metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C59⋊2D4, C22⋊D59, Dic59⋊C2, D118⋊2C2, C2.5D118, C118.5C22, (C2×C118)⋊2C2, SmallGroup(472,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C59⋊D4
G = < a,b,c | a59=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C59⋊D4
►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)
(1 162 82 180)(2 161 83 179)(3 160 84 178)(4 159 85 236)(5 158 86 235)(6 157 87 234)(7 156 88 233)(8 155 89 232)(9 154 90 231)(10 153 91 230)(11 152 92 229)(12 151 93 228)(13 150 94 227)(14 149 95 226)(15 148 96 225)(16 147 97 224)(17 146 98 223)(18 145 99 222)(19 144 100 221)(20 143 101 220)(21 142 102 219)(22 141 103 218)(23 140 104 217)(24 139 105 216)(25 138 106 215)(26 137 107 214)(27 136 108 213)(28 135 109 212)(29 134 110 211)(30 133 111 210)(31 132 112 209)(32 131 113 208)(33 130 114 207)(34 129 115 206)(35 128 116 205)(36 127 117 204)(37 126 118 203)(38 125 60 202)(39 124 61 201)(40 123 62 200)(41 122 63 199)(42 121 64 198)(43 120 65 197)(44 119 66 196)(45 177 67 195)(46 176 68 194)(47 175 69 193)(48 174 70 192)(49 173 71 191)(50 172 72 190)(51 171 73 189)(52 170 74 188)(53 169 75 187)(54 168 76 186)(55 167 77 185)(56 166 78 184)(57 165 79 183)(58 164 80 182)(59 163 81 181)
(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(60 104)(61 103)(62 102)(63 101)(64 100)(65 99)(66 98)(67 97)(68 96)(69 95)(70 94)(71 93)(72 92)(73 91)(74 90)(75 89)(76 88)(77 87)(78 86)(79 85)(80 84)(81 83)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(119 223)(120 222)(121 221)(122 220)(123 219)(124 218)(125 217)(126 216)(127 215)(128 214)(129 213)(130 212)(131 211)(132 210)(133 209)(134 208)(135 207)(136 206)(137 205)(138 204)(139 203)(140 202)(141 201)(142 200)(143 199)(144 198)(145 197)(146 196)(147 195)(148 194)(149 193)(150 192)(151 191)(152 190)(153 189)(154 188)(155 187)(156 186)(157 185)(158 184)(159 183)(160 182)(161 181)(162 180)(163 179)(164 178)(165 236)(166 235)(167 234)(168 233)(169 232)(170 231)(171 230)(172 229)(173 228)(174 227)(175 226)(176 225)(177 224)
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), (1,162,82,180)(2,161,83,179)(3,160,84,178)(4,159,85,236)(5,158,86,235)(6,157,87,234)(7,156,88,233)(8,155,89,232)(9,154,90,231)(10,153,91,230)(11,152,92,229)(12,151,93,228)(13,150,94,227)(14,149,95,226)(15,148,96,225)(16,147,97,224)(17,146,98,223)(18,145,99,222)(19,144,100,221)(20,143,101,220)(21,142,102,219)(22,141,103,218)(23,140,104,217)(24,139,105,216)(25,138,106,215)(26,137,107,214)(27,136,108,213)(28,135,109,212)(29,134,110,211)(30,133,111,210)(31,132,112,209)(32,131,113,208)(33,130,114,207)(34,129,115,206)(35,128,116,205)(36,127,117,204)(37,126,118,203)(38,125,60,202)(39,124,61,201)(40,123,62,200)(41,122,63,199)(42,121,64,198)(43,120,65,197)(44,119,66,196)(45,177,67,195)(46,176,68,194)(47,175,69,193)(48,174,70,192)(49,173,71,191)(50,172,72,190)(51,171,73,189)(52,170,74,188)(53,169,75,187)(54,168,76,186)(55,167,77,185)(56,166,78,184)(57,165,79,183)(58,164,80,182)(59,163,81,181), (2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(60,104)(61,103)(62,102)(63,101)(64,100)(65,99)(66,98)(67,97)(68,96)(69,95)(70,94)(71,93)(72,92)(73,91)(74,90)(75,89)(76,88)(77,87)(78,86)(79,85)(80,84)(81,83)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(119,223)(120,222)(121,221)(122,220)(123,219)(124,218)(125,217)(126,216)(127,215)(128,214)(129,213)(130,212)(131,211)(132,210)(133,209)(134,208)(135,207)(136,206)(137,205)(138,204)(139,203)(140,202)(141,201)(142,200)(143,199)(144,198)(145,197)(146,196)(147,195)(148,194)(149,193)(150,192)(151,191)(152,190)(153,189)(154,188)(155,187)(156,186)(157,185)(158,184)(159,183)(160,182)(161,181)(162,180)(163,179)(164,178)(165,236)(166,235)(167,234)(168,233)(169,232)(170,231)(171,230)(172,229)(173,228)(174,227)(175,226)(176,225)(177,224)>;
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), (1,162,82,180)(2,161,83,179)(3,160,84,178)(4,159,85,236)(5,158,86,235)(6,157,87,234)(7,156,88,233)(8,155,89,232)(9,154,90,231)(10,153,91,230)(11,152,92,229)(12,151,93,228)(13,150,94,227)(14,149,95,226)(15,148,96,225)(16,147,97,224)(17,146,98,223)(18,145,99,222)(19,144,100,221)(20,143,101,220)(21,142,102,219)(22,141,103,218)(23,140,104,217)(24,139,105,216)(25,138,106,215)(26,137,107,214)(27,136,108,213)(28,135,109,212)(29,134,110,211)(30,133,111,210)(31,132,112,209)(32,131,113,208)(33,130,114,207)(34,129,115,206)(35,128,116,205)(36,127,117,204)(37,126,118,203)(38,125,60,202)(39,124,61,201)(40,123,62,200)(41,122,63,199)(42,121,64,198)(43,120,65,197)(44,119,66,196)(45,177,67,195)(46,176,68,194)(47,175,69,193)(48,174,70,192)(49,173,71,191)(50,172,72,190)(51,171,73,189)(52,170,74,188)(53,169,75,187)(54,168,76,186)(55,167,77,185)(56,166,78,184)(57,165,79,183)(58,164,80,182)(59,163,81,181), (2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(60,104)(61,103)(62,102)(63,101)(64,100)(65,99)(66,98)(67,97)(68,96)(69,95)(70,94)(71,93)(72,92)(73,91)(74,90)(75,89)(76,88)(77,87)(78,86)(79,85)(80,84)(81,83)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(119,223)(120,222)(121,221)(122,220)(123,219)(124,218)(125,217)(126,216)(127,215)(128,214)(129,213)(130,212)(131,211)(132,210)(133,209)(134,208)(135,207)(136,206)(137,205)(138,204)(139,203)(140,202)(141,201)(142,200)(143,199)(144,198)(145,197)(146,196)(147,195)(148,194)(149,193)(150,192)(151,191)(152,190)(153,189)(154,188)(155,187)(156,186)(157,185)(158,184)(159,183)(160,182)(161,181)(162,180)(163,179)(164,178)(165,236)(166,235)(167,234)(168,233)(169,232)(170,231)(171,230)(172,229)(173,228)(174,227)(175,226)(176,225)(177,224) );
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)], [(1,162,82,180),(2,161,83,179),(3,160,84,178),(4,159,85,236),(5,158,86,235),(6,157,87,234),(7,156,88,233),(8,155,89,232),(9,154,90,231),(10,153,91,230),(11,152,92,229),(12,151,93,228),(13,150,94,227),(14,149,95,226),(15,148,96,225),(16,147,97,224),(17,146,98,223),(18,145,99,222),(19,144,100,221),(20,143,101,220),(21,142,102,219),(22,141,103,218),(23,140,104,217),(24,139,105,216),(25,138,106,215),(26,137,107,214),(27,136,108,213),(28,135,109,212),(29,134,110,211),(30,133,111,210),(31,132,112,209),(32,131,113,208),(33,130,114,207),(34,129,115,206),(35,128,116,205),(36,127,117,204),(37,126,118,203),(38,125,60,202),(39,124,61,201),(40,123,62,200),(41,122,63,199),(42,121,64,198),(43,120,65,197),(44,119,66,196),(45,177,67,195),(46,176,68,194),(47,175,69,193),(48,174,70,192),(49,173,71,191),(50,172,72,190),(51,171,73,189),(52,170,74,188),(53,169,75,187),(54,168,76,186),(55,167,77,185),(56,166,78,184),(57,165,79,183),(58,164,80,182),(59,163,81,181)], [(2,59),(3,58),(4,57),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(60,104),(61,103),(62,102),(63,101),(64,100),(65,99),(66,98),(67,97),(68,96),(69,95),(70,94),(71,93),(72,92),(73,91),(74,90),(75,89),(76,88),(77,87),(78,86),(79,85),(80,84),(81,83),(105,118),(106,117),(107,116),(108,115),(109,114),(110,113),(111,112),(119,223),(120,222),(121,221),(122,220),(123,219),(124,218),(125,217),(126,216),(127,215),(128,214),(129,213),(130,212),(131,211),(132,210),(133,209),(134,208),(135,207),(136,206),(137,205),(138,204),(139,203),(140,202),(141,201),(142,200),(143,199),(144,198),(145,197),(146,196),(147,195),(148,194),(149,193),(150,192),(151,191),(152,190),(153,189),(154,188),(155,187),(156,186),(157,185),(158,184),(159,183),(160,182),(161,181),(162,180),(163,179),(164,178),(165,236),(166,235),(167,234),(168,233),(169,232),(170,231),(171,230),(172,229),(173,228),(174,227),(175,226),(176,225),(177,224)]])
121 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 59A | ··· | 59AC | 118A | ··· | 118CI |
| order | 1 | 2 | 2 | 2 | 4 | 59 | ··· | 59 | 118 | ··· | 118 |
| size | 1 | 1 | 2 | 118 | 118 | 2 | ··· | 2 | 2 | ··· | 2 |
121 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D59 | D118 | C59⋊D4 |
| kernel | C59⋊D4 | Dic59 | D118 | C2×C118 | C59 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 29 | 29 | 58 |
Matrix representation of C59⋊D4 ►in GL2(𝔽709) generated by
| 490 | 1 |
| 135 | 226 |
| 634 | 416 |
| 508 | 75 |
| 394 | 142 |
| 180 | 315 |
G:=sub<GL(2,GF(709))| [490,135,1,226],[634,508,416,75],[394,180,142,315] >;
C59⋊D4 in GAP, Magma, Sage, TeX
C_{59}\rtimes D_4 % in TeX
G:=Group("C59:D4"); // GroupNames label
G:=SmallGroup(472,7);
// by ID
G=gap.SmallGroup(472,7);
# by ID
G:=PCGroup([4,-2,-2,-2,-59,49,7427]);
// Polycyclic
G:=Group<a,b,c|a^59=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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