direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×D33, C77⋊2S3, C33⋊1C14, C231⋊3C2, C21⋊3D11, C11⋊(S3×C7), C3⋊(C7×D11), SmallGroup(462,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — C7×D33 |
Generators and relations for C7×D33
G = < a,b,c | a7=b33=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D33
►On 231 points
Generators in S231
(1 215 174 161 116 75 41)(2 216 175 162 117 76 42)(3 217 176 163 118 77 43)(4 218 177 164 119 78 44)(5 219 178 165 120 79 45)(6 220 179 133 121 80 46)(7 221 180 134 122 81 47)(8 222 181 135 123 82 48)(9 223 182 136 124 83 49)(10 224 183 137 125 84 50)(11 225 184 138 126 85 51)(12 226 185 139 127 86 52)(13 227 186 140 128 87 53)(14 228 187 141 129 88 54)(15 229 188 142 130 89 55)(16 230 189 143 131 90 56)(17 231 190 144 132 91 57)(18 199 191 145 100 92 58)(19 200 192 146 101 93 59)(20 201 193 147 102 94 60)(21 202 194 148 103 95 61)(22 203 195 149 104 96 62)(23 204 196 150 105 97 63)(24 205 197 151 106 98 64)(25 206 198 152 107 99 65)(26 207 166 153 108 67 66)(27 208 167 154 109 68 34)(28 209 168 155 110 69 35)(29 210 169 156 111 70 36)(30 211 170 157 112 71 37)(31 212 171 158 113 72 38)(32 213 172 159 114 73 39)(33 214 173 160 115 74 40)
(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(48 66)(49 65)(50 64)(51 63)(52 62)(53 61)(54 60)(55 59)(56 58)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(100 131)(101 130)(102 129)(103 128)(104 127)(105 126)(106 125)(107 124)(108 123)(109 122)(110 121)(111 120)(112 119)(113 118)(114 117)(115 116)(133 155)(134 154)(135 153)(136 152)(137 151)(138 150)(139 149)(140 148)(141 147)(142 146)(143 145)(156 165)(157 164)(158 163)(159 162)(160 161)(166 181)(167 180)(168 179)(169 178)(170 177)(171 176)(172 175)(173 174)(182 198)(183 197)(184 196)(185 195)(186 194)(187 193)(188 192)(189 191)(199 230)(200 229)(201 228)(202 227)(203 226)(204 225)(205 224)(206 223)(207 222)(208 221)(209 220)(210 219)(211 218)(212 217)(213 216)(214 215)
G:=sub<Sym(231)| (1,215,174,161,116,75,41)(2,216,175,162,117,76,42)(3,217,176,163,118,77,43)(4,218,177,164,119,78,44)(5,219,178,165,120,79,45)(6,220,179,133,121,80,46)(7,221,180,134,122,81,47)(8,222,181,135,123,82,48)(9,223,182,136,124,83,49)(10,224,183,137,125,84,50)(11,225,184,138,126,85,51)(12,226,185,139,127,86,52)(13,227,186,140,128,87,53)(14,228,187,141,129,88,54)(15,229,188,142,130,89,55)(16,230,189,143,131,90,56)(17,231,190,144,132,91,57)(18,199,191,145,100,92,58)(19,200,192,146,101,93,59)(20,201,193,147,102,94,60)(21,202,194,148,103,95,61)(22,203,195,149,104,96,62)(23,204,196,150,105,97,63)(24,205,197,151,106,98,64)(25,206,198,152,107,99,65)(26,207,166,153,108,67,66)(27,208,167,154,109,68,34)(28,209,168,155,110,69,35)(29,210,169,156,111,70,36)(30,211,170,157,112,71,37)(31,212,171,158,113,72,38)(32,213,172,159,114,73,39)(33,214,173,160,115,74,40), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(100,131)(101,130)(102,129)(103,128)(104,127)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(111,120)(112,119)(113,118)(114,117)(115,116)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)(141,147)(142,146)(143,145)(156,165)(157,164)(158,163)(159,162)(160,161)(166,181)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)(182,198)(183,197)(184,196)(185,195)(186,194)(187,193)(188,192)(189,191)(199,230)(200,229)(201,228)(202,227)(203,226)(204,225)(205,224)(206,223)(207,222)(208,221)(209,220)(210,219)(211,218)(212,217)(213,216)(214,215)>;
G:=Group( (1,215,174,161,116,75,41)(2,216,175,162,117,76,42)(3,217,176,163,118,77,43)(4,218,177,164,119,78,44)(5,219,178,165,120,79,45)(6,220,179,133,121,80,46)(7,221,180,134,122,81,47)(8,222,181,135,123,82,48)(9,223,182,136,124,83,49)(10,224,183,137,125,84,50)(11,225,184,138,126,85,51)(12,226,185,139,127,86,52)(13,227,186,140,128,87,53)(14,228,187,141,129,88,54)(15,229,188,142,130,89,55)(16,230,189,143,131,90,56)(17,231,190,144,132,91,57)(18,199,191,145,100,92,58)(19,200,192,146,101,93,59)(20,201,193,147,102,94,60)(21,202,194,148,103,95,61)(22,203,195,149,104,96,62)(23,204,196,150,105,97,63)(24,205,197,151,106,98,64)(25,206,198,152,107,99,65)(26,207,166,153,108,67,66)(27,208,167,154,109,68,34)(28,209,168,155,110,69,35)(29,210,169,156,111,70,36)(30,211,170,157,112,71,37)(31,212,171,158,113,72,38)(32,213,172,159,114,73,39)(33,214,173,160,115,74,40), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(83,99)(84,98)(85,97)(86,96)(87,95)(88,94)(89,93)(90,92)(100,131)(101,130)(102,129)(103,128)(104,127)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(111,120)(112,119)(113,118)(114,117)(115,116)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)(141,147)(142,146)(143,145)(156,165)(157,164)(158,163)(159,162)(160,161)(166,181)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)(182,198)(183,197)(184,196)(185,195)(186,194)(187,193)(188,192)(189,191)(199,230)(200,229)(201,228)(202,227)(203,226)(204,225)(205,224)(206,223)(207,222)(208,221)(209,220)(210,219)(211,218)(212,217)(213,216)(214,215) );
G=PermutationGroup([[(1,215,174,161,116,75,41),(2,216,175,162,117,76,42),(3,217,176,163,118,77,43),(4,218,177,164,119,78,44),(5,219,178,165,120,79,45),(6,220,179,133,121,80,46),(7,221,180,134,122,81,47),(8,222,181,135,123,82,48),(9,223,182,136,124,83,49),(10,224,183,137,125,84,50),(11,225,184,138,126,85,51),(12,226,185,139,127,86,52),(13,227,186,140,128,87,53),(14,228,187,141,129,88,54),(15,229,188,142,130,89,55),(16,230,189,143,131,90,56),(17,231,190,144,132,91,57),(18,199,191,145,100,92,58),(19,200,192,146,101,93,59),(20,201,193,147,102,94,60),(21,202,194,148,103,95,61),(22,203,195,149,104,96,62),(23,204,196,150,105,97,63),(24,205,197,151,106,98,64),(25,206,198,152,107,99,65),(26,207,166,153,108,67,66),(27,208,167,154,109,68,34),(28,209,168,155,110,69,35),(29,210,169,156,111,70,36),(30,211,170,157,112,71,37),(31,212,171,158,113,72,38),(32,213,172,159,114,73,39),(33,214,173,160,115,74,40)], [(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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(48,66),(49,65),(50,64),(51,63),(52,62),(53,61),(54,60),(55,59),(56,58),(67,82),(68,81),(69,80),(70,79),(71,78),(72,77),(73,76),(74,75),(83,99),(84,98),(85,97),(86,96),(87,95),(88,94),(89,93),(90,92),(100,131),(101,130),(102,129),(103,128),(104,127),(105,126),(106,125),(107,124),(108,123),(109,122),(110,121),(111,120),(112,119),(113,118),(114,117),(115,116),(133,155),(134,154),(135,153),(136,152),(137,151),(138,150),(139,149),(140,148),(141,147),(142,146),(143,145),(156,165),(157,164),(158,163),(159,162),(160,161),(166,181),(167,180),(168,179),(169,178),(170,177),(171,176),(172,175),(173,174),(182,198),(183,197),(184,196),(185,195),(186,194),(187,193),(188,192),(189,191),(199,230),(200,229),(201,228),(202,227),(203,226),(204,225),(205,224),(206,223),(207,222),(208,221),(209,220),(210,219),(211,218),(212,217),(213,216),(214,215)]])
126 conjugacy classes
| class | 1 | 2 | 3 | 7A | ··· | 7F | 11A | ··· | 11E | 14A | ··· | 14F | 21A | ··· | 21F | 33A | ··· | 33J | 77A | ··· | 77AD | 231A | ··· | 231BH |
| order | 1 | 2 | 3 | 7 | ··· | 7 | 11 | ··· | 11 | 14 | ··· | 14 | 21 | ··· | 21 | 33 | ··· | 33 | 77 | ··· | 77 | 231 | ··· | 231 |
| size | 1 | 33 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 33 | ··· | 33 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C7 | C14 | S3 | D11 | S3×C7 | D33 | C7×D11 | C7×D33 |
| kernel | C7×D33 | C231 | D33 | C33 | C77 | C21 | C11 | C7 | C3 | C1 |
| # reps | 1 | 1 | 6 | 6 | 1 | 5 | 6 | 10 | 30 | 60 |
Matrix representation of C7×D33 ►in GL2(𝔽463) generated by
| 308 | 0 |
| 0 | 308 |
| 445 | 59 |
| 448 | 152 |
| 174 | 252 |
| 330 | 289 |
G:=sub<GL(2,GF(463))| [308,0,0,308],[445,448,59,152],[174,330,252,289] >;
C7×D33 in GAP, Magma, Sage, TeX
C_7\times D_{33} % in TeX
G:=Group("C7xD33"); // GroupNames label
G:=SmallGroup(462,9);
// by ID
G=gap.SmallGroup(462,9);
# by ID
G:=PCGroup([4,-2,-7,-3,-11,338,6723]);
// Polycyclic
G:=Group<a,b,c|a^7=b^33=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export