direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×D31, C31⋊C14, C217⋊2C2, SmallGroup(434,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C7×D31 |
Generators and relations for C7×D31
G = < a,b,c | a7=b31=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C7×D31
►On 217 points
Generators in S217
(1 215 170 143 99 92 51)(2 216 171 144 100 93 52)(3 217 172 145 101 63 53)(4 187 173 146 102 64 54)(5 188 174 147 103 65 55)(6 189 175 148 104 66 56)(7 190 176 149 105 67 57)(8 191 177 150 106 68 58)(9 192 178 151 107 69 59)(10 193 179 152 108 70 60)(11 194 180 153 109 71 61)(12 195 181 154 110 72 62)(13 196 182 155 111 73 32)(14 197 183 125 112 74 33)(15 198 184 126 113 75 34)(16 199 185 127 114 76 35)(17 200 186 128 115 77 36)(18 201 156 129 116 78 37)(19 202 157 130 117 79 38)(20 203 158 131 118 80 39)(21 204 159 132 119 81 40)(22 205 160 133 120 82 41)(23 206 161 134 121 83 42)(24 207 162 135 122 84 43)(25 208 163 136 123 85 44)(26 209 164 137 124 86 45)(27 210 165 138 94 87 46)(28 211 166 139 95 88 47)(29 212 167 140 96 89 48)(30 213 168 141 97 90 49)(31 214 169 142 98 91 50)
(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 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(32 38)(33 37)(34 36)(39 62)(40 61)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(63 89)(64 88)(65 87)(66 86)(67 85)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)(90 93)(91 92)(94 103)(95 102)(96 101)(97 100)(98 99)(104 124)(105 123)(106 122)(107 121)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)(125 129)(126 128)(130 155)(131 154)(132 153)(133 152)(134 151)(135 150)(136 149)(137 148)(138 147)(139 146)(140 145)(141 144)(142 143)(156 183)(157 182)(158 181)(159 180)(160 179)(161 178)(162 177)(163 176)(164 175)(165 174)(166 173)(167 172)(168 171)(169 170)(184 186)(187 211)(188 210)(189 209)(190 208)(191 207)(192 206)(193 205)(194 204)(195 203)(196 202)(197 201)(198 200)(212 217)(213 216)(214 215)
G:=sub<Sym(217)| (1,215,170,143,99,92,51)(2,216,171,144,100,93,52)(3,217,172,145,101,63,53)(4,187,173,146,102,64,54)(5,188,174,147,103,65,55)(6,189,175,148,104,66,56)(7,190,176,149,105,67,57)(8,191,177,150,106,68,58)(9,192,178,151,107,69,59)(10,193,179,152,108,70,60)(11,194,180,153,109,71,61)(12,195,181,154,110,72,62)(13,196,182,155,111,73,32)(14,197,183,125,112,74,33)(15,198,184,126,113,75,34)(16,199,185,127,114,76,35)(17,200,186,128,115,77,36)(18,201,156,129,116,78,37)(19,202,157,130,117,79,38)(20,203,158,131,118,80,39)(21,204,159,132,119,81,40)(22,205,160,133,120,82,41)(23,206,161,134,121,83,42)(24,207,162,135,122,84,43)(25,208,163,136,123,85,44)(26,209,164,137,124,86,45)(27,210,165,138,94,87,46)(28,211,166,139,95,88,47)(29,212,167,140,96,89,48)(30,213,168,141,97,90,49)(31,214,169,142,98,91,50), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(32,38)(33,37)(34,36)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(90,93)(91,92)(94,103)(95,102)(96,101)(97,100)(98,99)(104,124)(105,123)(106,122)(107,121)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)(125,129)(126,128)(130,155)(131,154)(132,153)(133,152)(134,151)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)(141,144)(142,143)(156,183)(157,182)(158,181)(159,180)(160,179)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)(184,186)(187,211)(188,210)(189,209)(190,208)(191,207)(192,206)(193,205)(194,204)(195,203)(196,202)(197,201)(198,200)(212,217)(213,216)(214,215)>;
G:=Group( (1,215,170,143,99,92,51)(2,216,171,144,100,93,52)(3,217,172,145,101,63,53)(4,187,173,146,102,64,54)(5,188,174,147,103,65,55)(6,189,175,148,104,66,56)(7,190,176,149,105,67,57)(8,191,177,150,106,68,58)(9,192,178,151,107,69,59)(10,193,179,152,108,70,60)(11,194,180,153,109,71,61)(12,195,181,154,110,72,62)(13,196,182,155,111,73,32)(14,197,183,125,112,74,33)(15,198,184,126,113,75,34)(16,199,185,127,114,76,35)(17,200,186,128,115,77,36)(18,201,156,129,116,78,37)(19,202,157,130,117,79,38)(20,203,158,131,118,80,39)(21,204,159,132,119,81,40)(22,205,160,133,120,82,41)(23,206,161,134,121,83,42)(24,207,162,135,122,84,43)(25,208,163,136,123,85,44)(26,209,164,137,124,86,45)(27,210,165,138,94,87,46)(28,211,166,139,95,88,47)(29,212,167,140,96,89,48)(30,213,168,141,97,90,49)(31,214,169,142,98,91,50), (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), (1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(32,38)(33,37)(34,36)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(90,93)(91,92)(94,103)(95,102)(96,101)(97,100)(98,99)(104,124)(105,123)(106,122)(107,121)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)(125,129)(126,128)(130,155)(131,154)(132,153)(133,152)(134,151)(135,150)(136,149)(137,148)(138,147)(139,146)(140,145)(141,144)(142,143)(156,183)(157,182)(158,181)(159,180)(160,179)(161,178)(162,177)(163,176)(164,175)(165,174)(166,173)(167,172)(168,171)(169,170)(184,186)(187,211)(188,210)(189,209)(190,208)(191,207)(192,206)(193,205)(194,204)(195,203)(196,202)(197,201)(198,200)(212,217)(213,216)(214,215) );
G=PermutationGroup([[(1,215,170,143,99,92,51),(2,216,171,144,100,93,52),(3,217,172,145,101,63,53),(4,187,173,146,102,64,54),(5,188,174,147,103,65,55),(6,189,175,148,104,66,56),(7,190,176,149,105,67,57),(8,191,177,150,106,68,58),(9,192,178,151,107,69,59),(10,193,179,152,108,70,60),(11,194,180,153,109,71,61),(12,195,181,154,110,72,62),(13,196,182,155,111,73,32),(14,197,183,125,112,74,33),(15,198,184,126,113,75,34),(16,199,185,127,114,76,35),(17,200,186,128,115,77,36),(18,201,156,129,116,78,37),(19,202,157,130,117,79,38),(20,203,158,131,118,80,39),(21,204,159,132,119,81,40),(22,205,160,133,120,82,41),(23,206,161,134,121,83,42),(24,207,162,135,122,84,43),(25,208,163,136,123,85,44),(26,209,164,137,124,86,45),(27,210,165,138,94,87,46),(28,211,166,139,95,88,47),(29,212,167,140,96,89,48),(30,213,168,141,97,90,49),(31,214,169,142,98,91,50)], [(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)], [(1,31),(2,30),(3,29),(4,28),(5,27),(6,26),(7,25),(8,24),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(15,17),(32,38),(33,37),(34,36),(39,62),(40,61),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(63,89),(64,88),(65,87),(66,86),(67,85),(68,84),(69,83),(70,82),(71,81),(72,80),(73,79),(74,78),(75,77),(90,93),(91,92),(94,103),(95,102),(96,101),(97,100),(98,99),(104,124),(105,123),(106,122),(107,121),(108,120),(109,119),(110,118),(111,117),(112,116),(113,115),(125,129),(126,128),(130,155),(131,154),(132,153),(133,152),(134,151),(135,150),(136,149),(137,148),(138,147),(139,146),(140,145),(141,144),(142,143),(156,183),(157,182),(158,181),(159,180),(160,179),(161,178),(162,177),(163,176),(164,175),(165,174),(166,173),(167,172),(168,171),(169,170),(184,186),(187,211),(188,210),(189,209),(190,208),(191,207),(192,206),(193,205),(194,204),(195,203),(196,202),(197,201),(198,200),(212,217),(213,216),(214,215)]])
119 conjugacy classes
| class | 1 | 2 | 7A | ··· | 7F | 14A | ··· | 14F | 31A | ··· | 31O | 217A | ··· | 217CL |
| order | 1 | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 31 | ··· | 31 | 217 | ··· | 217 |
| size | 1 | 31 | 1 | ··· | 1 | 31 | ··· | 31 | 2 | ··· | 2 | 2 | ··· | 2 |
119 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C7 | C14 | D31 | C7×D31 |
| kernel | C7×D31 | C217 | D31 | C31 | C7 | C1 |
| # reps | 1 | 1 | 6 | 6 | 15 | 90 |
Matrix representation of C7×D31 ►in GL2(𝔽1303) generated by
| 52 | 0 |
| 0 | 52 |
| 0 | 1 |
| 1302 | 534 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(1303))| [52,0,0,52],[0,1302,1,534],[0,1,1,0] >;
C7×D31 in GAP, Magma, Sage, TeX
C_7\times D_{31} % in TeX
G:=Group("C7xD31"); // GroupNames label
G:=SmallGroup(434,2);
// by ID
G=gap.SmallGroup(434,2);
# by ID
G:=PCGroup([3,-2,-7,-31,3782]);
// Polycyclic
G:=Group<a,b,c|a^7=b^31=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export