metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C31⋊C8, C62.C4, C4.2D31, C2.Dic31, C124.2C2, SmallGroup(248,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C31⋊C8 |
Generators and relations for C31⋊C8
G = < a,b | a31=b8=1, bab-1=a-1 >
Smallest permutation representation of C31⋊C8
►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)
(1 218 113 156 47 187 85 125)(2 248 114 186 48 217 86 155)(3 247 115 185 49 216 87 154)(4 246 116 184 50 215 88 153)(5 245 117 183 51 214 89 152)(6 244 118 182 52 213 90 151)(7 243 119 181 53 212 91 150)(8 242 120 180 54 211 92 149)(9 241 121 179 55 210 93 148)(10 240 122 178 56 209 63 147)(11 239 123 177 57 208 64 146)(12 238 124 176 58 207 65 145)(13 237 94 175 59 206 66 144)(14 236 95 174 60 205 67 143)(15 235 96 173 61 204 68 142)(16 234 97 172 62 203 69 141)(17 233 98 171 32 202 70 140)(18 232 99 170 33 201 71 139)(19 231 100 169 34 200 72 138)(20 230 101 168 35 199 73 137)(21 229 102 167 36 198 74 136)(22 228 103 166 37 197 75 135)(23 227 104 165 38 196 76 134)(24 226 105 164 39 195 77 133)(25 225 106 163 40 194 78 132)(26 224 107 162 41 193 79 131)(27 223 108 161 42 192 80 130)(28 222 109 160 43 191 81 129)(29 221 110 159 44 190 82 128)(30 220 111 158 45 189 83 127)(31 219 112 157 46 188 84 126)
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), (1,218,113,156,47,187,85,125)(2,248,114,186,48,217,86,155)(3,247,115,185,49,216,87,154)(4,246,116,184,50,215,88,153)(5,245,117,183,51,214,89,152)(6,244,118,182,52,213,90,151)(7,243,119,181,53,212,91,150)(8,242,120,180,54,211,92,149)(9,241,121,179,55,210,93,148)(10,240,122,178,56,209,63,147)(11,239,123,177,57,208,64,146)(12,238,124,176,58,207,65,145)(13,237,94,175,59,206,66,144)(14,236,95,174,60,205,67,143)(15,235,96,173,61,204,68,142)(16,234,97,172,62,203,69,141)(17,233,98,171,32,202,70,140)(18,232,99,170,33,201,71,139)(19,231,100,169,34,200,72,138)(20,230,101,168,35,199,73,137)(21,229,102,167,36,198,74,136)(22,228,103,166,37,197,75,135)(23,227,104,165,38,196,76,134)(24,226,105,164,39,195,77,133)(25,225,106,163,40,194,78,132)(26,224,107,162,41,193,79,131)(27,223,108,161,42,192,80,130)(28,222,109,160,43,191,81,129)(29,221,110,159,44,190,82,128)(30,220,111,158,45,189,83,127)(31,219,112,157,46,188,84,126)>;
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), (1,218,113,156,47,187,85,125)(2,248,114,186,48,217,86,155)(3,247,115,185,49,216,87,154)(4,246,116,184,50,215,88,153)(5,245,117,183,51,214,89,152)(6,244,118,182,52,213,90,151)(7,243,119,181,53,212,91,150)(8,242,120,180,54,211,92,149)(9,241,121,179,55,210,93,148)(10,240,122,178,56,209,63,147)(11,239,123,177,57,208,64,146)(12,238,124,176,58,207,65,145)(13,237,94,175,59,206,66,144)(14,236,95,174,60,205,67,143)(15,235,96,173,61,204,68,142)(16,234,97,172,62,203,69,141)(17,233,98,171,32,202,70,140)(18,232,99,170,33,201,71,139)(19,231,100,169,34,200,72,138)(20,230,101,168,35,199,73,137)(21,229,102,167,36,198,74,136)(22,228,103,166,37,197,75,135)(23,227,104,165,38,196,76,134)(24,226,105,164,39,195,77,133)(25,225,106,163,40,194,78,132)(26,224,107,162,41,193,79,131)(27,223,108,161,42,192,80,130)(28,222,109,160,43,191,81,129)(29,221,110,159,44,190,82,128)(30,220,111,158,45,189,83,127)(31,219,112,157,46,188,84,126) );
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)], [(1,218,113,156,47,187,85,125),(2,248,114,186,48,217,86,155),(3,247,115,185,49,216,87,154),(4,246,116,184,50,215,88,153),(5,245,117,183,51,214,89,152),(6,244,118,182,52,213,90,151),(7,243,119,181,53,212,91,150),(8,242,120,180,54,211,92,149),(9,241,121,179,55,210,93,148),(10,240,122,178,56,209,63,147),(11,239,123,177,57,208,64,146),(12,238,124,176,58,207,65,145),(13,237,94,175,59,206,66,144),(14,236,95,174,60,205,67,143),(15,235,96,173,61,204,68,142),(16,234,97,172,62,203,69,141),(17,233,98,171,32,202,70,140),(18,232,99,170,33,201,71,139),(19,231,100,169,34,200,72,138),(20,230,101,168,35,199,73,137),(21,229,102,167,36,198,74,136),(22,228,103,166,37,197,75,135),(23,227,104,165,38,196,76,134),(24,226,105,164,39,195,77,133),(25,225,106,163,40,194,78,132),(26,224,107,162,41,193,79,131),(27,223,108,161,42,192,80,130),(28,222,109,160,43,191,81,129),(29,221,110,159,44,190,82,128),(30,220,111,158,45,189,83,127),(31,219,112,157,46,188,84,126)]])
C31⋊C8 is a maximal subgroup of
C8×D31 C8⋊D31 C4.Dic31 D4⋊D31 D4.D31 Q8⋊D31 C31⋊Q16
C31⋊C8 is a maximal quotient of C31⋊C16
68 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 31A | ··· | 31O | 62A | ··· | 62O | 124A | ··· | 124AD |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 31 | ··· | 31 | 62 | ··· | 62 | 124 | ··· | 124 |
| size | 1 | 1 | 1 | 1 | 31 | 31 | 31 | 31 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | D31 | Dic31 | C31⋊C8 |
| kernel | C31⋊C8 | C124 | C62 | C31 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 15 | 15 | 30 |
Matrix representation of C31⋊C8 ►in GL2(𝔽1489) generated by
| 0 | 1 |
| 1488 | 1013 |
| 417 | 1085 |
| 630 | 1072 |
G:=sub<GL(2,GF(1489))| [0,1488,1,1013],[417,630,1085,1072] >;
C31⋊C8 in GAP, Magma, Sage, TeX
C_{31}\rtimes C_8 % in TeX
G:=Group("C31:C8"); // GroupNames label
G:=SmallGroup(248,1);
// by ID
G=gap.SmallGroup(248,1);
# by ID
G:=PCGroup([4,-2,-2,-2,-31,8,21,3843]);
// Polycyclic
G:=Group<a,b|a^31=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
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