direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: M4(2)×C29, C8⋊3C58, C4.C116, C232⋊7C2, C116.7C4, C22.C116, C116.22C22, (C2×C4).2C58, C4.6(C2×C58), (C2×C58).3C4, C58.19(C2×C4), (C2×C116).8C2, C2.3(C2×C116), SmallGroup(464,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×C29
G = < a,b,c | a29=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Smallest permutation representation of M4(2)×C29
►On 232 points
Generators in S232
(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 78 105 133 54 213 192 162)(2 79 106 134 55 214 193 163)(3 80 107 135 56 215 194 164)(4 81 108 136 57 216 195 165)(5 82 109 137 58 217 196 166)(6 83 110 138 30 218 197 167)(7 84 111 139 31 219 198 168)(8 85 112 140 32 220 199 169)(9 86 113 141 33 221 200 170)(10 87 114 142 34 222 201 171)(11 59 115 143 35 223 202 172)(12 60 116 144 36 224 203 173)(13 61 88 145 37 225 175 174)(14 62 89 117 38 226 176 146)(15 63 90 118 39 227 177 147)(16 64 91 119 40 228 178 148)(17 65 92 120 41 229 179 149)(18 66 93 121 42 230 180 150)(19 67 94 122 43 231 181 151)(20 68 95 123 44 232 182 152)(21 69 96 124 45 204 183 153)(22 70 97 125 46 205 184 154)(23 71 98 126 47 206 185 155)(24 72 99 127 48 207 186 156)(25 73 100 128 49 208 187 157)(26 74 101 129 50 209 188 158)(27 75 102 130 51 210 189 159)(28 76 103 131 52 211 190 160)(29 77 104 132 53 212 191 161)
(59 223)(60 224)(61 225)(62 226)(63 227)(64 228)(65 229)(66 230)(67 231)(68 232)(69 204)(70 205)(71 206)(72 207)(73 208)(74 209)(75 210)(76 211)(77 212)(78 213)(79 214)(80 215)(81 216)(82 217)(83 218)(84 219)(85 220)(86 221)(87 222)(117 146)(118 147)(119 148)(120 149)(121 150)(122 151)(123 152)(124 153)(125 154)(126 155)(127 156)(128 157)(129 158)(130 159)(131 160)(132 161)(133 162)(134 163)(135 164)(136 165)(137 166)(138 167)(139 168)(140 169)(141 170)(142 171)(143 172)(144 173)(145 174)
G:=sub<Sym(232)| (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), (1,78,105,133,54,213,192,162)(2,79,106,134,55,214,193,163)(3,80,107,135,56,215,194,164)(4,81,108,136,57,216,195,165)(5,82,109,137,58,217,196,166)(6,83,110,138,30,218,197,167)(7,84,111,139,31,219,198,168)(8,85,112,140,32,220,199,169)(9,86,113,141,33,221,200,170)(10,87,114,142,34,222,201,171)(11,59,115,143,35,223,202,172)(12,60,116,144,36,224,203,173)(13,61,88,145,37,225,175,174)(14,62,89,117,38,226,176,146)(15,63,90,118,39,227,177,147)(16,64,91,119,40,228,178,148)(17,65,92,120,41,229,179,149)(18,66,93,121,42,230,180,150)(19,67,94,122,43,231,181,151)(20,68,95,123,44,232,182,152)(21,69,96,124,45,204,183,153)(22,70,97,125,46,205,184,154)(23,71,98,126,47,206,185,155)(24,72,99,127,48,207,186,156)(25,73,100,128,49,208,187,157)(26,74,101,129,50,209,188,158)(27,75,102,130,51,210,189,159)(28,76,103,131,52,211,190,160)(29,77,104,132,53,212,191,161), (59,223)(60,224)(61,225)(62,226)(63,227)(64,228)(65,229)(66,230)(67,231)(68,232)(69,204)(70,205)(71,206)(72,207)(73,208)(74,209)(75,210)(76,211)(77,212)(78,213)(79,214)(80,215)(81,216)(82,217)(83,218)(84,219)(85,220)(86,221)(87,222)(117,146)(118,147)(119,148)(120,149)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(141,170)(142,171)(143,172)(144,173)(145,174)>;
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), (1,78,105,133,54,213,192,162)(2,79,106,134,55,214,193,163)(3,80,107,135,56,215,194,164)(4,81,108,136,57,216,195,165)(5,82,109,137,58,217,196,166)(6,83,110,138,30,218,197,167)(7,84,111,139,31,219,198,168)(8,85,112,140,32,220,199,169)(9,86,113,141,33,221,200,170)(10,87,114,142,34,222,201,171)(11,59,115,143,35,223,202,172)(12,60,116,144,36,224,203,173)(13,61,88,145,37,225,175,174)(14,62,89,117,38,226,176,146)(15,63,90,118,39,227,177,147)(16,64,91,119,40,228,178,148)(17,65,92,120,41,229,179,149)(18,66,93,121,42,230,180,150)(19,67,94,122,43,231,181,151)(20,68,95,123,44,232,182,152)(21,69,96,124,45,204,183,153)(22,70,97,125,46,205,184,154)(23,71,98,126,47,206,185,155)(24,72,99,127,48,207,186,156)(25,73,100,128,49,208,187,157)(26,74,101,129,50,209,188,158)(27,75,102,130,51,210,189,159)(28,76,103,131,52,211,190,160)(29,77,104,132,53,212,191,161), (59,223)(60,224)(61,225)(62,226)(63,227)(64,228)(65,229)(66,230)(67,231)(68,232)(69,204)(70,205)(71,206)(72,207)(73,208)(74,209)(75,210)(76,211)(77,212)(78,213)(79,214)(80,215)(81,216)(82,217)(83,218)(84,219)(85,220)(86,221)(87,222)(117,146)(118,147)(119,148)(120,149)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,158)(130,159)(131,160)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,169)(141,170)(142,171)(143,172)(144,173)(145,174) );
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)], [(1,78,105,133,54,213,192,162),(2,79,106,134,55,214,193,163),(3,80,107,135,56,215,194,164),(4,81,108,136,57,216,195,165),(5,82,109,137,58,217,196,166),(6,83,110,138,30,218,197,167),(7,84,111,139,31,219,198,168),(8,85,112,140,32,220,199,169),(9,86,113,141,33,221,200,170),(10,87,114,142,34,222,201,171),(11,59,115,143,35,223,202,172),(12,60,116,144,36,224,203,173),(13,61,88,145,37,225,175,174),(14,62,89,117,38,226,176,146),(15,63,90,118,39,227,177,147),(16,64,91,119,40,228,178,148),(17,65,92,120,41,229,179,149),(18,66,93,121,42,230,180,150),(19,67,94,122,43,231,181,151),(20,68,95,123,44,232,182,152),(21,69,96,124,45,204,183,153),(22,70,97,125,46,205,184,154),(23,71,98,126,47,206,185,155),(24,72,99,127,48,207,186,156),(25,73,100,128,49,208,187,157),(26,74,101,129,50,209,188,158),(27,75,102,130,51,210,189,159),(28,76,103,131,52,211,190,160),(29,77,104,132,53,212,191,161)], [(59,223),(60,224),(61,225),(62,226),(63,227),(64,228),(65,229),(66,230),(67,231),(68,232),(69,204),(70,205),(71,206),(72,207),(73,208),(74,209),(75,210),(76,211),(77,212),(78,213),(79,214),(80,215),(81,216),(82,217),(83,218),(84,219),(85,220),(86,221),(87,222),(117,146),(118,147),(119,148),(120,149),(121,150),(122,151),(123,152),(124,153),(125,154),(126,155),(127,156),(128,157),(129,158),(130,159),(131,160),(132,161),(133,162),(134,163),(135,164),(136,165),(137,166),(138,167),(139,168),(140,169),(141,170),(142,171),(143,172),(144,173),(145,174)]])
290 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 29A | ··· | 29AB | 58A | ··· | 58AB | 58AC | ··· | 58BD | 116A | ··· | 116BD | 116BE | ··· | 116CF | 232A | ··· | 232DH |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 29 | ··· | 29 | 58 | ··· | 58 | 58 | ··· | 58 | 116 | ··· | 116 | 116 | ··· | 116 | 232 | ··· | 232 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
290 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C29 | C58 | C58 | C116 | C116 | M4(2) | M4(2)×C29 |
| kernel | M4(2)×C29 | C232 | C2×C116 | C116 | C2×C58 | M4(2) | C8 | C2×C4 | C4 | C22 | C29 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 28 | 56 | 28 | 56 | 56 | 2 | 56 |
Matrix representation of M4(2)×C29 ►in GL2(𝔽233) generated by
| 63 | 0 |
| 0 | 63 |
| 140 | 197 |
| 178 | 93 |
| 1 | 111 |
| 0 | 232 |
G:=sub<GL(2,GF(233))| [63,0,0,63],[140,178,197,93],[1,0,111,232] >;
M4(2)×C29 in GAP, Magma, Sage, TeX
M_4(2)\times C_{29} % in TeX
G:=Group("M4(2)xC29"); // GroupNames label
G:=SmallGroup(464,24);
// by ID
G=gap.SmallGroup(464,24);
# by ID
G:=PCGroup([5,-2,-2,-29,-2,-2,580,2341,58]);
// Polycyclic
G:=Group<a,b,c|a^29=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations
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