metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C4.Dic29, C116.4C4, C4.15D58, C29⋊4M4(2), C22.Dic29, C116.15C22, C29⋊2C8⋊5C2, (C2×C58).5C4, (C2×C4).2D29, C58.14(C2×C4), (C2×C116).5C2, C2.3(C2×Dic29), SmallGroup(464,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.Dic29
G = < a,b,c | a4=1, b58=a2, c2=b29, ab=ba, cac-1=a-1, cbc-1=b57 >
Smallest permutation representation of C4.Dic29
►On 232 points
Generators in S232
(1 30 59 88)(2 31 60 89)(3 32 61 90)(4 33 62 91)(5 34 63 92)(6 35 64 93)(7 36 65 94)(8 37 66 95)(9 38 67 96)(10 39 68 97)(11 40 69 98)(12 41 70 99)(13 42 71 100)(14 43 72 101)(15 44 73 102)(16 45 74 103)(17 46 75 104)(18 47 76 105)(19 48 77 106)(20 49 78 107)(21 50 79 108)(22 51 80 109)(23 52 81 110)(24 53 82 111)(25 54 83 112)(26 55 84 113)(27 56 85 114)(28 57 86 115)(29 58 87 116)(117 204 175 146)(118 205 176 147)(119 206 177 148)(120 207 178 149)(121 208 179 150)(122 209 180 151)(123 210 181 152)(124 211 182 153)(125 212 183 154)(126 213 184 155)(127 214 185 156)(128 215 186 157)(129 216 187 158)(130 217 188 159)(131 218 189 160)(132 219 190 161)(133 220 191 162)(134 221 192 163)(135 222 193 164)(136 223 194 165)(137 224 195 166)(138 225 196 167)(139 226 197 168)(140 227 198 169)(141 228 199 170)(142 229 200 171)(143 230 201 172)(144 231 202 173)(145 232 203 174)
(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)
(1 204 30 117 59 146 88 175)(2 145 31 174 60 203 89 232)(3 202 32 231 61 144 90 173)(4 143 33 172 62 201 91 230)(5 200 34 229 63 142 92 171)(6 141 35 170 64 199 93 228)(7 198 36 227 65 140 94 169)(8 139 37 168 66 197 95 226)(9 196 38 225 67 138 96 167)(10 137 39 166 68 195 97 224)(11 194 40 223 69 136 98 165)(12 135 41 164 70 193 99 222)(13 192 42 221 71 134 100 163)(14 133 43 162 72 191 101 220)(15 190 44 219 73 132 102 161)(16 131 45 160 74 189 103 218)(17 188 46 217 75 130 104 159)(18 129 47 158 76 187 105 216)(19 186 48 215 77 128 106 157)(20 127 49 156 78 185 107 214)(21 184 50 213 79 126 108 155)(22 125 51 154 80 183 109 212)(23 182 52 211 81 124 110 153)(24 123 53 152 82 181 111 210)(25 180 54 209 83 122 112 151)(26 121 55 150 84 179 113 208)(27 178 56 207 85 120 114 149)(28 119 57 148 86 177 115 206)(29 176 58 205 87 118 116 147)
G:=sub<Sym(232)| (1,30,59,88)(2,31,60,89)(3,32,61,90)(4,33,62,91)(5,34,63,92)(6,35,64,93)(7,36,65,94)(8,37,66,95)(9,38,67,96)(10,39,68,97)(11,40,69,98)(12,41,70,99)(13,42,71,100)(14,43,72,101)(15,44,73,102)(16,45,74,103)(17,46,75,104)(18,47,76,105)(19,48,77,106)(20,49,78,107)(21,50,79,108)(22,51,80,109)(23,52,81,110)(24,53,82,111)(25,54,83,112)(26,55,84,113)(27,56,85,114)(28,57,86,115)(29,58,87,116)(117,204,175,146)(118,205,176,147)(119,206,177,148)(120,207,178,149)(121,208,179,150)(122,209,180,151)(123,210,181,152)(124,211,182,153)(125,212,183,154)(126,213,184,155)(127,214,185,156)(128,215,186,157)(129,216,187,158)(130,217,188,159)(131,218,189,160)(132,219,190,161)(133,220,191,162)(134,221,192,163)(135,222,193,164)(136,223,194,165)(137,224,195,166)(138,225,196,167)(139,226,197,168)(140,227,198,169)(141,228,199,170)(142,229,200,171)(143,230,201,172)(144,231,202,173)(145,232,203,174), (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), (1,204,30,117,59,146,88,175)(2,145,31,174,60,203,89,232)(3,202,32,231,61,144,90,173)(4,143,33,172,62,201,91,230)(5,200,34,229,63,142,92,171)(6,141,35,170,64,199,93,228)(7,198,36,227,65,140,94,169)(8,139,37,168,66,197,95,226)(9,196,38,225,67,138,96,167)(10,137,39,166,68,195,97,224)(11,194,40,223,69,136,98,165)(12,135,41,164,70,193,99,222)(13,192,42,221,71,134,100,163)(14,133,43,162,72,191,101,220)(15,190,44,219,73,132,102,161)(16,131,45,160,74,189,103,218)(17,188,46,217,75,130,104,159)(18,129,47,158,76,187,105,216)(19,186,48,215,77,128,106,157)(20,127,49,156,78,185,107,214)(21,184,50,213,79,126,108,155)(22,125,51,154,80,183,109,212)(23,182,52,211,81,124,110,153)(24,123,53,152,82,181,111,210)(25,180,54,209,83,122,112,151)(26,121,55,150,84,179,113,208)(27,178,56,207,85,120,114,149)(28,119,57,148,86,177,115,206)(29,176,58,205,87,118,116,147)>;
G:=Group( (1,30,59,88)(2,31,60,89)(3,32,61,90)(4,33,62,91)(5,34,63,92)(6,35,64,93)(7,36,65,94)(8,37,66,95)(9,38,67,96)(10,39,68,97)(11,40,69,98)(12,41,70,99)(13,42,71,100)(14,43,72,101)(15,44,73,102)(16,45,74,103)(17,46,75,104)(18,47,76,105)(19,48,77,106)(20,49,78,107)(21,50,79,108)(22,51,80,109)(23,52,81,110)(24,53,82,111)(25,54,83,112)(26,55,84,113)(27,56,85,114)(28,57,86,115)(29,58,87,116)(117,204,175,146)(118,205,176,147)(119,206,177,148)(120,207,178,149)(121,208,179,150)(122,209,180,151)(123,210,181,152)(124,211,182,153)(125,212,183,154)(126,213,184,155)(127,214,185,156)(128,215,186,157)(129,216,187,158)(130,217,188,159)(131,218,189,160)(132,219,190,161)(133,220,191,162)(134,221,192,163)(135,222,193,164)(136,223,194,165)(137,224,195,166)(138,225,196,167)(139,226,197,168)(140,227,198,169)(141,228,199,170)(142,229,200,171)(143,230,201,172)(144,231,202,173)(145,232,203,174), (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), (1,204,30,117,59,146,88,175)(2,145,31,174,60,203,89,232)(3,202,32,231,61,144,90,173)(4,143,33,172,62,201,91,230)(5,200,34,229,63,142,92,171)(6,141,35,170,64,199,93,228)(7,198,36,227,65,140,94,169)(8,139,37,168,66,197,95,226)(9,196,38,225,67,138,96,167)(10,137,39,166,68,195,97,224)(11,194,40,223,69,136,98,165)(12,135,41,164,70,193,99,222)(13,192,42,221,71,134,100,163)(14,133,43,162,72,191,101,220)(15,190,44,219,73,132,102,161)(16,131,45,160,74,189,103,218)(17,188,46,217,75,130,104,159)(18,129,47,158,76,187,105,216)(19,186,48,215,77,128,106,157)(20,127,49,156,78,185,107,214)(21,184,50,213,79,126,108,155)(22,125,51,154,80,183,109,212)(23,182,52,211,81,124,110,153)(24,123,53,152,82,181,111,210)(25,180,54,209,83,122,112,151)(26,121,55,150,84,179,113,208)(27,178,56,207,85,120,114,149)(28,119,57,148,86,177,115,206)(29,176,58,205,87,118,116,147) );
G=PermutationGroup([[(1,30,59,88),(2,31,60,89),(3,32,61,90),(4,33,62,91),(5,34,63,92),(6,35,64,93),(7,36,65,94),(8,37,66,95),(9,38,67,96),(10,39,68,97),(11,40,69,98),(12,41,70,99),(13,42,71,100),(14,43,72,101),(15,44,73,102),(16,45,74,103),(17,46,75,104),(18,47,76,105),(19,48,77,106),(20,49,78,107),(21,50,79,108),(22,51,80,109),(23,52,81,110),(24,53,82,111),(25,54,83,112),(26,55,84,113),(27,56,85,114),(28,57,86,115),(29,58,87,116),(117,204,175,146),(118,205,176,147),(119,206,177,148),(120,207,178,149),(121,208,179,150),(122,209,180,151),(123,210,181,152),(124,211,182,153),(125,212,183,154),(126,213,184,155),(127,214,185,156),(128,215,186,157),(129,216,187,158),(130,217,188,159),(131,218,189,160),(132,219,190,161),(133,220,191,162),(134,221,192,163),(135,222,193,164),(136,223,194,165),(137,224,195,166),(138,225,196,167),(139,226,197,168),(140,227,198,169),(141,228,199,170),(142,229,200,171),(143,230,201,172),(144,231,202,173),(145,232,203,174)], [(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)], [(1,204,30,117,59,146,88,175),(2,145,31,174,60,203,89,232),(3,202,32,231,61,144,90,173),(4,143,33,172,62,201,91,230),(5,200,34,229,63,142,92,171),(6,141,35,170,64,199,93,228),(7,198,36,227,65,140,94,169),(8,139,37,168,66,197,95,226),(9,196,38,225,67,138,96,167),(10,137,39,166,68,195,97,224),(11,194,40,223,69,136,98,165),(12,135,41,164,70,193,99,222),(13,192,42,221,71,134,100,163),(14,133,43,162,72,191,101,220),(15,190,44,219,73,132,102,161),(16,131,45,160,74,189,103,218),(17,188,46,217,75,130,104,159),(18,129,47,158,76,187,105,216),(19,186,48,215,77,128,106,157),(20,127,49,156,78,185,107,214),(21,184,50,213,79,126,108,155),(22,125,51,154,80,183,109,212),(23,182,52,211,81,124,110,153),(24,123,53,152,82,181,111,210),(25,180,54,209,83,122,112,151),(26,121,55,150,84,179,113,208),(27,178,56,207,85,120,114,149),(28,119,57,148,86,177,115,206),(29,176,58,205,87,118,116,147)]])
122 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 29A | ··· | 29N | 58A | ··· | 58AP | 116A | ··· | 116BD |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 58 | 58 | 58 | 58 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
122 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | M4(2) | D29 | Dic29 | D58 | Dic29 | C4.Dic29 |
| kernel | C4.Dic29 | C29⋊2C8 | C2×C116 | C116 | C2×C58 | C29 | C2×C4 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 56 |
Matrix representation of C4.Dic29 ►in GL2(𝔽233) generated by
| 89 | 0 |
| 0 | 144 |
| 224 | 0 |
| 0 | 26 |
| 0 | 1 |
| 89 | 0 |
G:=sub<GL(2,GF(233))| [89,0,0,144],[224,0,0,26],[0,89,1,0] >;
C4.Dic29 in GAP, Magma, Sage, TeX
C_4.{\rm Dic}_{29} % in TeX
G:=Group("C4.Dic29"); // GroupNames label
G:=SmallGroup(464,10);
// by ID
G=gap.SmallGroup(464,10);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,20,101,42,11204]);
// Polycyclic
G:=Group<a,b,c|a^4=1,b^58=a^2,c^2=b^29,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^57>;
// generators/relations
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