direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: M4(2)×C31, C8⋊3C62, C4.C124, C248⋊7C2, C124.4C4, C22.C124, C124.22C22, (C2×C62).1C4, (C2×C4).2C62, C4.6(C2×C62), C62.12(C2×C4), (C2×C124).8C2, C2.3(C2×C124), SmallGroup(496,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×C31
G = < a,b,c | a31=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Smallest permutation representation of M4(2)×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)
(1 138 62 187 168 235 112 64)(2 139 32 188 169 236 113 65)(3 140 33 189 170 237 114 66)(4 141 34 190 171 238 115 67)(5 142 35 191 172 239 116 68)(6 143 36 192 173 240 117 69)(7 144 37 193 174 241 118 70)(8 145 38 194 175 242 119 71)(9 146 39 195 176 243 120 72)(10 147 40 196 177 244 121 73)(11 148 41 197 178 245 122 74)(12 149 42 198 179 246 123 75)(13 150 43 199 180 247 124 76)(14 151 44 200 181 248 94 77)(15 152 45 201 182 218 95 78)(16 153 46 202 183 219 96 79)(17 154 47 203 184 220 97 80)(18 155 48 204 185 221 98 81)(19 125 49 205 186 222 99 82)(20 126 50 206 156 223 100 83)(21 127 51 207 157 224 101 84)(22 128 52 208 158 225 102 85)(23 129 53 209 159 226 103 86)(24 130 54 210 160 227 104 87)(25 131 55 211 161 228 105 88)(26 132 56 212 162 229 106 89)(27 133 57 213 163 230 107 90)(28 134 58 214 164 231 108 91)(29 135 59 215 165 232 109 92)(30 136 60 216 166 233 110 93)(31 137 61 217 167 234 111 63)
(63 217)(64 187)(65 188)(66 189)(67 190)(68 191)(69 192)(70 193)(71 194)(72 195)(73 196)(74 197)(75 198)(76 199)(77 200)(78 201)(79 202)(80 203)(81 204)(82 205)(83 206)(84 207)(85 208)(86 209)(87 210)(88 211)(89 212)(90 213)(91 214)(92 215)(93 216)(125 222)(126 223)(127 224)(128 225)(129 226)(130 227)(131 228)(132 229)(133 230)(134 231)(135 232)(136 233)(137 234)(138 235)(139 236)(140 237)(141 238)(142 239)(143 240)(144 241)(145 242)(146 243)(147 244)(148 245)(149 246)(150 247)(151 248)(152 218)(153 219)(154 220)(155 221)
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), (1,138,62,187,168,235,112,64)(2,139,32,188,169,236,113,65)(3,140,33,189,170,237,114,66)(4,141,34,190,171,238,115,67)(5,142,35,191,172,239,116,68)(6,143,36,192,173,240,117,69)(7,144,37,193,174,241,118,70)(8,145,38,194,175,242,119,71)(9,146,39,195,176,243,120,72)(10,147,40,196,177,244,121,73)(11,148,41,197,178,245,122,74)(12,149,42,198,179,246,123,75)(13,150,43,199,180,247,124,76)(14,151,44,200,181,248,94,77)(15,152,45,201,182,218,95,78)(16,153,46,202,183,219,96,79)(17,154,47,203,184,220,97,80)(18,155,48,204,185,221,98,81)(19,125,49,205,186,222,99,82)(20,126,50,206,156,223,100,83)(21,127,51,207,157,224,101,84)(22,128,52,208,158,225,102,85)(23,129,53,209,159,226,103,86)(24,130,54,210,160,227,104,87)(25,131,55,211,161,228,105,88)(26,132,56,212,162,229,106,89)(27,133,57,213,163,230,107,90)(28,134,58,214,164,231,108,91)(29,135,59,215,165,232,109,92)(30,136,60,216,166,233,110,93)(31,137,61,217,167,234,111,63), (63,217)(64,187)(65,188)(66,189)(67,190)(68,191)(69,192)(70,193)(71,194)(72,195)(73,196)(74,197)(75,198)(76,199)(77,200)(78,201)(79,202)(80,203)(81,204)(82,205)(83,206)(84,207)(85,208)(86,209)(87,210)(88,211)(89,212)(90,213)(91,214)(92,215)(93,216)(125,222)(126,223)(127,224)(128,225)(129,226)(130,227)(131,228)(132,229)(133,230)(134,231)(135,232)(136,233)(137,234)(138,235)(139,236)(140,237)(141,238)(142,239)(143,240)(144,241)(145,242)(146,243)(147,244)(148,245)(149,246)(150,247)(151,248)(152,218)(153,219)(154,220)(155,221)>;
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), (1,138,62,187,168,235,112,64)(2,139,32,188,169,236,113,65)(3,140,33,189,170,237,114,66)(4,141,34,190,171,238,115,67)(5,142,35,191,172,239,116,68)(6,143,36,192,173,240,117,69)(7,144,37,193,174,241,118,70)(8,145,38,194,175,242,119,71)(9,146,39,195,176,243,120,72)(10,147,40,196,177,244,121,73)(11,148,41,197,178,245,122,74)(12,149,42,198,179,246,123,75)(13,150,43,199,180,247,124,76)(14,151,44,200,181,248,94,77)(15,152,45,201,182,218,95,78)(16,153,46,202,183,219,96,79)(17,154,47,203,184,220,97,80)(18,155,48,204,185,221,98,81)(19,125,49,205,186,222,99,82)(20,126,50,206,156,223,100,83)(21,127,51,207,157,224,101,84)(22,128,52,208,158,225,102,85)(23,129,53,209,159,226,103,86)(24,130,54,210,160,227,104,87)(25,131,55,211,161,228,105,88)(26,132,56,212,162,229,106,89)(27,133,57,213,163,230,107,90)(28,134,58,214,164,231,108,91)(29,135,59,215,165,232,109,92)(30,136,60,216,166,233,110,93)(31,137,61,217,167,234,111,63), (63,217)(64,187)(65,188)(66,189)(67,190)(68,191)(69,192)(70,193)(71,194)(72,195)(73,196)(74,197)(75,198)(76,199)(77,200)(78,201)(79,202)(80,203)(81,204)(82,205)(83,206)(84,207)(85,208)(86,209)(87,210)(88,211)(89,212)(90,213)(91,214)(92,215)(93,216)(125,222)(126,223)(127,224)(128,225)(129,226)(130,227)(131,228)(132,229)(133,230)(134,231)(135,232)(136,233)(137,234)(138,235)(139,236)(140,237)(141,238)(142,239)(143,240)(144,241)(145,242)(146,243)(147,244)(148,245)(149,246)(150,247)(151,248)(152,218)(153,219)(154,220)(155,221) );
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)], [(1,138,62,187,168,235,112,64),(2,139,32,188,169,236,113,65),(3,140,33,189,170,237,114,66),(4,141,34,190,171,238,115,67),(5,142,35,191,172,239,116,68),(6,143,36,192,173,240,117,69),(7,144,37,193,174,241,118,70),(8,145,38,194,175,242,119,71),(9,146,39,195,176,243,120,72),(10,147,40,196,177,244,121,73),(11,148,41,197,178,245,122,74),(12,149,42,198,179,246,123,75),(13,150,43,199,180,247,124,76),(14,151,44,200,181,248,94,77),(15,152,45,201,182,218,95,78),(16,153,46,202,183,219,96,79),(17,154,47,203,184,220,97,80),(18,155,48,204,185,221,98,81),(19,125,49,205,186,222,99,82),(20,126,50,206,156,223,100,83),(21,127,51,207,157,224,101,84),(22,128,52,208,158,225,102,85),(23,129,53,209,159,226,103,86),(24,130,54,210,160,227,104,87),(25,131,55,211,161,228,105,88),(26,132,56,212,162,229,106,89),(27,133,57,213,163,230,107,90),(28,134,58,214,164,231,108,91),(29,135,59,215,165,232,109,92),(30,136,60,216,166,233,110,93),(31,137,61,217,167,234,111,63)], [(63,217),(64,187),(65,188),(66,189),(67,190),(68,191),(69,192),(70,193),(71,194),(72,195),(73,196),(74,197),(75,198),(76,199),(77,200),(78,201),(79,202),(80,203),(81,204),(82,205),(83,206),(84,207),(85,208),(86,209),(87,210),(88,211),(89,212),(90,213),(91,214),(92,215),(93,216),(125,222),(126,223),(127,224),(128,225),(129,226),(130,227),(131,228),(132,229),(133,230),(134,231),(135,232),(136,233),(137,234),(138,235),(139,236),(140,237),(141,238),(142,239),(143,240),(144,241),(145,242),(146,243),(147,244),(148,245),(149,246),(150,247),(151,248),(152,218),(153,219),(154,220),(155,221)]])
310 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 31A | ··· | 31AD | 62A | ··· | 62AD | 62AE | ··· | 62BH | 124A | ··· | 124BH | 124BI | ··· | 124CL | 248A | ··· | 248DP |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 31 | ··· | 31 | 62 | ··· | 62 | 62 | ··· | 62 | 124 | ··· | 124 | 124 | ··· | 124 | 248 | ··· | 248 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
310 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C31 | C62 | C62 | C124 | C124 | M4(2) | M4(2)×C31 |
| kernel | M4(2)×C31 | C248 | C2×C124 | C124 | C2×C62 | M4(2) | C8 | C2×C4 | C4 | C22 | C31 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 30 | 60 | 30 | 60 | 60 | 2 | 60 |
Matrix representation of M4(2)×C31 ►in GL2(𝔽1489) generated by
| 992 | 0 |
| 0 | 992 |
| 0 | 1 |
| 1264 | 0 |
| 1 | 0 |
| 0 | 1488 |
G:=sub<GL(2,GF(1489))| [992,0,0,992],[0,1264,1,0],[1,0,0,1488] >;
M4(2)×C31 in GAP, Magma, Sage, TeX
M_4(2)\times C_{31} % in TeX
G:=Group("M4(2)xC31"); // GroupNames label
G:=SmallGroup(496,23);
// by ID
G=gap.SmallGroup(496,23);
# by ID
G:=PCGroup([5,-2,-2,-31,-2,-2,620,2501,58]);
// 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^5>;
// generators/relations
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