direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D7×C33, C7⋊3C66, C77⋊7C6, C231⋊6C2, C21⋊2C22, SmallGroup(462,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C33 |
Generators and relations for D7×C33
G = < a,b,c | a33=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C33
►On 231 points
Generators in S231
(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)
(1 181 43 74 160 114 206)(2 182 44 75 161 115 207)(3 183 45 76 162 116 208)(4 184 46 77 163 117 209)(5 185 47 78 164 118 210)(6 186 48 79 165 119 211)(7 187 49 80 133 120 212)(8 188 50 81 134 121 213)(9 189 51 82 135 122 214)(10 190 52 83 136 123 215)(11 191 53 84 137 124 216)(12 192 54 85 138 125 217)(13 193 55 86 139 126 218)(14 194 56 87 140 127 219)(15 195 57 88 141 128 220)(16 196 58 89 142 129 221)(17 197 59 90 143 130 222)(18 198 60 91 144 131 223)(19 166 61 92 145 132 224)(20 167 62 93 146 100 225)(21 168 63 94 147 101 226)(22 169 64 95 148 102 227)(23 170 65 96 149 103 228)(24 171 66 97 150 104 229)(25 172 34 98 151 105 230)(26 173 35 99 152 106 231)(27 174 36 67 153 107 199)(28 175 37 68 154 108 200)(29 176 38 69 155 109 201)(30 177 39 70 156 110 202)(31 178 40 71 157 111 203)(32 179 41 72 158 112 204)(33 180 42 73 159 113 205)
(1 206)(2 207)(3 208)(4 209)(5 210)(6 211)(7 212)(8 213)(9 214)(10 215)(11 216)(12 217)(13 218)(14 219)(15 220)(16 221)(17 222)(18 223)(19 224)(20 225)(21 226)(22 227)(23 228)(24 229)(25 230)(26 231)(27 199)(28 200)(29 201)(30 202)(31 203)(32 204)(33 205)(34 151)(35 152)(36 153)(37 154)(38 155)(39 156)(40 157)(41 158)(42 159)(43 160)(44 161)(45 162)(46 163)(47 164)(48 165)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 139)(56 140)(57 141)(58 142)(59 143)(60 144)(61 145)(62 146)(63 147)(64 148)(65 149)(66 150)(100 167)(101 168)(102 169)(103 170)(104 171)(105 172)(106 173)(107 174)(108 175)(109 176)(110 177)(111 178)(112 179)(113 180)(114 181)(115 182)(116 183)(117 184)(118 185)(119 186)(120 187)(121 188)(122 189)(123 190)(124 191)(125 192)(126 193)(127 194)(128 195)(129 196)(130 197)(131 198)(132 166)
G:=sub<Sym(231)| (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), (1,181,43,74,160,114,206)(2,182,44,75,161,115,207)(3,183,45,76,162,116,208)(4,184,46,77,163,117,209)(5,185,47,78,164,118,210)(6,186,48,79,165,119,211)(7,187,49,80,133,120,212)(8,188,50,81,134,121,213)(9,189,51,82,135,122,214)(10,190,52,83,136,123,215)(11,191,53,84,137,124,216)(12,192,54,85,138,125,217)(13,193,55,86,139,126,218)(14,194,56,87,140,127,219)(15,195,57,88,141,128,220)(16,196,58,89,142,129,221)(17,197,59,90,143,130,222)(18,198,60,91,144,131,223)(19,166,61,92,145,132,224)(20,167,62,93,146,100,225)(21,168,63,94,147,101,226)(22,169,64,95,148,102,227)(23,170,65,96,149,103,228)(24,171,66,97,150,104,229)(25,172,34,98,151,105,230)(26,173,35,99,152,106,231)(27,174,36,67,153,107,199)(28,175,37,68,154,108,200)(29,176,38,69,155,109,201)(30,177,39,70,156,110,202)(31,178,40,71,157,111,203)(32,179,41,72,158,112,204)(33,180,42,73,159,113,205), (1,206)(2,207)(3,208)(4,209)(5,210)(6,211)(7,212)(8,213)(9,214)(10,215)(11,216)(12,217)(13,218)(14,219)(15,220)(16,221)(17,222)(18,223)(19,224)(20,225)(21,226)(22,227)(23,228)(24,229)(25,230)(26,231)(27,199)(28,200)(29,201)(30,202)(31,203)(32,204)(33,205)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,157)(41,158)(42,159)(43,160)(44,161)(45,162)(46,163)(47,164)(48,165)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,139)(56,140)(57,141)(58,142)(59,143)(60,144)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(100,167)(101,168)(102,169)(103,170)(104,171)(105,172)(106,173)(107,174)(108,175)(109,176)(110,177)(111,178)(112,179)(113,180)(114,181)(115,182)(116,183)(117,184)(118,185)(119,186)(120,187)(121,188)(122,189)(123,190)(124,191)(125,192)(126,193)(127,194)(128,195)(129,196)(130,197)(131,198)(132,166)>;
G:=Group( (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), (1,181,43,74,160,114,206)(2,182,44,75,161,115,207)(3,183,45,76,162,116,208)(4,184,46,77,163,117,209)(5,185,47,78,164,118,210)(6,186,48,79,165,119,211)(7,187,49,80,133,120,212)(8,188,50,81,134,121,213)(9,189,51,82,135,122,214)(10,190,52,83,136,123,215)(11,191,53,84,137,124,216)(12,192,54,85,138,125,217)(13,193,55,86,139,126,218)(14,194,56,87,140,127,219)(15,195,57,88,141,128,220)(16,196,58,89,142,129,221)(17,197,59,90,143,130,222)(18,198,60,91,144,131,223)(19,166,61,92,145,132,224)(20,167,62,93,146,100,225)(21,168,63,94,147,101,226)(22,169,64,95,148,102,227)(23,170,65,96,149,103,228)(24,171,66,97,150,104,229)(25,172,34,98,151,105,230)(26,173,35,99,152,106,231)(27,174,36,67,153,107,199)(28,175,37,68,154,108,200)(29,176,38,69,155,109,201)(30,177,39,70,156,110,202)(31,178,40,71,157,111,203)(32,179,41,72,158,112,204)(33,180,42,73,159,113,205), (1,206)(2,207)(3,208)(4,209)(5,210)(6,211)(7,212)(8,213)(9,214)(10,215)(11,216)(12,217)(13,218)(14,219)(15,220)(16,221)(17,222)(18,223)(19,224)(20,225)(21,226)(22,227)(23,228)(24,229)(25,230)(26,231)(27,199)(28,200)(29,201)(30,202)(31,203)(32,204)(33,205)(34,151)(35,152)(36,153)(37,154)(38,155)(39,156)(40,157)(41,158)(42,159)(43,160)(44,161)(45,162)(46,163)(47,164)(48,165)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,139)(56,140)(57,141)(58,142)(59,143)(60,144)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(100,167)(101,168)(102,169)(103,170)(104,171)(105,172)(106,173)(107,174)(108,175)(109,176)(110,177)(111,178)(112,179)(113,180)(114,181)(115,182)(116,183)(117,184)(118,185)(119,186)(120,187)(121,188)(122,189)(123,190)(124,191)(125,192)(126,193)(127,194)(128,195)(129,196)(130,197)(131,198)(132,166) );
G=PermutationGroup([[(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)], [(1,181,43,74,160,114,206),(2,182,44,75,161,115,207),(3,183,45,76,162,116,208),(4,184,46,77,163,117,209),(5,185,47,78,164,118,210),(6,186,48,79,165,119,211),(7,187,49,80,133,120,212),(8,188,50,81,134,121,213),(9,189,51,82,135,122,214),(10,190,52,83,136,123,215),(11,191,53,84,137,124,216),(12,192,54,85,138,125,217),(13,193,55,86,139,126,218),(14,194,56,87,140,127,219),(15,195,57,88,141,128,220),(16,196,58,89,142,129,221),(17,197,59,90,143,130,222),(18,198,60,91,144,131,223),(19,166,61,92,145,132,224),(20,167,62,93,146,100,225),(21,168,63,94,147,101,226),(22,169,64,95,148,102,227),(23,170,65,96,149,103,228),(24,171,66,97,150,104,229),(25,172,34,98,151,105,230),(26,173,35,99,152,106,231),(27,174,36,67,153,107,199),(28,175,37,68,154,108,200),(29,176,38,69,155,109,201),(30,177,39,70,156,110,202),(31,178,40,71,157,111,203),(32,179,41,72,158,112,204),(33,180,42,73,159,113,205)], [(1,206),(2,207),(3,208),(4,209),(5,210),(6,211),(7,212),(8,213),(9,214),(10,215),(11,216),(12,217),(13,218),(14,219),(15,220),(16,221),(17,222),(18,223),(19,224),(20,225),(21,226),(22,227),(23,228),(24,229),(25,230),(26,231),(27,199),(28,200),(29,201),(30,202),(31,203),(32,204),(33,205),(34,151),(35,152),(36,153),(37,154),(38,155),(39,156),(40,157),(41,158),(42,159),(43,160),(44,161),(45,162),(46,163),(47,164),(48,165),(49,133),(50,134),(51,135),(52,136),(53,137),(54,138),(55,139),(56,140),(57,141),(58,142),(59,143),(60,144),(61,145),(62,146),(63,147),(64,148),(65,149),(66,150),(100,167),(101,168),(102,169),(103,170),(104,171),(105,172),(106,173),(107,174),(108,175),(109,176),(110,177),(111,178),(112,179),(113,180),(114,181),(115,182),(116,183),(117,184),(118,185),(119,186),(120,187),(121,188),(122,189),(123,190),(124,191),(125,192),(126,193),(127,194),(128,195),(129,196),(130,197),(131,198),(132,166)]])
165 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | 7B | 7C | 11A | ··· | 11J | 21A | ··· | 21F | 22A | ··· | 22J | 33A | ··· | 33T | 66A | ··· | 66T | 77A | ··· | 77AD | 231A | ··· | 231BH |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | 7 | 7 | 11 | ··· | 11 | 21 | ··· | 21 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 | 77 | ··· | 77 | 231 | ··· | 231 |
| size | 1 | 7 | 1 | 1 | 7 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 |
165 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C6 | C11 | C22 | C33 | C66 | D7 | C3×D7 | C11×D7 | D7×C33 |
| kernel | D7×C33 | C231 | C11×D7 | C77 | C3×D7 | C21 | D7 | C7 | C33 | C11 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 10 | 10 | 20 | 20 | 3 | 6 | 30 | 60 |
Matrix representation of D7×C33 ►in GL3(𝔽463) generated by
| 94 | 0 | 0 |
| 0 | 158 | 0 |
| 0 | 0 | 158 |
| 1 | 0 | 0 |
| 0 | 320 | 1 |
| 0 | 462 | 0 |
| 462 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(463))| [94,0,0,0,158,0,0,0,158],[1,0,0,0,320,462,0,1,0],[462,0,0,0,0,1,0,1,0] >;
D7×C33 in GAP, Magma, Sage, TeX
D_7\times C_{33} % in TeX
G:=Group("D7xC33"); // GroupNames label
G:=SmallGroup(462,6);
// by ID
G=gap.SmallGroup(462,6);
# by ID
G:=PCGroup([4,-2,-3,-11,-7,6339]);
// Polycyclic
G:=Group<a,b,c|a^33=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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