direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C13×D15, C65⋊3S3, C39⋊3D5, C195⋊4C2, C15⋊1C26, C5⋊(S3×C13), C3⋊(D5×C13), SmallGroup(390,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C13×D15 |
Generators and relations for C13×D15
G = < a,b,c | a13=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C13×D15
►On 195 points
Generators in S195
(1 181 174 152 148 125 106 99 80 73 58 43 28)(2 182 175 153 149 126 107 100 81 74 59 44 29)(3 183 176 154 150 127 108 101 82 75 60 45 30)(4 184 177 155 136 128 109 102 83 61 46 31 16)(5 185 178 156 137 129 110 103 84 62 47 32 17)(6 186 179 157 138 130 111 104 85 63 48 33 18)(7 187 180 158 139 131 112 105 86 64 49 34 19)(8 188 166 159 140 132 113 91 87 65 50 35 20)(9 189 167 160 141 133 114 92 88 66 51 36 21)(10 190 168 161 142 134 115 93 89 67 52 37 22)(11 191 169 162 143 135 116 94 90 68 53 38 23)(12 192 170 163 144 121 117 95 76 69 54 39 24)(13 193 171 164 145 122 118 96 77 70 55 40 25)(14 194 172 165 146 123 119 97 78 71 56 41 26)(15 195 173 151 147 124 120 98 79 72 57 42 27)
(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 24)(17 23)(18 22)(19 21)(25 30)(26 29)(27 28)(31 39)(32 38)(33 37)(34 36)(40 45)(41 44)(42 43)(46 54)(47 53)(48 52)(49 51)(55 60)(56 59)(57 58)(61 69)(62 68)(63 67)(64 66)(70 75)(71 74)(72 73)(76 83)(77 82)(78 81)(79 80)(84 90)(85 89)(86 88)(92 105)(93 104)(94 103)(95 102)(96 101)(97 100)(98 99)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(121 128)(122 127)(123 126)(124 125)(129 135)(130 134)(131 133)(136 144)(137 143)(138 142)(139 141)(145 150)(146 149)(147 148)(151 152)(153 165)(154 164)(155 163)(156 162)(157 161)(158 160)(167 180)(168 179)(169 178)(170 177)(171 176)(172 175)(173 174)(181 195)(182 194)(183 193)(184 192)(185 191)(186 190)(187 189)
G:=sub<Sym(195)| (1,181,174,152,148,125,106,99,80,73,58,43,28)(2,182,175,153,149,126,107,100,81,74,59,44,29)(3,183,176,154,150,127,108,101,82,75,60,45,30)(4,184,177,155,136,128,109,102,83,61,46,31,16)(5,185,178,156,137,129,110,103,84,62,47,32,17)(6,186,179,157,138,130,111,104,85,63,48,33,18)(7,187,180,158,139,131,112,105,86,64,49,34,19)(8,188,166,159,140,132,113,91,87,65,50,35,20)(9,189,167,160,141,133,114,92,88,66,51,36,21)(10,190,168,161,142,134,115,93,89,67,52,37,22)(11,191,169,162,143,135,116,94,90,68,53,38,23)(12,192,170,163,144,121,117,95,76,69,54,39,24)(13,193,171,164,145,122,118,96,77,70,55,40,25)(14,194,172,165,146,123,119,97,78,71,56,41,26)(15,195,173,151,147,124,120,98,79,72,57,42,27), (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)(16,24)(17,23)(18,22)(19,21)(25,30)(26,29)(27,28)(31,39)(32,38)(33,37)(34,36)(40,45)(41,44)(42,43)(46,54)(47,53)(48,52)(49,51)(55,60)(56,59)(57,58)(61,69)(62,68)(63,67)(64,66)(70,75)(71,74)(72,73)(76,83)(77,82)(78,81)(79,80)(84,90)(85,89)(86,88)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(121,128)(122,127)(123,126)(124,125)(129,135)(130,134)(131,133)(136,144)(137,143)(138,142)(139,141)(145,150)(146,149)(147,148)(151,152)(153,165)(154,164)(155,163)(156,162)(157,161)(158,160)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)(181,195)(182,194)(183,193)(184,192)(185,191)(186,190)(187,189)>;
G:=Group( (1,181,174,152,148,125,106,99,80,73,58,43,28)(2,182,175,153,149,126,107,100,81,74,59,44,29)(3,183,176,154,150,127,108,101,82,75,60,45,30)(4,184,177,155,136,128,109,102,83,61,46,31,16)(5,185,178,156,137,129,110,103,84,62,47,32,17)(6,186,179,157,138,130,111,104,85,63,48,33,18)(7,187,180,158,139,131,112,105,86,64,49,34,19)(8,188,166,159,140,132,113,91,87,65,50,35,20)(9,189,167,160,141,133,114,92,88,66,51,36,21)(10,190,168,161,142,134,115,93,89,67,52,37,22)(11,191,169,162,143,135,116,94,90,68,53,38,23)(12,192,170,163,144,121,117,95,76,69,54,39,24)(13,193,171,164,145,122,118,96,77,70,55,40,25)(14,194,172,165,146,123,119,97,78,71,56,41,26)(15,195,173,151,147,124,120,98,79,72,57,42,27), (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)(16,24)(17,23)(18,22)(19,21)(25,30)(26,29)(27,28)(31,39)(32,38)(33,37)(34,36)(40,45)(41,44)(42,43)(46,54)(47,53)(48,52)(49,51)(55,60)(56,59)(57,58)(61,69)(62,68)(63,67)(64,66)(70,75)(71,74)(72,73)(76,83)(77,82)(78,81)(79,80)(84,90)(85,89)(86,88)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(121,128)(122,127)(123,126)(124,125)(129,135)(130,134)(131,133)(136,144)(137,143)(138,142)(139,141)(145,150)(146,149)(147,148)(151,152)(153,165)(154,164)(155,163)(156,162)(157,161)(158,160)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)(181,195)(182,194)(183,193)(184,192)(185,191)(186,190)(187,189) );
G=PermutationGroup([[(1,181,174,152,148,125,106,99,80,73,58,43,28),(2,182,175,153,149,126,107,100,81,74,59,44,29),(3,183,176,154,150,127,108,101,82,75,60,45,30),(4,184,177,155,136,128,109,102,83,61,46,31,16),(5,185,178,156,137,129,110,103,84,62,47,32,17),(6,186,179,157,138,130,111,104,85,63,48,33,18),(7,187,180,158,139,131,112,105,86,64,49,34,19),(8,188,166,159,140,132,113,91,87,65,50,35,20),(9,189,167,160,141,133,114,92,88,66,51,36,21),(10,190,168,161,142,134,115,93,89,67,52,37,22),(11,191,169,162,143,135,116,94,90,68,53,38,23),(12,192,170,163,144,121,117,95,76,69,54,39,24),(13,193,171,164,145,122,118,96,77,70,55,40,25),(14,194,172,165,146,123,119,97,78,71,56,41,26),(15,195,173,151,147,124,120,98,79,72,57,42,27)], [(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),(16,24),(17,23),(18,22),(19,21),(25,30),(26,29),(27,28),(31,39),(32,38),(33,37),(34,36),(40,45),(41,44),(42,43),(46,54),(47,53),(48,52),(49,51),(55,60),(56,59),(57,58),(61,69),(62,68),(63,67),(64,66),(70,75),(71,74),(72,73),(76,83),(77,82),(78,81),(79,80),(84,90),(85,89),(86,88),(92,105),(93,104),(94,103),(95,102),(96,101),(97,100),(98,99),(106,120),(107,119),(108,118),(109,117),(110,116),(111,115),(112,114),(121,128),(122,127),(123,126),(124,125),(129,135),(130,134),(131,133),(136,144),(137,143),(138,142),(139,141),(145,150),(146,149),(147,148),(151,152),(153,165),(154,164),(155,163),(156,162),(157,161),(158,160),(167,180),(168,179),(169,178),(170,177),(171,176),(172,175),(173,174),(181,195),(182,194),(183,193),(184,192),(185,191),(186,190),(187,189)]])
117 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 13A | ··· | 13L | 15A | 15B | 15C | 15D | 26A | ··· | 26L | 39A | ··· | 39L | 65A | ··· | 65X | 195A | ··· | 195AV |
| order | 1 | 2 | 3 | 5 | 5 | 13 | ··· | 13 | 15 | 15 | 15 | 15 | 26 | ··· | 26 | 39 | ··· | 39 | 65 | ··· | 65 | 195 | ··· | 195 |
| size | 1 | 15 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 15 | ··· | 15 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C13 | C26 | S3 | D5 | D15 | S3×C13 | D5×C13 | C13×D15 |
| kernel | C13×D15 | C195 | D15 | C15 | C65 | C39 | C13 | C5 | C3 | C1 |
| # reps | 1 | 1 | 12 | 12 | 1 | 2 | 4 | 12 | 24 | 48 |
Matrix representation of C13×D15 ►in GL2(𝔽1171) generated by
| 224 | 0 |
| 0 | 224 |
| 764 | 911 |
| 625 | 218 |
| 364 | 322 |
| 1025 | 807 |
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