direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D7×C31, C7⋊C62, C217⋊3C2, SmallGroup(434,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C31 |
Generators and relations for D7×C31
G = < a,b,c | a31=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C31
►On 217 points
Generators in S217
(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)
(1 178 117 206 45 152 81)(2 179 118 207 46 153 82)(3 180 119 208 47 154 83)(4 181 120 209 48 155 84)(5 182 121 210 49 125 85)(6 183 122 211 50 126 86)(7 184 123 212 51 127 87)(8 185 124 213 52 128 88)(9 186 94 214 53 129 89)(10 156 95 215 54 130 90)(11 157 96 216 55 131 91)(12 158 97 217 56 132 92)(13 159 98 187 57 133 93)(14 160 99 188 58 134 63)(15 161 100 189 59 135 64)(16 162 101 190 60 136 65)(17 163 102 191 61 137 66)(18 164 103 192 62 138 67)(19 165 104 193 32 139 68)(20 166 105 194 33 140 69)(21 167 106 195 34 141 70)(22 168 107 196 35 142 71)(23 169 108 197 36 143 72)(24 170 109 198 37 144 73)(25 171 110 199 38 145 74)(26 172 111 200 39 146 75)(27 173 112 201 40 147 76)(28 174 113 202 41 148 77)(29 175 114 203 42 149 78)(30 176 115 204 43 150 79)(31 177 116 205 44 151 80)
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 93)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 94)(54 95)(55 96)(56 97)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(125 182)(126 183)(127 184)(128 185)(129 186)(130 156)(131 157)(132 158)(133 159)(134 160)(135 161)(136 162)(137 163)(138 164)(139 165)(140 166)(141 167)(142 168)(143 169)(144 170)(145 171)(146 172)(147 173)(148 174)(149 175)(150 176)(151 177)(152 178)(153 179)(154 180)(155 181)
G:=sub<Sym(217)| (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), (1,178,117,206,45,152,81)(2,179,118,207,46,153,82)(3,180,119,208,47,154,83)(4,181,120,209,48,155,84)(5,182,121,210,49,125,85)(6,183,122,211,50,126,86)(7,184,123,212,51,127,87)(8,185,124,213,52,128,88)(9,186,94,214,53,129,89)(10,156,95,215,54,130,90)(11,157,96,216,55,131,91)(12,158,97,217,56,132,92)(13,159,98,187,57,133,93)(14,160,99,188,58,134,63)(15,161,100,189,59,135,64)(16,162,101,190,60,136,65)(17,163,102,191,61,137,66)(18,164,103,192,62,138,67)(19,165,104,193,32,139,68)(20,166,105,194,33,140,69)(21,167,106,195,34,141,70)(22,168,107,196,35,142,71)(23,169,108,197,36,143,72)(24,170,109,198,37,144,73)(25,171,110,199,38,145,74)(26,172,111,200,39,146,75)(27,173,112,201,40,147,76)(28,174,113,202,41,148,77)(29,175,114,203,42,149,78)(30,176,115,204,43,150,79)(31,177,116,205,44,151,80), (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(125,182)(126,183)(127,184)(128,185)(129,186)(130,156)(131,157)(132,158)(133,159)(134,160)(135,161)(136,162)(137,163)(138,164)(139,165)(140,166)(141,167)(142,168)(143,169)(144,170)(145,171)(146,172)(147,173)(148,174)(149,175)(150,176)(151,177)(152,178)(153,179)(154,180)(155,181)>;
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), (1,178,117,206,45,152,81)(2,179,118,207,46,153,82)(3,180,119,208,47,154,83)(4,181,120,209,48,155,84)(5,182,121,210,49,125,85)(6,183,122,211,50,126,86)(7,184,123,212,51,127,87)(8,185,124,213,52,128,88)(9,186,94,214,53,129,89)(10,156,95,215,54,130,90)(11,157,96,216,55,131,91)(12,158,97,217,56,132,92)(13,159,98,187,57,133,93)(14,160,99,188,58,134,63)(15,161,100,189,59,135,64)(16,162,101,190,60,136,65)(17,163,102,191,61,137,66)(18,164,103,192,62,138,67)(19,165,104,193,32,139,68)(20,166,105,194,33,140,69)(21,167,106,195,34,141,70)(22,168,107,196,35,142,71)(23,169,108,197,36,143,72)(24,170,109,198,37,144,73)(25,171,110,199,38,145,74)(26,172,111,200,39,146,75)(27,173,112,201,40,147,76)(28,174,113,202,41,148,77)(29,175,114,203,42,149,78)(30,176,115,204,43,150,79)(31,177,116,205,44,151,80), (1,81)(2,82)(3,83)(4,84)(5,85)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,113)(42,114)(43,115)(44,116)(45,117)(46,118)(47,119)(48,120)(49,121)(50,122)(51,123)(52,124)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(125,182)(126,183)(127,184)(128,185)(129,186)(130,156)(131,157)(132,158)(133,159)(134,160)(135,161)(136,162)(137,163)(138,164)(139,165)(140,166)(141,167)(142,168)(143,169)(144,170)(145,171)(146,172)(147,173)(148,174)(149,175)(150,176)(151,177)(152,178)(153,179)(154,180)(155,181) );
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)], [(1,178,117,206,45,152,81),(2,179,118,207,46,153,82),(3,180,119,208,47,154,83),(4,181,120,209,48,155,84),(5,182,121,210,49,125,85),(6,183,122,211,50,126,86),(7,184,123,212,51,127,87),(8,185,124,213,52,128,88),(9,186,94,214,53,129,89),(10,156,95,215,54,130,90),(11,157,96,216,55,131,91),(12,158,97,217,56,132,92),(13,159,98,187,57,133,93),(14,160,99,188,58,134,63),(15,161,100,189,59,135,64),(16,162,101,190,60,136,65),(17,163,102,191,61,137,66),(18,164,103,192,62,138,67),(19,165,104,193,32,139,68),(20,166,105,194,33,140,69),(21,167,106,195,34,141,70),(22,168,107,196,35,142,71),(23,169,108,197,36,143,72),(24,170,109,198,37,144,73),(25,171,110,199,38,145,74),(26,172,111,200,39,146,75),(27,173,112,201,40,147,76),(28,174,113,202,41,148,77),(29,175,114,203,42,149,78),(30,176,115,204,43,150,79),(31,177,116,205,44,151,80)], [(1,81),(2,82),(3,83),(4,84),(5,85),(6,86),(7,87),(8,88),(9,89),(10,90),(11,91),(12,92),(13,93),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,104),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,113),(42,114),(43,115),(44,116),(45,117),(46,118),(47,119),(48,120),(49,121),(50,122),(51,123),(52,124),(53,94),(54,95),(55,96),(56,97),(57,98),(58,99),(59,100),(60,101),(61,102),(62,103),(125,182),(126,183),(127,184),(128,185),(129,186),(130,156),(131,157),(132,158),(133,159),(134,160),(135,161),(136,162),(137,163),(138,164),(139,165),(140,166),(141,167),(142,168),(143,169),(144,170),(145,171),(146,172),(147,173),(148,174),(149,175),(150,176),(151,177),(152,178),(153,179),(154,180),(155,181)]])
155 conjugacy classes
| class | 1 | 2 | 7A | 7B | 7C | 31A | ··· | 31AD | 62A | ··· | 62AD | 217A | ··· | 217CL |
| order | 1 | 2 | 7 | 7 | 7 | 31 | ··· | 31 | 62 | ··· | 62 | 217 | ··· | 217 |
| size | 1 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 |
155 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C31 | C62 | D7 | D7×C31 |
| kernel | D7×C31 | C217 | D7 | C7 | C31 | C1 |
| # reps | 1 | 1 | 30 | 30 | 3 | 90 |
Matrix representation of D7×C31 ►in GL2(𝔽1303) generated by
| 784 | 0 |
| 0 | 784 |
| 936 | 1 |
| 1190 | 824 |
| 824 | 1302 |
| 112 | 479 |
G:=sub<GL(2,GF(1303))| [784,0,0,784],[936,1190,1,824],[824,112,1302,479] >;
D7×C31 in GAP, Magma, Sage, TeX
D_7\times C_{31} % in TeX
G:=Group("D7xC31"); // GroupNames label
G:=SmallGroup(434,1);
// by ID
G=gap.SmallGroup(434,1);
# by ID
G:=PCGroup([3,-2,-31,-7,3350]);
// Polycyclic
G:=Group<a,b,c|a^31=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export