direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C22⋊C4×C29, C22⋊C116, C23.C58, C58.12D4, (C2×C58)⋊3C4, (C2×C4)⋊1C58, (C2×C116)⋊2C2, C2.1(D4×C29), C58.17(C2×C4), C2.1(C2×C116), (C22×C58).1C2, C22.2(C2×C58), (C2×C58).13C22, SmallGroup(464,21)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4×C29
G = < a,b,c,d | a29=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >
Smallest permutation representation of C22⋊C4×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)
(1 115)(2 116)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 99)(15 100)(16 101)(17 102)(18 103)(19 104)(20 105)(21 106)(22 107)(23 108)(24 109)(25 110)(26 111)(27 112)(28 113)(29 114)(30 186)(31 187)(32 188)(33 189)(34 190)(35 191)(36 192)(37 193)(38 194)(39 195)(40 196)(41 197)(42 198)(43 199)(44 200)(45 201)(46 202)(47 203)(48 175)(49 176)(50 177)(51 178)(52 179)(53 180)(54 181)(55 182)(56 183)(57 184)(58 185)(59 222)(60 223)(61 224)(62 225)(63 226)(64 227)(65 228)(66 229)(67 230)(68 231)(69 232)(70 204)(71 205)(72 206)(73 207)(74 208)(75 209)(76 210)(77 211)(78 212)(79 213)(80 214)(81 215)(82 216)(83 217)(84 218)(85 219)(86 220)(87 221)(117 147)(118 148)(119 149)(120 150)(121 151)(122 152)(123 153)(124 154)(125 155)(126 156)(127 157)(128 158)(129 159)(130 160)(131 161)(132 162)(133 163)(134 164)(135 165)(136 166)(137 167)(138 168)(139 169)(140 170)(141 171)(142 172)(143 173)(144 174)(145 146)
(1 190)(2 191)(3 192)(4 193)(5 194)(6 195)(7 196)(8 197)(9 198)(10 199)(11 200)(12 201)(13 202)(14 203)(15 175)(16 176)(17 177)(18 178)(19 179)(20 180)(21 181)(22 182)(23 183)(24 184)(25 185)(26 186)(27 187)(28 188)(29 189)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 88)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(46 98)(47 99)(48 100)(49 101)(50 102)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 109)(58 110)(59 165)(60 166)(61 167)(62 168)(63 169)(64 170)(65 171)(66 172)(67 173)(68 174)(69 146)(70 147)(71 148)(72 149)(73 150)(74 151)(75 152)(76 153)(77 154)(78 155)(79 156)(80 157)(81 158)(82 159)(83 160)(84 161)(85 162)(86 163)(87 164)(117 204)(118 205)(119 206)(120 207)(121 208)(122 209)(123 210)(124 211)(125 212)(126 213)(127 214)(128 215)(129 216)(130 217)(131 218)(132 219)(133 220)(134 221)(135 222)(136 223)(137 224)(138 225)(139 226)(140 227)(141 228)(142 229)(143 230)(144 231)(145 232)
(1 225 115 168)(2 226 116 169)(3 227 88 170)(4 228 89 171)(5 229 90 172)(6 230 91 173)(7 231 92 174)(8 232 93 146)(9 204 94 147)(10 205 95 148)(11 206 96 149)(12 207 97 150)(13 208 98 151)(14 209 99 152)(15 210 100 153)(16 211 101 154)(17 212 102 155)(18 213 103 156)(19 214 104 157)(20 215 105 158)(21 216 106 159)(22 217 107 160)(23 218 108 161)(24 219 109 162)(25 220 110 163)(26 221 111 164)(27 222 112 165)(28 223 113 166)(29 224 114 167)(30 87 186 134)(31 59 187 135)(32 60 188 136)(33 61 189 137)(34 62 190 138)(35 63 191 139)(36 64 192 140)(37 65 193 141)(38 66 194 142)(39 67 195 143)(40 68 196 144)(41 69 197 145)(42 70 198 117)(43 71 199 118)(44 72 200 119)(45 73 201 120)(46 74 202 121)(47 75 203 122)(48 76 175 123)(49 77 176 124)(50 78 177 125)(51 79 178 126)(52 80 179 127)(53 81 180 128)(54 82 181 129)(55 83 182 130)(56 84 183 131)(57 85 184 132)(58 86 185 133)
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), (1,115)(2,116)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,99)(15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,113)(29,114)(30,186)(31,187)(32,188)(33,189)(34,190)(35,191)(36,192)(37,193)(38,194)(39,195)(40,196)(41,197)(42,198)(43,199)(44,200)(45,201)(46,202)(47,203)(48,175)(49,176)(50,177)(51,178)(52,179)(53,180)(54,181)(55,182)(56,183)(57,184)(58,185)(59,222)(60,223)(61,224)(62,225)(63,226)(64,227)(65,228)(66,229)(67,230)(68,231)(69,232)(70,204)(71,205)(72,206)(73,207)(74,208)(75,209)(76,210)(77,211)(78,212)(79,213)(80,214)(81,215)(82,216)(83,217)(84,218)(85,219)(86,220)(87,221)(117,147)(118,148)(119,149)(120,150)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,146), (1,190)(2,191)(3,192)(4,193)(5,194)(6,195)(7,196)(8,197)(9,198)(10,199)(11,200)(12,201)(13,202)(14,203)(15,175)(16,176)(17,177)(18,178)(19,179)(20,180)(21,181)(22,182)(23,183)(24,184)(25,185)(26,186)(27,187)(28,188)(29,189)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,165)(60,166)(61,167)(62,168)(63,169)(64,170)(65,171)(66,172)(67,173)(68,174)(69,146)(70,147)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,155)(79,156)(80,157)(81,158)(82,159)(83,160)(84,161)(85,162)(86,163)(87,164)(117,204)(118,205)(119,206)(120,207)(121,208)(122,209)(123,210)(124,211)(125,212)(126,213)(127,214)(128,215)(129,216)(130,217)(131,218)(132,219)(133,220)(134,221)(135,222)(136,223)(137,224)(138,225)(139,226)(140,227)(141,228)(142,229)(143,230)(144,231)(145,232), (1,225,115,168)(2,226,116,169)(3,227,88,170)(4,228,89,171)(5,229,90,172)(6,230,91,173)(7,231,92,174)(8,232,93,146)(9,204,94,147)(10,205,95,148)(11,206,96,149)(12,207,97,150)(13,208,98,151)(14,209,99,152)(15,210,100,153)(16,211,101,154)(17,212,102,155)(18,213,103,156)(19,214,104,157)(20,215,105,158)(21,216,106,159)(22,217,107,160)(23,218,108,161)(24,219,109,162)(25,220,110,163)(26,221,111,164)(27,222,112,165)(28,223,113,166)(29,224,114,167)(30,87,186,134)(31,59,187,135)(32,60,188,136)(33,61,189,137)(34,62,190,138)(35,63,191,139)(36,64,192,140)(37,65,193,141)(38,66,194,142)(39,67,195,143)(40,68,196,144)(41,69,197,145)(42,70,198,117)(43,71,199,118)(44,72,200,119)(45,73,201,120)(46,74,202,121)(47,75,203,122)(48,76,175,123)(49,77,176,124)(50,78,177,125)(51,79,178,126)(52,80,179,127)(53,81,180,128)(54,82,181,129)(55,83,182,130)(56,84,183,131)(57,85,184,132)(58,86,185,133)>;
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), (1,115)(2,116)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,99)(15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,113)(29,114)(30,186)(31,187)(32,188)(33,189)(34,190)(35,191)(36,192)(37,193)(38,194)(39,195)(40,196)(41,197)(42,198)(43,199)(44,200)(45,201)(46,202)(47,203)(48,175)(49,176)(50,177)(51,178)(52,179)(53,180)(54,181)(55,182)(56,183)(57,184)(58,185)(59,222)(60,223)(61,224)(62,225)(63,226)(64,227)(65,228)(66,229)(67,230)(68,231)(69,232)(70,204)(71,205)(72,206)(73,207)(74,208)(75,209)(76,210)(77,211)(78,212)(79,213)(80,214)(81,215)(82,216)(83,217)(84,218)(85,219)(86,220)(87,221)(117,147)(118,148)(119,149)(120,150)(121,151)(122,152)(123,153)(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166)(137,167)(138,168)(139,169)(140,170)(141,171)(142,172)(143,173)(144,174)(145,146), (1,190)(2,191)(3,192)(4,193)(5,194)(6,195)(7,196)(8,197)(9,198)(10,199)(11,200)(12,201)(13,202)(14,203)(15,175)(16,176)(17,177)(18,178)(19,179)(20,180)(21,181)(22,182)(23,183)(24,184)(25,185)(26,186)(27,187)(28,188)(29,189)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,88)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,97)(46,98)(47,99)(48,100)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,109)(58,110)(59,165)(60,166)(61,167)(62,168)(63,169)(64,170)(65,171)(66,172)(67,173)(68,174)(69,146)(70,147)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,155)(79,156)(80,157)(81,158)(82,159)(83,160)(84,161)(85,162)(86,163)(87,164)(117,204)(118,205)(119,206)(120,207)(121,208)(122,209)(123,210)(124,211)(125,212)(126,213)(127,214)(128,215)(129,216)(130,217)(131,218)(132,219)(133,220)(134,221)(135,222)(136,223)(137,224)(138,225)(139,226)(140,227)(141,228)(142,229)(143,230)(144,231)(145,232), (1,225,115,168)(2,226,116,169)(3,227,88,170)(4,228,89,171)(5,229,90,172)(6,230,91,173)(7,231,92,174)(8,232,93,146)(9,204,94,147)(10,205,95,148)(11,206,96,149)(12,207,97,150)(13,208,98,151)(14,209,99,152)(15,210,100,153)(16,211,101,154)(17,212,102,155)(18,213,103,156)(19,214,104,157)(20,215,105,158)(21,216,106,159)(22,217,107,160)(23,218,108,161)(24,219,109,162)(25,220,110,163)(26,221,111,164)(27,222,112,165)(28,223,113,166)(29,224,114,167)(30,87,186,134)(31,59,187,135)(32,60,188,136)(33,61,189,137)(34,62,190,138)(35,63,191,139)(36,64,192,140)(37,65,193,141)(38,66,194,142)(39,67,195,143)(40,68,196,144)(41,69,197,145)(42,70,198,117)(43,71,199,118)(44,72,200,119)(45,73,201,120)(46,74,202,121)(47,75,203,122)(48,76,175,123)(49,77,176,124)(50,78,177,125)(51,79,178,126)(52,80,179,127)(53,81,180,128)(54,82,181,129)(55,83,182,130)(56,84,183,131)(57,85,184,132)(58,86,185,133) );
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)], [(1,115),(2,116),(3,88),(4,89),(5,90),(6,91),(7,92),(8,93),(9,94),(10,95),(11,96),(12,97),(13,98),(14,99),(15,100),(16,101),(17,102),(18,103),(19,104),(20,105),(21,106),(22,107),(23,108),(24,109),(25,110),(26,111),(27,112),(28,113),(29,114),(30,186),(31,187),(32,188),(33,189),(34,190),(35,191),(36,192),(37,193),(38,194),(39,195),(40,196),(41,197),(42,198),(43,199),(44,200),(45,201),(46,202),(47,203),(48,175),(49,176),(50,177),(51,178),(52,179),(53,180),(54,181),(55,182),(56,183),(57,184),(58,185),(59,222),(60,223),(61,224),(62,225),(63,226),(64,227),(65,228),(66,229),(67,230),(68,231),(69,232),(70,204),(71,205),(72,206),(73,207),(74,208),(75,209),(76,210),(77,211),(78,212),(79,213),(80,214),(81,215),(82,216),(83,217),(84,218),(85,219),(86,220),(87,221),(117,147),(118,148),(119,149),(120,150),(121,151),(122,152),(123,153),(124,154),(125,155),(126,156),(127,157),(128,158),(129,159),(130,160),(131,161),(132,162),(133,163),(134,164),(135,165),(136,166),(137,167),(138,168),(139,169),(140,170),(141,171),(142,172),(143,173),(144,174),(145,146)], [(1,190),(2,191),(3,192),(4,193),(5,194),(6,195),(7,196),(8,197),(9,198),(10,199),(11,200),(12,201),(13,202),(14,203),(15,175),(16,176),(17,177),(18,178),(19,179),(20,180),(21,181),(22,182),(23,183),(24,184),(25,185),(26,186),(27,187),(28,188),(29,189),(30,111),(31,112),(32,113),(33,114),(34,115),(35,116),(36,88),(37,89),(38,90),(39,91),(40,92),(41,93),(42,94),(43,95),(44,96),(45,97),(46,98),(47,99),(48,100),(49,101),(50,102),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108),(57,109),(58,110),(59,165),(60,166),(61,167),(62,168),(63,169),(64,170),(65,171),(66,172),(67,173),(68,174),(69,146),(70,147),(71,148),(72,149),(73,150),(74,151),(75,152),(76,153),(77,154),(78,155),(79,156),(80,157),(81,158),(82,159),(83,160),(84,161),(85,162),(86,163),(87,164),(117,204),(118,205),(119,206),(120,207),(121,208),(122,209),(123,210),(124,211),(125,212),(126,213),(127,214),(128,215),(129,216),(130,217),(131,218),(132,219),(133,220),(134,221),(135,222),(136,223),(137,224),(138,225),(139,226),(140,227),(141,228),(142,229),(143,230),(144,231),(145,232)], [(1,225,115,168),(2,226,116,169),(3,227,88,170),(4,228,89,171),(5,229,90,172),(6,230,91,173),(7,231,92,174),(8,232,93,146),(9,204,94,147),(10,205,95,148),(11,206,96,149),(12,207,97,150),(13,208,98,151),(14,209,99,152),(15,210,100,153),(16,211,101,154),(17,212,102,155),(18,213,103,156),(19,214,104,157),(20,215,105,158),(21,216,106,159),(22,217,107,160),(23,218,108,161),(24,219,109,162),(25,220,110,163),(26,221,111,164),(27,222,112,165),(28,223,113,166),(29,224,114,167),(30,87,186,134),(31,59,187,135),(32,60,188,136),(33,61,189,137),(34,62,190,138),(35,63,191,139),(36,64,192,140),(37,65,193,141),(38,66,194,142),(39,67,195,143),(40,68,196,144),(41,69,197,145),(42,70,198,117),(43,71,199,118),(44,72,200,119),(45,73,201,120),(46,74,202,121),(47,75,203,122),(48,76,175,123),(49,77,176,124),(50,78,177,125),(51,79,178,126),(52,80,179,127),(53,81,180,128),(54,82,181,129),(55,83,182,130),(56,84,183,131),(57,85,184,132),(58,86,185,133)]])
290 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 29A | ··· | 29AB | 58A | ··· | 58CF | 58CG | ··· | 58EJ | 116A | ··· | 116DH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
290 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C29 | C58 | C58 | C116 | D4 | D4×C29 |
| kernel | C22⋊C4×C29 | C2×C116 | C22×C58 | C2×C58 | C22⋊C4 | C2×C4 | C23 | C22 | C58 | C2 |
| # reps | 1 | 2 | 1 | 4 | 28 | 56 | 28 | 112 | 2 | 56 |
Matrix representation of C22⋊C4×C29 ►in GL3(𝔽233) generated by
| 1 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 0 | 46 |
| 1 | 0 | 0 |
| 0 | 1 | 69 |
| 0 | 0 | 232 |
| 1 | 0 | 0 |
| 0 | 232 | 0 |
| 0 | 0 | 232 |
| 144 | 0 | 0 |
| 0 | 69 | 50 |
| 0 | 231 | 164 |
G:=sub<GL(3,GF(233))| [1,0,0,0,46,0,0,0,46],[1,0,0,0,1,0,0,69,232],[1,0,0,0,232,0,0,0,232],[144,0,0,0,69,231,0,50,164] >;
C22⋊C4×C29 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times C_{29} % in TeX
G:=Group("C2^2:C4xC29"); // GroupNames label
G:=SmallGroup(464,21);
// by ID
G=gap.SmallGroup(464,21);
# by ID
G:=PCGroup([5,-2,-2,-29,-2,-2,1160,1181]);
// Polycyclic
G:=Group<a,b,c,d|a^29=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations
Export