direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: Q8×C31, C4.C62, C124.3C2, C62.7C22, C2.2(C2×C62), SmallGroup(248,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C31
G = < a,b,c | a31=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Q8×C31
►Regular action on 248 points
Generators in S248
(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)(218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248)
(1 109 152 199)(2 110 153 200)(3 111 154 201)(4 112 155 202)(5 113 125 203)(6 114 126 204)(7 115 127 205)(8 116 128 206)(9 117 129 207)(10 118 130 208)(11 119 131 209)(12 120 132 210)(13 121 133 211)(14 122 134 212)(15 123 135 213)(16 124 136 214)(17 94 137 215)(18 95 138 216)(19 96 139 217)(20 97 140 187)(21 98 141 188)(22 99 142 189)(23 100 143 190)(24 101 144 191)(25 102 145 192)(26 103 146 193)(27 104 147 194)(28 105 148 195)(29 106 149 196)(30 107 150 197)(31 108 151 198)(32 90 169 223)(33 91 170 224)(34 92 171 225)(35 93 172 226)(36 63 173 227)(37 64 174 228)(38 65 175 229)(39 66 176 230)(40 67 177 231)(41 68 178 232)(42 69 179 233)(43 70 180 234)(44 71 181 235)(45 72 182 236)(46 73 183 237)(47 74 184 238)(48 75 185 239)(49 76 186 240)(50 77 156 241)(51 78 157 242)(52 79 158 243)(53 80 159 244)(54 81 160 245)(55 82 161 246)(56 83 162 247)(57 84 163 248)(58 85 164 218)(59 86 165 219)(60 87 166 220)(61 88 167 221)(62 89 168 222)
(1 162 152 56)(2 163 153 57)(3 164 154 58)(4 165 155 59)(5 166 125 60)(6 167 126 61)(7 168 127 62)(8 169 128 32)(9 170 129 33)(10 171 130 34)(11 172 131 35)(12 173 132 36)(13 174 133 37)(14 175 134 38)(15 176 135 39)(16 177 136 40)(17 178 137 41)(18 179 138 42)(19 180 139 43)(20 181 140 44)(21 182 141 45)(22 183 142 46)(23 184 143 47)(24 185 144 48)(25 186 145 49)(26 156 146 50)(27 157 147 51)(28 158 148 52)(29 159 149 53)(30 160 150 54)(31 161 151 55)(63 210 227 120)(64 211 228 121)(65 212 229 122)(66 213 230 123)(67 214 231 124)(68 215 232 94)(69 216 233 95)(70 217 234 96)(71 187 235 97)(72 188 236 98)(73 189 237 99)(74 190 238 100)(75 191 239 101)(76 192 240 102)(77 193 241 103)(78 194 242 104)(79 195 243 105)(80 196 244 106)(81 197 245 107)(82 198 246 108)(83 199 247 109)(84 200 248 110)(85 201 218 111)(86 202 219 112)(87 203 220 113)(88 204 221 114)(89 205 222 115)(90 206 223 116)(91 207 224 117)(92 208 225 118)(93 209 226 119)
G:=sub<Sym(248)| (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)(218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248), (1,109,152,199)(2,110,153,200)(3,111,154,201)(4,112,155,202)(5,113,125,203)(6,114,126,204)(7,115,127,205)(8,116,128,206)(9,117,129,207)(10,118,130,208)(11,119,131,209)(12,120,132,210)(13,121,133,211)(14,122,134,212)(15,123,135,213)(16,124,136,214)(17,94,137,215)(18,95,138,216)(19,96,139,217)(20,97,140,187)(21,98,141,188)(22,99,142,189)(23,100,143,190)(24,101,144,191)(25,102,145,192)(26,103,146,193)(27,104,147,194)(28,105,148,195)(29,106,149,196)(30,107,150,197)(31,108,151,198)(32,90,169,223)(33,91,170,224)(34,92,171,225)(35,93,172,226)(36,63,173,227)(37,64,174,228)(38,65,175,229)(39,66,176,230)(40,67,177,231)(41,68,178,232)(42,69,179,233)(43,70,180,234)(44,71,181,235)(45,72,182,236)(46,73,183,237)(47,74,184,238)(48,75,185,239)(49,76,186,240)(50,77,156,241)(51,78,157,242)(52,79,158,243)(53,80,159,244)(54,81,160,245)(55,82,161,246)(56,83,162,247)(57,84,163,248)(58,85,164,218)(59,86,165,219)(60,87,166,220)(61,88,167,221)(62,89,168,222), (1,162,152,56)(2,163,153,57)(3,164,154,58)(4,165,155,59)(5,166,125,60)(6,167,126,61)(7,168,127,62)(8,169,128,32)(9,170,129,33)(10,171,130,34)(11,172,131,35)(12,173,132,36)(13,174,133,37)(14,175,134,38)(15,176,135,39)(16,177,136,40)(17,178,137,41)(18,179,138,42)(19,180,139,43)(20,181,140,44)(21,182,141,45)(22,183,142,46)(23,184,143,47)(24,185,144,48)(25,186,145,49)(26,156,146,50)(27,157,147,51)(28,158,148,52)(29,159,149,53)(30,160,150,54)(31,161,151,55)(63,210,227,120)(64,211,228,121)(65,212,229,122)(66,213,230,123)(67,214,231,124)(68,215,232,94)(69,216,233,95)(70,217,234,96)(71,187,235,97)(72,188,236,98)(73,189,237,99)(74,190,238,100)(75,191,239,101)(76,192,240,102)(77,193,241,103)(78,194,242,104)(79,195,243,105)(80,196,244,106)(81,197,245,107)(82,198,246,108)(83,199,247,109)(84,200,248,110)(85,201,218,111)(86,202,219,112)(87,203,220,113)(88,204,221,114)(89,205,222,115)(90,206,223,116)(91,207,224,117)(92,208,225,118)(93,209,226,119)>;
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)(218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248), (1,109,152,199)(2,110,153,200)(3,111,154,201)(4,112,155,202)(5,113,125,203)(6,114,126,204)(7,115,127,205)(8,116,128,206)(9,117,129,207)(10,118,130,208)(11,119,131,209)(12,120,132,210)(13,121,133,211)(14,122,134,212)(15,123,135,213)(16,124,136,214)(17,94,137,215)(18,95,138,216)(19,96,139,217)(20,97,140,187)(21,98,141,188)(22,99,142,189)(23,100,143,190)(24,101,144,191)(25,102,145,192)(26,103,146,193)(27,104,147,194)(28,105,148,195)(29,106,149,196)(30,107,150,197)(31,108,151,198)(32,90,169,223)(33,91,170,224)(34,92,171,225)(35,93,172,226)(36,63,173,227)(37,64,174,228)(38,65,175,229)(39,66,176,230)(40,67,177,231)(41,68,178,232)(42,69,179,233)(43,70,180,234)(44,71,181,235)(45,72,182,236)(46,73,183,237)(47,74,184,238)(48,75,185,239)(49,76,186,240)(50,77,156,241)(51,78,157,242)(52,79,158,243)(53,80,159,244)(54,81,160,245)(55,82,161,246)(56,83,162,247)(57,84,163,248)(58,85,164,218)(59,86,165,219)(60,87,166,220)(61,88,167,221)(62,89,168,222), (1,162,152,56)(2,163,153,57)(3,164,154,58)(4,165,155,59)(5,166,125,60)(6,167,126,61)(7,168,127,62)(8,169,128,32)(9,170,129,33)(10,171,130,34)(11,172,131,35)(12,173,132,36)(13,174,133,37)(14,175,134,38)(15,176,135,39)(16,177,136,40)(17,178,137,41)(18,179,138,42)(19,180,139,43)(20,181,140,44)(21,182,141,45)(22,183,142,46)(23,184,143,47)(24,185,144,48)(25,186,145,49)(26,156,146,50)(27,157,147,51)(28,158,148,52)(29,159,149,53)(30,160,150,54)(31,161,151,55)(63,210,227,120)(64,211,228,121)(65,212,229,122)(66,213,230,123)(67,214,231,124)(68,215,232,94)(69,216,233,95)(70,217,234,96)(71,187,235,97)(72,188,236,98)(73,189,237,99)(74,190,238,100)(75,191,239,101)(76,192,240,102)(77,193,241,103)(78,194,242,104)(79,195,243,105)(80,196,244,106)(81,197,245,107)(82,198,246,108)(83,199,247,109)(84,200,248,110)(85,201,218,111)(86,202,219,112)(87,203,220,113)(88,204,221,114)(89,205,222,115)(90,206,223,116)(91,207,224,117)(92,208,225,118)(93,209,226,119) );
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),(218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248)], [(1,109,152,199),(2,110,153,200),(3,111,154,201),(4,112,155,202),(5,113,125,203),(6,114,126,204),(7,115,127,205),(8,116,128,206),(9,117,129,207),(10,118,130,208),(11,119,131,209),(12,120,132,210),(13,121,133,211),(14,122,134,212),(15,123,135,213),(16,124,136,214),(17,94,137,215),(18,95,138,216),(19,96,139,217),(20,97,140,187),(21,98,141,188),(22,99,142,189),(23,100,143,190),(24,101,144,191),(25,102,145,192),(26,103,146,193),(27,104,147,194),(28,105,148,195),(29,106,149,196),(30,107,150,197),(31,108,151,198),(32,90,169,223),(33,91,170,224),(34,92,171,225),(35,93,172,226),(36,63,173,227),(37,64,174,228),(38,65,175,229),(39,66,176,230),(40,67,177,231),(41,68,178,232),(42,69,179,233),(43,70,180,234),(44,71,181,235),(45,72,182,236),(46,73,183,237),(47,74,184,238),(48,75,185,239),(49,76,186,240),(50,77,156,241),(51,78,157,242),(52,79,158,243),(53,80,159,244),(54,81,160,245),(55,82,161,246),(56,83,162,247),(57,84,163,248),(58,85,164,218),(59,86,165,219),(60,87,166,220),(61,88,167,221),(62,89,168,222)], [(1,162,152,56),(2,163,153,57),(3,164,154,58),(4,165,155,59),(5,166,125,60),(6,167,126,61),(7,168,127,62),(8,169,128,32),(9,170,129,33),(10,171,130,34),(11,172,131,35),(12,173,132,36),(13,174,133,37),(14,175,134,38),(15,176,135,39),(16,177,136,40),(17,178,137,41),(18,179,138,42),(19,180,139,43),(20,181,140,44),(21,182,141,45),(22,183,142,46),(23,184,143,47),(24,185,144,48),(25,186,145,49),(26,156,146,50),(27,157,147,51),(28,158,148,52),(29,159,149,53),(30,160,150,54),(31,161,151,55),(63,210,227,120),(64,211,228,121),(65,212,229,122),(66,213,230,123),(67,214,231,124),(68,215,232,94),(69,216,233,95),(70,217,234,96),(71,187,235,97),(72,188,236,98),(73,189,237,99),(74,190,238,100),(75,191,239,101),(76,192,240,102),(77,193,241,103),(78,194,242,104),(79,195,243,105),(80,196,244,106),(81,197,245,107),(82,198,246,108),(83,199,247,109),(84,200,248,110),(85,201,218,111),(86,202,219,112),(87,203,220,113),(88,204,221,114),(89,205,222,115),(90,206,223,116),(91,207,224,117),(92,208,225,118),(93,209,226,119)]])
Q8×C31 is a maximal subgroup of
Q8⋊D31 C31⋊Q16 Q8⋊2D31
155 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 31A | ··· | 31AD | 62A | ··· | 62AD | 124A | ··· | 124CL |
| order | 1 | 2 | 4 | 4 | 4 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
155 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | - | |||
| image | C1 | C2 | C31 | C62 | Q8 | Q8×C31 |
| kernel | Q8×C31 | C124 | Q8 | C4 | C31 | C1 |
| # reps | 1 | 3 | 30 | 90 | 1 | 30 |
Matrix representation of Q8×C31 ►in GL2(𝔽373) generated by
| 119 | 0 |
| 0 | 119 |
| 372 | 371 |
| 1 | 1 |
| 5 | 220 |
| 105 | 368 |
G:=sub<GL(2,GF(373))| [119,0,0,119],[372,1,371,1],[5,105,220,368] >;
Q8×C31 in GAP, Magma, Sage, TeX
Q_8\times C_{31} % in TeX
G:=Group("Q8xC31"); // GroupNames label
G:=SmallGroup(248,10);
// by ID
G=gap.SmallGroup(248,10);
# by ID
G:=PCGroup([4,-2,-2,-31,-2,496,1009,501]);
// Polycyclic
G:=Group<a,b,c|a^31=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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