Copied to
clipboard

G = C13×D17order 442 = 2·13·17

Direct product of C13 and D17

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

Aliases: C13×D17, C17⋊C26, C2212C2, SmallGroup(442,2)

Series: Derived Chief Lower central Upper central

C1C17 — C13×D17
C1C17C221 — C13×D17
C17 — C13×D17
C1C13

Generators and relations for C13×D17
 G = < a,b,c | a13=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >

17C2
17C26

Smallest permutation representation of C13×D17
On 221 points
Generators in S221
(1 216 199 180 161 137 120 114 92 69 56 49 34)(2 217 200 181 162 138 121 115 93 70 57 50 18)(3 218 201 182 163 139 122 116 94 71 58 51 19)(4 219 202 183 164 140 123 117 95 72 59 35 20)(5 220 203 184 165 141 124 118 96 73 60 36 21)(6 221 204 185 166 142 125 119 97 74 61 37 22)(7 205 188 186 167 143 126 103 98 75 62 38 23)(8 206 189 187 168 144 127 104 99 76 63 39 24)(9 207 190 171 169 145 128 105 100 77 64 40 25)(10 208 191 172 170 146 129 106 101 78 65 41 26)(11 209 192 173 154 147 130 107 102 79 66 42 27)(12 210 193 174 155 148 131 108 86 80 67 43 28)(13 211 194 175 156 149 132 109 87 81 68 44 29)(14 212 195 176 157 150 133 110 88 82 52 45 30)(15 213 196 177 158 151 134 111 89 83 53 46 31)(16 214 197 178 159 152 135 112 90 84 54 47 32)(17 215 198 179 160 153 136 113 91 85 55 48 33)
(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)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)(33 34)(35 45)(36 44)(37 43)(38 42)(39 41)(46 51)(47 50)(48 49)(52 59)(53 58)(54 57)(55 56)(60 68)(61 67)(62 66)(63 65)(69 85)(70 84)(71 83)(72 82)(73 81)(74 80)(75 79)(76 78)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(98 102)(99 101)(103 107)(104 106)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)(120 136)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(137 153)(138 152)(139 151)(140 150)(141 149)(142 148)(143 147)(144 146)(154 167)(155 166)(156 165)(157 164)(158 163)(159 162)(160 161)(168 170)(172 187)(173 186)(174 185)(175 184)(176 183)(177 182)(178 181)(179 180)(188 192)(189 191)(193 204)(194 203)(195 202)(196 201)(197 200)(198 199)(205 209)(206 208)(210 221)(211 220)(212 219)(213 218)(214 217)(215 216)

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

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

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

130 conjugacy classes

class 1  2 13A···13L17A···17H26A···26L221A···221CR
order1213···1317···1726···26221···221
size1171···12···217···172···2

130 irreducible representations

dim111122
type+++
imageC1C2C13C26D17C13×D17
kernelC13×D17C221D17C17C13C1
# reps111212896

Matrix representation of C13×D17 in GL2(𝔽443) generated by

3560
0356
,
01
44244
,
01
10
G:=sub<GL(2,GF(443))| [356,0,0,356],[0,442,1,44],[0,1,1,0] >;

C13×D17 in GAP, Magma, Sage, TeX

C_{13}\times D_{17}
% in TeX

G:=Group("C13xD17");
// GroupNames label

G:=SmallGroup(442,2);
// by ID

G=gap.SmallGroup(442,2);
# by ID

G:=PCGroup([3,-2,-13,-17,3746]);
// Polycyclic

G:=Group<a,b,c|a^13=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C13×D17 in TeX

׿
×
𝔽