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G = D8×C31order 496 = 24·31

Direct product of C31 and D8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: D8×C31, D4⋊C62, C81C62, C2485C2, C62.14D4, C124.17C22, (D4×C31)⋊4C2, C4.1(C2×C62), C2.3(D4×C31), SmallGroup(496,24)

Series: Derived Chief Lower central Upper central

C1C4 — D8×C31
C1C2C4C124D4×C31 — D8×C31
C1C2C4 — D8×C31
C1C62C124 — D8×C31

Generators and relations for D8×C31
 G = < a,b,c | a31=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

4C2
4C2
2C22
2C22
4C62
4C62
2C2×C62
2C2×C62

Smallest permutation representation of D8×C31
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)
(1 224 198 184 54 143 85 96)(2 225 199 185 55 144 86 97)(3 226 200 186 56 145 87 98)(4 227 201 156 57 146 88 99)(5 228 202 157 58 147 89 100)(6 229 203 158 59 148 90 101)(7 230 204 159 60 149 91 102)(8 231 205 160 61 150 92 103)(9 232 206 161 62 151 93 104)(10 233 207 162 32 152 63 105)(11 234 208 163 33 153 64 106)(12 235 209 164 34 154 65 107)(13 236 210 165 35 155 66 108)(14 237 211 166 36 125 67 109)(15 238 212 167 37 126 68 110)(16 239 213 168 38 127 69 111)(17 240 214 169 39 128 70 112)(18 241 215 170 40 129 71 113)(19 242 216 171 41 130 72 114)(20 243 217 172 42 131 73 115)(21 244 187 173 43 132 74 116)(22 245 188 174 44 133 75 117)(23 246 189 175 45 134 76 118)(24 247 190 176 46 135 77 119)(25 248 191 177 47 136 78 120)(26 218 192 178 48 137 79 121)(27 219 193 179 49 138 80 122)(28 220 194 180 50 139 81 123)(29 221 195 181 51 140 82 124)(30 222 196 182 52 141 83 94)(31 223 197 183 53 142 84 95)
(63 207)(64 208)(65 209)(66 210)(67 211)(68 212)(69 213)(70 214)(71 215)(72 216)(73 217)(74 187)(75 188)(76 189)(77 190)(78 191)(79 192)(80 193)(81 194)(82 195)(83 196)(84 197)(85 198)(86 199)(87 200)(88 201)(89 202)(90 203)(91 204)(92 205)(93 206)(94 222)(95 223)(96 224)(97 225)(98 226)(99 227)(100 228)(101 229)(102 230)(103 231)(104 232)(105 233)(106 234)(107 235)(108 236)(109 237)(110 238)(111 239)(112 240)(113 241)(114 242)(115 243)(116 244)(117 245)(118 246)(119 247)(120 248)(121 218)(122 219)(123 220)(124 221)(125 166)(126 167)(127 168)(128 169)(129 170)(130 171)(131 172)(132 173)(133 174)(134 175)(135 176)(136 177)(137 178)(138 179)(139 180)(140 181)(141 182)(142 183)(143 184)(144 185)(145 186)(146 156)(147 157)(148 158)(149 159)(150 160)(151 161)(152 162)(153 163)(154 164)(155 165)

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

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

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

217 conjugacy classes

class 1 2A2B2C 4 8A8B31A···31AD62A···62AD62AE···62CL124A···124AD248A···248BH
order122248831···3162···6262···62124···124248···248
size11442221···11···14···42···22···2

217 irreducible representations

dim1111112222
type+++++
imageC1C2C2C31C62C62D4D8D4×C31D8×C31
kernelD8×C31C248D4×C31D8C8D4C62C31C2C1
# reps112303060123060

Matrix representation of D8×C31 in GL2(𝔽1489) generated by

12540
01254
,
1911298
191191
,
10
01488
G:=sub<GL(2,GF(1489))| [1254,0,0,1254],[191,191,1298,191],[1,0,0,1488] >;

D8×C31 in GAP, Magma, Sage, TeX

D_8\times C_{31}
% in TeX

G:=Group("D8xC31");
// GroupNames label

G:=SmallGroup(496,24);
// by ID

G=gap.SmallGroup(496,24);
# by ID

G:=PCGroup([5,-2,-2,-31,-2,-2,1261,7443,3728,58]);
// Polycyclic

G:=Group<a,b,c|a^31=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D8×C31 in TeX

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