)(3 195 172 165 139 135 120 94 86 71 48 34 19)(4 181 173 151 140 121 106 95 87 72 49 35 20)(5 182 174 152 141 122 107 96 88 73 50 36 21)(6 183 175 153 142 123 108 97 89 74 51 37 22)(7 184 176 154 143 124 109 98 90 75 52 38 23)(8 185 177 155 144 125 110 99 76 61 53 39 24)(9 186 178 156 145 126 111 100 77 62 54 40 25)(10 187 179 157 146 127 112 101 78 63 55 41 26)(11 188 180 158 147 128 113 102 79 64 56 42 27)(12 189 166 159 148 129 114 103 80 65 57 43 28)(13 190 167 160 149 130 115 104 81 66 58 44 29)(14 191 168 161 150 131 116 105 82 67 59 45 30)(15 192 169 162 136 132 117 91 83 68 60 31 16)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 17)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(31 32)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)(91 92)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)(106 114)(107 113)(108 112)(109 111)(115 120)(116 119)(117 118)(121 129)(122 128)(123 127)(124 126)(130 135)(131 134)(132 133)(136 137)(138 150)(139 149)(140 148)(141 147)(142 146)(143 145)(151 159)(152 158)(153 157)(154 156)(160 165)(161 164)(162 163)(166 173)(167 172)(168 171)(169 170)(174 180)(175 179)(176 178)(181 189)(182 188)(183 187)(184 186)(190 195)(191 194)(192 193)

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

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

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

117 conjugacy classes

class 1  2  3 5A5B13A···13L15A15B15C15D26A···26L39A···39L65A···65X195A···195AV
order1235513···131515151526···2639···3965···65195···195
size1152221···1222215···152···22···22···2

117 irreducible representations

dim1111222222
type+++++
imageC1C2C13C26S3D5D15S3×C13D5×C13C13×D15
kernelC13×D15C195D15C15C65C39C13C5C3C1
# reps111212124122448

Matrix representation of C13×D15 in GL2(𝔽1171) generated by

2240
0224
,
764911
625218
,
364322
1025807
G:=sub<GL(2,GF(1171))| [224,0,0,224],[764,625,911,218],[364,1025,322,807] >;

C13×D15 in GAP, Magma, Sage, TeX

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

G:=Group("C13xD15");
// GroupNames label

G:=SmallGroup(390,10);
// by ID

G=gap.SmallGroup(390,10);
# by ID

G:=PCGroup([4,-2,-13,-3,-5,626,4995]);
// Polycyclic

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

Export

Subgroup lattice of C13×D15 in TeX

׿
×
𝔽