metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D236, C4⋊D59, C59⋊1D4, C236⋊1C2, D118⋊1C2, C2.4D118, C118.3C22, sometimes denoted D472 or Dih236 or Dih472, SmallGroup(472,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D236
G = < a,b | a236=b2=1, bab=a-1 >
Smallest permutation representation of D236
►On 236 points
Generators in S236
(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)
(1 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 50)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(60 236)(61 235)(62 234)(63 233)(64 232)(65 231)(66 230)(67 229)(68 228)(69 227)(70 226)(71 225)(72 224)(73 223)(74 222)(75 221)(76 220)(77 219)(78 218)(79 217)(80 216)(81 215)(82 214)(83 213)(84 212)(85 211)(86 210)(87 209)(88 208)(89 207)(90 206)(91 205)(92 204)(93 203)(94 202)(95 201)(96 200)(97 199)(98 198)(99 197)(100 196)(101 195)(102 194)(103 193)(104 192)(105 191)(106 190)(107 189)(108 188)(109 187)(110 186)(111 185)(112 184)(113 183)(114 182)(115 181)(116 180)(117 179)(118 178)(119 177)(120 176)(121 175)(122 174)(123 173)(124 172)(125 171)(126 170)(127 169)(128 168)(129 167)(130 166)(131 165)(132 164)(133 163)(134 162)(135 161)(136 160)(137 159)(138 158)(139 157)(140 156)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)
G:=sub<Sym(236)| (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), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,50)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(60,236)(61,235)(62,234)(63,233)(64,232)(65,231)(66,230)(67,229)(68,228)(69,227)(70,226)(71,225)(72,224)(73,223)(74,222)(75,221)(76,220)(77,219)(78,218)(79,217)(80,216)(81,215)(82,214)(83,213)(84,212)(85,211)(86,210)(87,209)(88,208)(89,207)(90,206)(91,205)(92,204)(93,203)(94,202)(95,201)(96,200)(97,199)(98,198)(99,197)(100,196)(101,195)(102,194)(103,193)(104,192)(105,191)(106,190)(107,189)(108,188)(109,187)(110,186)(111,185)(112,184)(113,183)(114,182)(115,181)(116,180)(117,179)(118,178)(119,177)(120,176)(121,175)(122,174)(123,173)(124,172)(125,171)(126,170)(127,169)(128,168)(129,167)(130,166)(131,165)(132,164)(133,163)(134,162)(135,161)(136,160)(137,159)(138,158)(139,157)(140,156)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)>;
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), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,50)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(60,236)(61,235)(62,234)(63,233)(64,232)(65,231)(66,230)(67,229)(68,228)(69,227)(70,226)(71,225)(72,224)(73,223)(74,222)(75,221)(76,220)(77,219)(78,218)(79,217)(80,216)(81,215)(82,214)(83,213)(84,212)(85,211)(86,210)(87,209)(88,208)(89,207)(90,206)(91,205)(92,204)(93,203)(94,202)(95,201)(96,200)(97,199)(98,198)(99,197)(100,196)(101,195)(102,194)(103,193)(104,192)(105,191)(106,190)(107,189)(108,188)(109,187)(110,186)(111,185)(112,184)(113,183)(114,182)(115,181)(116,180)(117,179)(118,178)(119,177)(120,176)(121,175)(122,174)(123,173)(124,172)(125,171)(126,170)(127,169)(128,168)(129,167)(130,166)(131,165)(132,164)(133,163)(134,162)(135,161)(136,160)(137,159)(138,158)(139,157)(140,156)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149) );
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)], [(1,59),(2,58),(3,57),(4,56),(5,55),(6,54),(7,53),(8,52),(9,51),(10,50),(11,49),(12,48),(13,47),(14,46),(15,45),(16,44),(17,43),(18,42),(19,41),(20,40),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(60,236),(61,235),(62,234),(63,233),(64,232),(65,231),(66,230),(67,229),(68,228),(69,227),(70,226),(71,225),(72,224),(73,223),(74,222),(75,221),(76,220),(77,219),(78,218),(79,217),(80,216),(81,215),(82,214),(83,213),(84,212),(85,211),(86,210),(87,209),(88,208),(89,207),(90,206),(91,205),(92,204),(93,203),(94,202),(95,201),(96,200),(97,199),(98,198),(99,197),(100,196),(101,195),(102,194),(103,193),(104,192),(105,191),(106,190),(107,189),(108,188),(109,187),(110,186),(111,185),(112,184),(113,183),(114,182),(115,181),(116,180),(117,179),(118,178),(119,177),(120,176),(121,175),(122,174),(123,173),(124,172),(125,171),(126,170),(127,169),(128,168),(129,167),(130,166),(131,165),(132,164),(133,163),(134,162),(135,161),(136,160),(137,159),(138,158),(139,157),(140,156),(141,155),(142,154),(143,153),(144,152),(145,151),(146,150),(147,149)]])
121 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 59A | ··· | 59AC | 118A | ··· | 118AC | 236A | ··· | 236BF |
| order | 1 | 2 | 2 | 2 | 4 | 59 | ··· | 59 | 118 | ··· | 118 | 236 | ··· | 236 |
| size | 1 | 1 | 118 | 118 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
121 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D59 | D118 | D236 |
| kernel | D236 | C236 | D118 | C59 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 29 | 29 | 58 |
Matrix representation of D236 ►in GL2(𝔽709) generated by
| 301 | 353 |
| 311 | 287 |
| 168 | 72 |
| 189 | 541 |
G:=sub<GL(2,GF(709))| [301,311,353,287],[168,189,72,541] >;
D236 in GAP, Magma, Sage, TeX
D_{236} % in TeX
G:=Group("D236"); // GroupNames label
G:=SmallGroup(472,5);
// by ID
G=gap.SmallGroup(472,5);
# by ID
G:=PCGroup([4,-2,-2,-2,-59,49,21,7427]);
// Polycyclic
G:=Group<a,b|a^236=b^2=1,b*a*b=a^-1>;
// generators/relations
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