metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C4.Dic31, C124.1C4, C4.15D62, C31⋊2M4(2), C22.Dic31, C124.15C22, C31⋊C8⋊5C2, C62.7(C2×C4), (C2×C62).3C4, (C2×C4).2D31, (C2×C124).5C2, C2.3(C2×Dic31), SmallGroup(496,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.Dic31
G = < a,b,c | a4=1, b62=a2, c2=a2b31, ab=ba, cac-1=a-1, cbc-1=b61 >
Smallest permutation representation of C4.Dic31
►On 248 points
Generators in S248
(1 94 63 32)(2 95 64 33)(3 96 65 34)(4 97 66 35)(5 98 67 36)(6 99 68 37)(7 100 69 38)(8 101 70 39)(9 102 71 40)(10 103 72 41)(11 104 73 42)(12 105 74 43)(13 106 75 44)(14 107 76 45)(15 108 77 46)(16 109 78 47)(17 110 79 48)(18 111 80 49)(19 112 81 50)(20 113 82 51)(21 114 83 52)(22 115 84 53)(23 116 85 54)(24 117 86 55)(25 118 87 56)(26 119 88 57)(27 120 89 58)(28 121 90 59)(29 122 91 60)(30 123 92 61)(31 124 93 62)(125 156 187 218)(126 157 188 219)(127 158 189 220)(128 159 190 221)(129 160 191 222)(130 161 192 223)(131 162 193 224)(132 163 194 225)(133 164 195 226)(134 165 196 227)(135 166 197 228)(136 167 198 229)(137 168 199 230)(138 169 200 231)(139 170 201 232)(140 171 202 233)(141 172 203 234)(142 173 204 235)(143 174 205 236)(144 175 206 237)(145 176 207 238)(146 177 208 239)(147 178 209 240)(148 179 210 241)(149 180 211 242)(150 181 212 243)(151 182 213 244)(152 183 214 245)(153 184 215 246)(154 185 216 247)(155 186 217 248)
(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)
(1 125 94 218 63 187 32 156)(2 186 95 155 64 248 33 217)(3 247 96 216 65 185 34 154)(4 184 97 153 66 246 35 215)(5 245 98 214 67 183 36 152)(6 182 99 151 68 244 37 213)(7 243 100 212 69 181 38 150)(8 180 101 149 70 242 39 211)(9 241 102 210 71 179 40 148)(10 178 103 147 72 240 41 209)(11 239 104 208 73 177 42 146)(12 176 105 145 74 238 43 207)(13 237 106 206 75 175 44 144)(14 174 107 143 76 236 45 205)(15 235 108 204 77 173 46 142)(16 172 109 141 78 234 47 203)(17 233 110 202 79 171 48 140)(18 170 111 139 80 232 49 201)(19 231 112 200 81 169 50 138)(20 168 113 137 82 230 51 199)(21 229 114 198 83 167 52 136)(22 166 115 135 84 228 53 197)(23 227 116 196 85 165 54 134)(24 164 117 133 86 226 55 195)(25 225 118 194 87 163 56 132)(26 162 119 131 88 224 57 193)(27 223 120 192 89 161 58 130)(28 160 121 129 90 222 59 191)(29 221 122 190 91 159 60 128)(30 158 123 127 92 220 61 189)(31 219 124 188 93 157 62 126)
G:=sub<Sym(248)| (1,94,63,32)(2,95,64,33)(3,96,65,34)(4,97,66,35)(5,98,67,36)(6,99,68,37)(7,100,69,38)(8,101,70,39)(9,102,71,40)(10,103,72,41)(11,104,73,42)(12,105,74,43)(13,106,75,44)(14,107,76,45)(15,108,77,46)(16,109,78,47)(17,110,79,48)(18,111,80,49)(19,112,81,50)(20,113,82,51)(21,114,83,52)(22,115,84,53)(23,116,85,54)(24,117,86,55)(25,118,87,56)(26,119,88,57)(27,120,89,58)(28,121,90,59)(29,122,91,60)(30,123,92,61)(31,124,93,62)(125,156,187,218)(126,157,188,219)(127,158,189,220)(128,159,190,221)(129,160,191,222)(130,161,192,223)(131,162,193,224)(132,163,194,225)(133,164,195,226)(134,165,196,227)(135,166,197,228)(136,167,198,229)(137,168,199,230)(138,169,200,231)(139,170,201,232)(140,171,202,233)(141,172,203,234)(142,173,204,235)(143,174,205,236)(144,175,206,237)(145,176,207,238)(146,177,208,239)(147,178,209,240)(148,179,210,241)(149,180,211,242)(150,181,212,243)(151,182,213,244)(152,183,214,245)(153,184,215,246)(154,185,216,247)(155,186,217,248), (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), (1,125,94,218,63,187,32,156)(2,186,95,155,64,248,33,217)(3,247,96,216,65,185,34,154)(4,184,97,153,66,246,35,215)(5,245,98,214,67,183,36,152)(6,182,99,151,68,244,37,213)(7,243,100,212,69,181,38,150)(8,180,101,149,70,242,39,211)(9,241,102,210,71,179,40,148)(10,178,103,147,72,240,41,209)(11,239,104,208,73,177,42,146)(12,176,105,145,74,238,43,207)(13,237,106,206,75,175,44,144)(14,174,107,143,76,236,45,205)(15,235,108,204,77,173,46,142)(16,172,109,141,78,234,47,203)(17,233,110,202,79,171,48,140)(18,170,111,139,80,232,49,201)(19,231,112,200,81,169,50,138)(20,168,113,137,82,230,51,199)(21,229,114,198,83,167,52,136)(22,166,115,135,84,228,53,197)(23,227,116,196,85,165,54,134)(24,164,117,133,86,226,55,195)(25,225,118,194,87,163,56,132)(26,162,119,131,88,224,57,193)(27,223,120,192,89,161,58,130)(28,160,121,129,90,222,59,191)(29,221,122,190,91,159,60,128)(30,158,123,127,92,220,61,189)(31,219,124,188,93,157,62,126)>;
G:=Group( (1,94,63,32)(2,95,64,33)(3,96,65,34)(4,97,66,35)(5,98,67,36)(6,99,68,37)(7,100,69,38)(8,101,70,39)(9,102,71,40)(10,103,72,41)(11,104,73,42)(12,105,74,43)(13,106,75,44)(14,107,76,45)(15,108,77,46)(16,109,78,47)(17,110,79,48)(18,111,80,49)(19,112,81,50)(20,113,82,51)(21,114,83,52)(22,115,84,53)(23,116,85,54)(24,117,86,55)(25,118,87,56)(26,119,88,57)(27,120,89,58)(28,121,90,59)(29,122,91,60)(30,123,92,61)(31,124,93,62)(125,156,187,218)(126,157,188,219)(127,158,189,220)(128,159,190,221)(129,160,191,222)(130,161,192,223)(131,162,193,224)(132,163,194,225)(133,164,195,226)(134,165,196,227)(135,166,197,228)(136,167,198,229)(137,168,199,230)(138,169,200,231)(139,170,201,232)(140,171,202,233)(141,172,203,234)(142,173,204,235)(143,174,205,236)(144,175,206,237)(145,176,207,238)(146,177,208,239)(147,178,209,240)(148,179,210,241)(149,180,211,242)(150,181,212,243)(151,182,213,244)(152,183,214,245)(153,184,215,246)(154,185,216,247)(155,186,217,248), (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), (1,125,94,218,63,187,32,156)(2,186,95,155,64,248,33,217)(3,247,96,216,65,185,34,154)(4,184,97,153,66,246,35,215)(5,245,98,214,67,183,36,152)(6,182,99,151,68,244,37,213)(7,243,100,212,69,181,38,150)(8,180,101,149,70,242,39,211)(9,241,102,210,71,179,40,148)(10,178,103,147,72,240,41,209)(11,239,104,208,73,177,42,146)(12,176,105,145,74,238,43,207)(13,237,106,206,75,175,44,144)(14,174,107,143,76,236,45,205)(15,235,108,204,77,173,46,142)(16,172,109,141,78,234,47,203)(17,233,110,202,79,171,48,140)(18,170,111,139,80,232,49,201)(19,231,112,200,81,169,50,138)(20,168,113,137,82,230,51,199)(21,229,114,198,83,167,52,136)(22,166,115,135,84,228,53,197)(23,227,116,196,85,165,54,134)(24,164,117,133,86,226,55,195)(25,225,118,194,87,163,56,132)(26,162,119,131,88,224,57,193)(27,223,120,192,89,161,58,130)(28,160,121,129,90,222,59,191)(29,221,122,190,91,159,60,128)(30,158,123,127,92,220,61,189)(31,219,124,188,93,157,62,126) );
G=PermutationGroup([[(1,94,63,32),(2,95,64,33),(3,96,65,34),(4,97,66,35),(5,98,67,36),(6,99,68,37),(7,100,69,38),(8,101,70,39),(9,102,71,40),(10,103,72,41),(11,104,73,42),(12,105,74,43),(13,106,75,44),(14,107,76,45),(15,108,77,46),(16,109,78,47),(17,110,79,48),(18,111,80,49),(19,112,81,50),(20,113,82,51),(21,114,83,52),(22,115,84,53),(23,116,85,54),(24,117,86,55),(25,118,87,56),(26,119,88,57),(27,120,89,58),(28,121,90,59),(29,122,91,60),(30,123,92,61),(31,124,93,62),(125,156,187,218),(126,157,188,219),(127,158,189,220),(128,159,190,221),(129,160,191,222),(130,161,192,223),(131,162,193,224),(132,163,194,225),(133,164,195,226),(134,165,196,227),(135,166,197,228),(136,167,198,229),(137,168,199,230),(138,169,200,231),(139,170,201,232),(140,171,202,233),(141,172,203,234),(142,173,204,235),(143,174,205,236),(144,175,206,237),(145,176,207,238),(146,177,208,239),(147,178,209,240),(148,179,210,241),(149,180,211,242),(150,181,212,243),(151,182,213,244),(152,183,214,245),(153,184,215,246),(154,185,216,247),(155,186,217,248)], [(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)], [(1,125,94,218,63,187,32,156),(2,186,95,155,64,248,33,217),(3,247,96,216,65,185,34,154),(4,184,97,153,66,246,35,215),(5,245,98,214,67,183,36,152),(6,182,99,151,68,244,37,213),(7,243,100,212,69,181,38,150),(8,180,101,149,70,242,39,211),(9,241,102,210,71,179,40,148),(10,178,103,147,72,240,41,209),(11,239,104,208,73,177,42,146),(12,176,105,145,74,238,43,207),(13,237,106,206,75,175,44,144),(14,174,107,143,76,236,45,205),(15,235,108,204,77,173,46,142),(16,172,109,141,78,234,47,203),(17,233,110,202,79,171,48,140),(18,170,111,139,80,232,49,201),(19,231,112,200,81,169,50,138),(20,168,113,137,82,230,51,199),(21,229,114,198,83,167,52,136),(22,166,115,135,84,228,53,197),(23,227,116,196,85,165,54,134),(24,164,117,133,86,226,55,195),(25,225,118,194,87,163,56,132),(26,162,119,131,88,224,57,193),(27,223,120,192,89,161,58,130),(28,160,121,129,90,222,59,191),(29,221,122,190,91,159,60,128),(30,158,123,127,92,220,61,189),(31,219,124,188,93,157,62,126)]])
130 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 31A | ··· | 31O | 62A | ··· | 62AS | 124A | ··· | 124BH |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 62 | 62 | 62 | 62 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | M4(2) | D31 | Dic31 | D62 | Dic31 | C4.Dic31 |
| kernel | C4.Dic31 | C31⋊C8 | C2×C124 | C124 | C2×C62 | C31 | C2×C4 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 15 | 15 | 15 | 15 | 60 |
Matrix representation of C4.Dic31 ►in GL2(𝔽1489) generated by
| 225 | 475 |
| 0 | 1264 |
| 1076 | 205 |
| 0 | 1352 |
| 1324 | 98 |
| 940 | 165 |
G:=sub<GL(2,GF(1489))| [225,0,475,1264],[1076,0,205,1352],[1324,940,98,165] >;
C4.Dic31 in GAP, Magma, Sage, TeX
C_4.{\rm Dic}_{31} % in TeX
G:=Group("C4.Dic31"); // GroupNames label
G:=SmallGroup(496,9);
// by ID
G=gap.SmallGroup(496,9);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-31,20,101,42,12004]);
// Polycyclic
G:=Group<a,b,c|a^4=1,b^62=a^2,c^2=a^2*b^31,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^61>;
// generators/relations
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