Copied to
clipboard

G = C13×D19order 494 = 2·13·19

Direct product of C13 and D19

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

Aliases: C13×D19, C19⋊C26, C2472C2, SmallGroup(494,2)

Series: Derived Chief Lower central Upper central

C1C19 — C13×D19
C1C19C247 — C13×D19
C19 — C13×D19
C1C13

Generators and relations for C13×D19
 G = < a,b,c | a13=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >

19C2
19C26

Smallest permutation representation of C13×D19
On 247 points
Generators in S247
(1 247 223 191 189 157 151 117 111 77 69 53 29)(2 229 224 192 190 158 152 118 112 78 70 54 30)(3 230 225 193 172 159 134 119 113 79 71 55 31)(4 231 226 194 173 160 135 120 114 80 72 56 32)(5 232 227 195 174 161 136 121 96 81 73 57 33)(6 233 228 196 175 162 137 122 97 82 74 39 34)(7 234 210 197 176 163 138 123 98 83 75 40 35)(8 235 211 198 177 164 139 124 99 84 76 41 36)(9 236 212 199 178 165 140 125 100 85 58 42 37)(10 237 213 200 179 166 141 126 101 86 59 43 38)(11 238 214 201 180 167 142 127 102 87 60 44 20)(12 239 215 202 181 168 143 128 103 88 61 45 21)(13 240 216 203 182 169 144 129 104 89 62 46 22)(14 241 217 204 183 170 145 130 105 90 63 47 23)(15 242 218 205 184 171 146 131 106 91 64 48 24)(16 243 219 206 185 153 147 132 107 92 65 49 25)(17 244 220 207 186 154 148 133 108 93 66 50 26)(18 245 221 208 187 155 149 115 109 94 67 51 27)(19 246 222 209 188 156 150 116 110 95 68 52 28)
(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)(191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209)(210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228)(229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(39 47)(40 46)(41 45)(42 44)(48 57)(49 56)(50 55)(51 54)(52 53)(58 60)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 95)(78 94)(79 93)(80 92)(81 91)(82 90)(83 89)(84 88)(85 87)(96 106)(97 105)(98 104)(99 103)(100 102)(107 114)(108 113)(109 112)(110 111)(115 118)(116 117)(119 133)(120 132)(121 131)(122 130)(123 129)(124 128)(125 127)(134 148)(135 147)(136 146)(137 145)(138 144)(139 143)(140 142)(149 152)(150 151)(153 160)(154 159)(155 158)(156 157)(161 171)(162 170)(163 169)(164 168)(165 167)(172 186)(173 185)(174 184)(175 183)(176 182)(177 181)(178 180)(187 190)(188 189)(191 209)(192 208)(193 207)(194 206)(195 205)(196 204)(197 203)(198 202)(199 201)(210 216)(211 215)(212 214)(217 228)(218 227)(219 226)(220 225)(221 224)(222 223)(229 245)(230 244)(231 243)(232 242)(233 241)(234 240)(235 239)(236 238)(246 247)

G:=sub<Sym(247)| (1,247,223,191,189,157,151,117,111,77,69,53,29)(2,229,224,192,190,158,152,118,112,78,70,54,30)(3,230,225,193,172,159,134,119,113,79,71,55,31)(4,231,226,194,173,160,135,120,114,80,72,56,32)(5,232,227,195,174,161,136,121,96,81,73,57,33)(6,233,228,196,175,162,137,122,97,82,74,39,34)(7,234,210,197,176,163,138,123,98,83,75,40,35)(8,235,211,198,177,164,139,124,99,84,76,41,36)(9,236,212,199,178,165,140,125,100,85,58,42,37)(10,237,213,200,179,166,141,126,101,86,59,43,38)(11,238,214,201,180,167,142,127,102,87,60,44,20)(12,239,215,202,181,168,143,128,103,88,61,45,21)(13,240,216,203,182,169,144,129,104,89,62,46,22)(14,241,217,204,183,170,145,130,105,90,63,47,23)(15,242,218,205,184,171,146,131,106,91,64,48,24)(16,243,219,206,185,153,147,132,107,92,65,49,25)(17,244,220,207,186,154,148,133,108,93,66,50,26)(18,245,221,208,187,155,149,115,109,94,67,51,27)(19,246,222,209,188,156,150,116,110,95,68,52,28), (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)(191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209)(210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228)(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(39,47)(40,46)(41,45)(42,44)(48,57)(49,56)(50,55)(51,54)(52,53)(58,60)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,95)(78,94)(79,93)(80,92)(81,91)(82,90)(83,89)(84,88)(85,87)(96,106)(97,105)(98,104)(99,103)(100,102)(107,114)(108,113)(109,112)(110,111)(115,118)(116,117)(119,133)(120,132)(121,131)(122,130)(123,129)(124,128)(125,127)(134,148)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(149,152)(150,151)(153,160)(154,159)(155,158)(156,157)(161,171)(162,170)(163,169)(164,168)(165,167)(172,186)(173,185)(174,184)(175,183)(176,182)(177,181)(178,180)(187,190)(188,189)(191,209)(192,208)(193,207)(194,206)(195,205)(196,204)(197,203)(198,202)(199,201)(210,216)(211,215)(212,214)(217,228)(218,227)(219,226)(220,225)(221,224)(222,223)(229,245)(230,244)(231,243)(232,242)(233,241)(234,240)(235,239)(236,238)(246,247)>;

G:=Group( (1,247,223,191,189,157,151,117,111,77,69,53,29)(2,229,224,192,190,158,152,118,112,78,70,54,30)(3,230,225,193,172,159,134,119,113,79,71,55,31)(4,231,226,194,173,160,135,120,114,80,72,56,32)(5,232,227,195,174,161,136,121,96,81,73,57,33)(6,233,228,196,175,162,137,122,97,82,74,39,34)(7,234,210,197,176,163,138,123,98,83,75,40,35)(8,235,211,198,177,164,139,124,99,84,76,41,36)(9,236,212,199,178,165,140,125,100,85,58,42,37)(10,237,213,200,179,166,141,126,101,86,59,43,38)(11,238,214,201,180,167,142,127,102,87,60,44,20)(12,239,215,202,181,168,143,128,103,88,61,45,21)(13,240,216,203,182,169,144,129,104,89,62,46,22)(14,241,217,204,183,170,145,130,105,90,63,47,23)(15,242,218,205,184,171,146,131,106,91,64,48,24)(16,243,219,206,185,153,147,132,107,92,65,49,25)(17,244,220,207,186,154,148,133,108,93,66,50,26)(18,245,221,208,187,155,149,115,109,94,67,51,27)(19,246,222,209,188,156,150,116,110,95,68,52,28), (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)(191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209)(210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228)(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(39,47)(40,46)(41,45)(42,44)(48,57)(49,56)(50,55)(51,54)(52,53)(58,60)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)(77,95)(78,94)(79,93)(80,92)(81,91)(82,90)(83,89)(84,88)(85,87)(96,106)(97,105)(98,104)(99,103)(100,102)(107,114)(108,113)(109,112)(110,111)(115,118)(116,117)(119,133)(120,132)(121,131)(122,130)(123,129)(124,128)(125,127)(134,148)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(149,152)(150,151)(153,160)(154,159)(155,158)(156,157)(161,171)(162,170)(163,169)(164,168)(165,167)(172,186)(173,185)(174,184)(175,183)(176,182)(177,181)(178,180)(187,190)(188,189)(191,209)(192,208)(193,207)(194,206)(195,205)(196,204)(197,203)(198,202)(199,201)(210,216)(211,215)(212,214)(217,228)(218,227)(219,226)(220,225)(221,224)(222,223)(229,245)(230,244)(231,243)(232,242)(233,241)(234,240)(235,239)(236,238)(246,247) );

G=PermutationGroup([(1,247,223,191,189,157,151,117,111,77,69,53,29),(2,229,224,192,190,158,152,118,112,78,70,54,30),(3,230,225,193,172,159,134,119,113,79,71,55,31),(4,231,226,194,173,160,135,120,114,80,72,56,32),(5,232,227,195,174,161,136,121,96,81,73,57,33),(6,233,228,196,175,162,137,122,97,82,74,39,34),(7,234,210,197,176,163,138,123,98,83,75,40,35),(8,235,211,198,177,164,139,124,99,84,76,41,36),(9,236,212,199,178,165,140,125,100,85,58,42,37),(10,237,213,200,179,166,141,126,101,86,59,43,38),(11,238,214,201,180,167,142,127,102,87,60,44,20),(12,239,215,202,181,168,143,128,103,88,61,45,21),(13,240,216,203,182,169,144,129,104,89,62,46,22),(14,241,217,204,183,170,145,130,105,90,63,47,23),(15,242,218,205,184,171,146,131,106,91,64,48,24),(16,243,219,206,185,153,147,132,107,92,65,49,25),(17,244,220,207,186,154,148,133,108,93,66,50,26),(18,245,221,208,187,155,149,115,109,94,67,51,27),(19,246,222,209,188,156,150,116,110,95,68,52,28)], [(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),(191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209),(210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228),(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,37),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,30),(28,29),(39,47),(40,46),(41,45),(42,44),(48,57),(49,56),(50,55),(51,54),(52,53),(58,60),(61,76),(62,75),(63,74),(64,73),(65,72),(66,71),(67,70),(68,69),(77,95),(78,94),(79,93),(80,92),(81,91),(82,90),(83,89),(84,88),(85,87),(96,106),(97,105),(98,104),(99,103),(100,102),(107,114),(108,113),(109,112),(110,111),(115,118),(116,117),(119,133),(120,132),(121,131),(122,130),(123,129),(124,128),(125,127),(134,148),(135,147),(136,146),(137,145),(138,144),(139,143),(140,142),(149,152),(150,151),(153,160),(154,159),(155,158),(156,157),(161,171),(162,170),(163,169),(164,168),(165,167),(172,186),(173,185),(174,184),(175,183),(176,182),(177,181),(178,180),(187,190),(188,189),(191,209),(192,208),(193,207),(194,206),(195,205),(196,204),(197,203),(198,202),(199,201),(210,216),(211,215),(212,214),(217,228),(218,227),(219,226),(220,225),(221,224),(222,223),(229,245),(230,244),(231,243),(232,242),(233,241),(234,240),(235,239),(236,238),(246,247)])

143 conjugacy classes

class 1  2 13A···13L19A···19I26A···26L247A···247DD
order1213···1319···1926···26247···247
size1191···12···219···192···2

143 irreducible representations

dim111122
type+++
imageC1C2C13C26D19C13×D19
kernelC13×D19C247D19C19C13C1
# reps1112129108

Matrix representation of C13×D19 in GL2(𝔽1483) generated by

13180
01318
,
62482
14827
,
9671116
677516
G:=sub<GL(2,GF(1483))| [1318,0,0,1318],[624,1482,82,7],[967,677,1116,516] >;

C13×D19 in GAP, Magma, Sage, TeX

C_{13}\times D_{19}
% in TeX

G:=Group("C13xD19");
// GroupNames label

G:=SmallGroup(494,2);
// by ID

G=gap.SmallGroup(494,2);
# by ID

G:=PCGroup([3,-2,-13,-19,4214]);
// Polycyclic

G:=Group<a,b,c|a^13=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C13×D19 in TeX

׿
×
𝔽