direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×D29, C203⋊2C2, C29⋊3C14, SmallGroup(406,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C29 — C7×D29 |
Generators and relations for C7×D29
G = < a,b,c | a7=b29=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D29
►On 203 points
Generators in S203
(1 181 168 132 113 73 32)(2 182 169 133 114 74 33)(3 183 170 134 115 75 34)(4 184 171 135 116 76 35)(5 185 172 136 88 77 36)(6 186 173 137 89 78 37)(7 187 174 138 90 79 38)(8 188 146 139 91 80 39)(9 189 147 140 92 81 40)(10 190 148 141 93 82 41)(11 191 149 142 94 83 42)(12 192 150 143 95 84 43)(13 193 151 144 96 85 44)(14 194 152 145 97 86 45)(15 195 153 117 98 87 46)(16 196 154 118 99 59 47)(17 197 155 119 100 60 48)(18 198 156 120 101 61 49)(19 199 157 121 102 62 50)(20 200 158 122 103 63 51)(21 201 159 123 104 64 52)(22 202 160 124 105 65 53)(23 203 161 125 106 66 54)(24 175 162 126 107 67 55)(25 176 163 127 108 68 56)(26 177 164 128 109 69 57)(27 178 165 129 110 70 58)(28 179 166 130 111 71 30)(29 180 167 131 112 72 31)
(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)
(1 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(30 33)(31 32)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(59 86)(60 85)(61 84)(62 83)(63 82)(64 81)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(88 108)(89 107)(90 106)(91 105)(92 104)(93 103)(94 102)(95 101)(96 100)(97 99)(109 116)(110 115)(111 114)(112 113)(118 145)(119 144)(120 143)(121 142)(122 141)(123 140)(124 139)(125 138)(126 137)(127 136)(128 135)(129 134)(130 133)(131 132)(146 160)(147 159)(148 158)(149 157)(150 156)(151 155)(152 154)(161 174)(162 173)(163 172)(164 171)(165 170)(166 169)(167 168)(175 186)(176 185)(177 184)(178 183)(179 182)(180 181)(187 203)(188 202)(189 201)(190 200)(191 199)(192 198)(193 197)(194 196)
G:=sub<Sym(203)| (1,181,168,132,113,73,32)(2,182,169,133,114,74,33)(3,183,170,134,115,75,34)(4,184,171,135,116,76,35)(5,185,172,136,88,77,36)(6,186,173,137,89,78,37)(7,187,174,138,90,79,38)(8,188,146,139,91,80,39)(9,189,147,140,92,81,40)(10,190,148,141,93,82,41)(11,191,149,142,94,83,42)(12,192,150,143,95,84,43)(13,193,151,144,96,85,44)(14,194,152,145,97,86,45)(15,195,153,117,98,87,46)(16,196,154,118,99,59,47)(17,197,155,119,100,60,48)(18,198,156,120,101,61,49)(19,199,157,121,102,62,50)(20,200,158,122,103,63,51)(21,201,159,123,104,64,52)(22,202,160,124,105,65,53)(23,203,161,125,106,66,54)(24,175,162,126,107,67,55)(25,176,163,127,108,68,56)(26,177,164,128,109,69,57)(27,178,165,129,110,70,58)(28,179,166,130,111,71,30)(29,180,167,131,112,72,31), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(30,33)(31,32)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(88,108)(89,107)(90,106)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(109,116)(110,115)(111,114)(112,113)(118,145)(119,144)(120,143)(121,142)(122,141)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)(146,160)(147,159)(148,158)(149,157)(150,156)(151,155)(152,154)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(175,186)(176,185)(177,184)(178,183)(179,182)(180,181)(187,203)(188,202)(189,201)(190,200)(191,199)(192,198)(193,197)(194,196)>;
G:=Group( (1,181,168,132,113,73,32)(2,182,169,133,114,74,33)(3,183,170,134,115,75,34)(4,184,171,135,116,76,35)(5,185,172,136,88,77,36)(6,186,173,137,89,78,37)(7,187,174,138,90,79,38)(8,188,146,139,91,80,39)(9,189,147,140,92,81,40)(10,190,148,141,93,82,41)(11,191,149,142,94,83,42)(12,192,150,143,95,84,43)(13,193,151,144,96,85,44)(14,194,152,145,97,86,45)(15,195,153,117,98,87,46)(16,196,154,118,99,59,47)(17,197,155,119,100,60,48)(18,198,156,120,101,61,49)(19,199,157,121,102,62,50)(20,200,158,122,103,63,51)(21,201,159,123,104,64,52)(22,202,160,124,105,65,53)(23,203,161,125,106,66,54)(24,175,162,126,107,67,55)(25,176,163,127,108,68,56)(26,177,164,128,109,69,57)(27,178,165,129,110,70,58)(28,179,166,130,111,71,30)(29,180,167,131,112,72,31), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(30,33)(31,32)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)(88,108)(89,107)(90,106)(91,105)(92,104)(93,103)(94,102)(95,101)(96,100)(97,99)(109,116)(110,115)(111,114)(112,113)(118,145)(119,144)(120,143)(121,142)(122,141)(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(131,132)(146,160)(147,159)(148,158)(149,157)(150,156)(151,155)(152,154)(161,174)(162,173)(163,172)(164,171)(165,170)(166,169)(167,168)(175,186)(176,185)(177,184)(178,183)(179,182)(180,181)(187,203)(188,202)(189,201)(190,200)(191,199)(192,198)(193,197)(194,196) );
G=PermutationGroup([[(1,181,168,132,113,73,32),(2,182,169,133,114,74,33),(3,183,170,134,115,75,34),(4,184,171,135,116,76,35),(5,185,172,136,88,77,36),(6,186,173,137,89,78,37),(7,187,174,138,90,79,38),(8,188,146,139,91,80,39),(9,189,147,140,92,81,40),(10,190,148,141,93,82,41),(11,191,149,142,94,83,42),(12,192,150,143,95,84,43),(13,193,151,144,96,85,44),(14,194,152,145,97,86,45),(15,195,153,117,98,87,46),(16,196,154,118,99,59,47),(17,197,155,119,100,60,48),(18,198,156,120,101,61,49),(19,199,157,121,102,62,50),(20,200,158,122,103,63,51),(21,201,159,123,104,64,52),(22,202,160,124,105,65,53),(23,203,161,125,106,66,54),(24,175,162,126,107,67,55),(25,176,163,127,108,68,56),(26,177,164,128,109,69,57),(27,178,165,129,110,70,58),(28,179,166,130,111,71,30),(29,180,167,131,112,72,31)], [(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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(30,33),(31,32),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(59,86),(60,85),(61,84),(62,83),(63,82),(64,81),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73),(88,108),(89,107),(90,106),(91,105),(92,104),(93,103),(94,102),(95,101),(96,100),(97,99),(109,116),(110,115),(111,114),(112,113),(118,145),(119,144),(120,143),(121,142),(122,141),(123,140),(124,139),(125,138),(126,137),(127,136),(128,135),(129,134),(130,133),(131,132),(146,160),(147,159),(148,158),(149,157),(150,156),(151,155),(152,154),(161,174),(162,173),(163,172),(164,171),(165,170),(166,169),(167,168),(175,186),(176,185),(177,184),(178,183),(179,182),(180,181),(187,203),(188,202),(189,201),(190,200),(191,199),(192,198),(193,197),(194,196)]])
112 conjugacy classes
| class | 1 | 2 | 7A | ··· | 7F | 14A | ··· | 14F | 29A | ··· | 29N | 203A | ··· | 203CF |
| order | 1 | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 29 | ··· | 29 | 203 | ··· | 203 |
| size | 1 | 29 | 1 | ··· | 1 | 29 | ··· | 29 | 2 | ··· | 2 | 2 | ··· | 2 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C7 | C14 | D29 | C7×D29 |
| kernel | C7×D29 | C203 | D29 | C29 | C7 | C1 |
| # reps | 1 | 1 | 6 | 6 | 14 | 84 |
Matrix representation of C7×D29 ►in GL2(𝔽2437) generated by
| 645 | 0 |
| 0 | 645 |
| 583 | 1 |
| 2436 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(2437))| [645,0,0,645],[583,2436,1,0],[0,1,1,0] >;
C7×D29 in GAP, Magma, Sage, TeX
C_7\times D_{29} % in TeX
G:=Group("C7xD29"); // GroupNames label
G:=SmallGroup(406,4);
// by ID
G=gap.SmallGroup(406,4);
# by ID
G:=PCGroup([3,-2,-7,-29,3530]);
// Polycyclic
G:=Group<a,b,c|a^7=b^29=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export