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G = Q8×C31order 248 = 23·31

Direct product of C31 and Q8

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

 Derived series C1 — C2 — Q8×C31
 Chief series C1 — C2 — C62 — C124 — Q8×C31
 Lower central C1 — C2 — Q8×C31
 Upper central C1 — C62 — Q8×C31

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 52 135 70)(2 53 136 71)(3 54 137 72)(4 55 138 73)(5 56 139 74)(6 57 140 75)(7 58 141 76)(8 59 142 77)(9 60 143 78)(10 61 144 79)(11 62 145 80)(12 32 146 81)(13 33 147 82)(14 34 148 83)(15 35 149 84)(16 36 150 85)(17 37 151 86)(18 38 152 87)(19 39 153 88)(20 40 154 89)(21 41 155 90)(22 42 125 91)(23 43 126 92)(24 44 127 93)(25 45 128 63)(26 46 129 64)(27 47 130 65)(28 48 131 66)(29 49 132 67)(30 50 133 68)(31 51 134 69)(94 248 213 170)(95 218 214 171)(96 219 215 172)(97 220 216 173)(98 221 217 174)(99 222 187 175)(100 223 188 176)(101 224 189 177)(102 225 190 178)(103 226 191 179)(104 227 192 180)(105 228 193 181)(106 229 194 182)(107 230 195 183)(108 231 196 184)(109 232 197 185)(110 233 198 186)(111 234 199 156)(112 235 200 157)(113 236 201 158)(114 237 202 159)(115 238 203 160)(116 239 204 161)(117 240 205 162)(118 241 206 163)(119 242 207 164)(120 243 208 165)(121 244 209 166)(122 245 210 167)(123 246 211 168)(124 247 212 169)
(1 103 135 191)(2 104 136 192)(3 105 137 193)(4 106 138 194)(5 107 139 195)(6 108 140 196)(7 109 141 197)(8 110 142 198)(9 111 143 199)(10 112 144 200)(11 113 145 201)(12 114 146 202)(13 115 147 203)(14 116 148 204)(15 117 149 205)(16 118 150 206)(17 119 151 207)(18 120 152 208)(19 121 153 209)(20 122 154 210)(21 123 155 211)(22 124 125 212)(23 94 126 213)(24 95 127 214)(25 96 128 215)(26 97 129 216)(27 98 130 217)(28 99 131 187)(29 100 132 188)(30 101 133 189)(31 102 134 190)(32 159 81 237)(33 160 82 238)(34 161 83 239)(35 162 84 240)(36 163 85 241)(37 164 86 242)(38 165 87 243)(39 166 88 244)(40 167 89 245)(41 168 90 246)(42 169 91 247)(43 170 92 248)(44 171 93 218)(45 172 63 219)(46 173 64 220)(47 174 65 221)(48 175 66 222)(49 176 67 223)(50 177 68 224)(51 178 69 225)(52 179 70 226)(53 180 71 227)(54 181 72 228)(55 182 73 229)(56 183 74 230)(57 184 75 231)(58 185 76 232)(59 186 77 233)(60 156 78 234)(61 157 79 235)(62 158 80 236)

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), (1,52,135,70)(2,53,136,71)(3,54,137,72)(4,55,138,73)(5,56,139,74)(6,57,140,75)(7,58,141,76)(8,59,142,77)(9,60,143,78)(10,61,144,79)(11,62,145,80)(12,32,146,81)(13,33,147,82)(14,34,148,83)(15,35,149,84)(16,36,150,85)(17,37,151,86)(18,38,152,87)(19,39,153,88)(20,40,154,89)(21,41,155,90)(22,42,125,91)(23,43,126,92)(24,44,127,93)(25,45,128,63)(26,46,129,64)(27,47,130,65)(28,48,131,66)(29,49,132,67)(30,50,133,68)(31,51,134,69)(94,248,213,170)(95,218,214,171)(96,219,215,172)(97,220,216,173)(98,221,217,174)(99,222,187,175)(100,223,188,176)(101,224,189,177)(102,225,190,178)(103,226,191,179)(104,227,192,180)(105,228,193,181)(106,229,194,182)(107,230,195,183)(108,231,196,184)(109,232,197,185)(110,233,198,186)(111,234,199,156)(112,235,200,157)(113,236,201,158)(114,237,202,159)(115,238,203,160)(116,239,204,161)(117,240,205,162)(118,241,206,163)(119,242,207,164)(120,243,208,165)(121,244,209,166)(122,245,210,167)(123,246,211,168)(124,247,212,169), (1,103,135,191)(2,104,136,192)(3,105,137,193)(4,106,138,194)(5,107,139,195)(6,108,140,196)(7,109,141,197)(8,110,142,198)(9,111,143,199)(10,112,144,200)(11,113,145,201)(12,114,146,202)(13,115,147,203)(14,116,148,204)(15,117,149,205)(16,118,150,206)(17,119,151,207)(18,120,152,208)(19,121,153,209)(20,122,154,210)(21,123,155,211)(22,124,125,212)(23,94,126,213)(24,95,127,214)(25,96,128,215)(26,97,129,216)(27,98,130,217)(28,99,131,187)(29,100,132,188)(30,101,133,189)(31,102,134,190)(32,159,81,237)(33,160,82,238)(34,161,83,239)(35,162,84,240)(36,163,85,241)(37,164,86,242)(38,165,87,243)(39,166,88,244)(40,167,89,245)(41,168,90,246)(42,169,91,247)(43,170,92,248)(44,171,93,218)(45,172,63,219)(46,173,64,220)(47,174,65,221)(48,175,66,222)(49,176,67,223)(50,177,68,224)(51,178,69,225)(52,179,70,226)(53,180,71,227)(54,181,72,228)(55,182,73,229)(56,183,74,230)(57,184,75,231)(58,185,76,232)(59,186,77,233)(60,156,78,234)(61,157,79,235)(62,158,80,236)>;

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), (1,52,135,70)(2,53,136,71)(3,54,137,72)(4,55,138,73)(5,56,139,74)(6,57,140,75)(7,58,141,76)(8,59,142,77)(9,60,143,78)(10,61,144,79)(11,62,145,80)(12,32,146,81)(13,33,147,82)(14,34,148,83)(15,35,149,84)(16,36,150,85)(17,37,151,86)(18,38,152,87)(19,39,153,88)(20,40,154,89)(21,41,155,90)(22,42,125,91)(23,43,126,92)(24,44,127,93)(25,45,128,63)(26,46,129,64)(27,47,130,65)(28,48,131,66)(29,49,132,67)(30,50,133,68)(31,51,134,69)(94,248,213,170)(95,218,214,171)(96,219,215,172)(97,220,216,173)(98,221,217,174)(99,222,187,175)(100,223,188,176)(101,224,189,177)(102,225,190,178)(103,226,191,179)(104,227,192,180)(105,228,193,181)(106,229,194,182)(107,230,195,183)(108,231,196,184)(109,232,197,185)(110,233,198,186)(111,234,199,156)(112,235,200,157)(113,236,201,158)(114,237,202,159)(115,238,203,160)(116,239,204,161)(117,240,205,162)(118,241,206,163)(119,242,207,164)(120,243,208,165)(121,244,209,166)(122,245,210,167)(123,246,211,168)(124,247,212,169), (1,103,135,191)(2,104,136,192)(3,105,137,193)(4,106,138,194)(5,107,139,195)(6,108,140,196)(7,109,141,197)(8,110,142,198)(9,111,143,199)(10,112,144,200)(11,113,145,201)(12,114,146,202)(13,115,147,203)(14,116,148,204)(15,117,149,205)(16,118,150,206)(17,119,151,207)(18,120,152,208)(19,121,153,209)(20,122,154,210)(21,123,155,211)(22,124,125,212)(23,94,126,213)(24,95,127,214)(25,96,128,215)(26,97,129,216)(27,98,130,217)(28,99,131,187)(29,100,132,188)(30,101,133,189)(31,102,134,190)(32,159,81,237)(33,160,82,238)(34,161,83,239)(35,162,84,240)(36,163,85,241)(37,164,86,242)(38,165,87,243)(39,166,88,244)(40,167,89,245)(41,168,90,246)(42,169,91,247)(43,170,92,248)(44,171,93,218)(45,172,63,219)(46,173,64,220)(47,174,65,221)(48,175,66,222)(49,176,67,223)(50,177,68,224)(51,178,69,225)(52,179,70,226)(53,180,71,227)(54,181,72,228)(55,182,73,229)(56,183,74,230)(57,184,75,231)(58,185,76,232)(59,186,77,233)(60,156,78,234)(61,157,79,235)(62,158,80,236) );

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)], [(1,52,135,70),(2,53,136,71),(3,54,137,72),(4,55,138,73),(5,56,139,74),(6,57,140,75),(7,58,141,76),(8,59,142,77),(9,60,143,78),(10,61,144,79),(11,62,145,80),(12,32,146,81),(13,33,147,82),(14,34,148,83),(15,35,149,84),(16,36,150,85),(17,37,151,86),(18,38,152,87),(19,39,153,88),(20,40,154,89),(21,41,155,90),(22,42,125,91),(23,43,126,92),(24,44,127,93),(25,45,128,63),(26,46,129,64),(27,47,130,65),(28,48,131,66),(29,49,132,67),(30,50,133,68),(31,51,134,69),(94,248,213,170),(95,218,214,171),(96,219,215,172),(97,220,216,173),(98,221,217,174),(99,222,187,175),(100,223,188,176),(101,224,189,177),(102,225,190,178),(103,226,191,179),(104,227,192,180),(105,228,193,181),(106,229,194,182),(107,230,195,183),(108,231,196,184),(109,232,197,185),(110,233,198,186),(111,234,199,156),(112,235,200,157),(113,236,201,158),(114,237,202,159),(115,238,203,160),(116,239,204,161),(117,240,205,162),(118,241,206,163),(119,242,207,164),(120,243,208,165),(121,244,209,166),(122,245,210,167),(123,246,211,168),(124,247,212,169)], [(1,103,135,191),(2,104,136,192),(3,105,137,193),(4,106,138,194),(5,107,139,195),(6,108,140,196),(7,109,141,197),(8,110,142,198),(9,111,143,199),(10,112,144,200),(11,113,145,201),(12,114,146,202),(13,115,147,203),(14,116,148,204),(15,117,149,205),(16,118,150,206),(17,119,151,207),(18,120,152,208),(19,121,153,209),(20,122,154,210),(21,123,155,211),(22,124,125,212),(23,94,126,213),(24,95,127,214),(25,96,128,215),(26,97,129,216),(27,98,130,217),(28,99,131,187),(29,100,132,188),(30,101,133,189),(31,102,134,190),(32,159,81,237),(33,160,82,238),(34,161,83,239),(35,162,84,240),(36,163,85,241),(37,164,86,242),(38,165,87,243),(39,166,88,244),(40,167,89,245),(41,168,90,246),(42,169,91,247),(43,170,92,248),(44,171,93,218),(45,172,63,219),(46,173,64,220),(47,174,65,221),(48,175,66,222),(49,176,67,223),(50,177,68,224),(51,178,69,225),(52,179,70,226),(53,180,71,227),(54,181,72,228),(55,182,73,229),(56,183,74,230),(57,184,75,231),(58,185,76,232),(59,186,77,233),(60,156,78,234),(61,157,79,235),(62,158,80,236)])

Q8×C31 is a maximal subgroup of   Q8⋊D31  C31⋊Q16  Q82D31

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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