metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.185D14, C8⋊C4⋊9D7, D14⋊C4.8C4, Dic7⋊C8⋊37C2, D14⋊C8.16C2, (C2×C8).158D14, Dic7⋊C4.8C4, C14.23(C8○D4), C2.8(D28.C4), C28.247(C4○D4), C4.131(C4○D28), (C2×C28).815C23, (C2×C56).312C22, (C4×C28).230C22, C42⋊D7.12C2, C2.13(C42⋊D7), C7⋊2(C42.7C22), C14.10(C42⋊C2), (C4×Dic7).181C22, (C4×C7⋊C8)⋊22C2, (C7×C8⋊C4)⋊18C2, (C2×C4).62(C4×D7), C22.100(C2×C4×D7), (C2×C28).149(C2×C4), (C2×C7⋊C8).297C22, (C2×C4×D7).178C22, (C2×C14).70(C22×C4), (C2×Dic7).15(C2×C4), (C22×D7).11(C2×C4), (C2×C4).757(C22×D7), SmallGroup(448,243)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.185D14
G = < a,b,c,d | a4=b4=1, c14=b-1, d2=a2b, ab=ba, cac-1=ab2, ad=da, bc=cb, bd=db, dcd-1=a2b2c13 >
Subgroups: 388 in 96 conjugacy classes, 47 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, D7, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic7, C28, C28, D14, C2×C14, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C42⋊C2, C7⋊C8, C56, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C42.7C22, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, D14⋊C4, C4×C28, C2×C56, C2×C4×D7, C4×C7⋊C8, Dic7⋊C8, D14⋊C8, C7×C8⋊C4, C42⋊D7, C42.185D14
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, D14, C42⋊C2, C8○D4, C4×D7, C22×D7, C42.7C22, C2×C4×D7, C4○D28, C42⋊D7, D28.C4, C42.185D14
►On 224 points
Generators in S224
(1 97 182 118)(2 70 183 147)(3 99 184 120)(4 72 185 149)(5 101 186 122)(6 74 187 151)(7 103 188 124)(8 76 189 153)(9 105 190 126)(10 78 191 155)(11 107 192 128)(12 80 193 157)(13 109 194 130)(14 82 195 159)(15 111 196 132)(16 84 197 161)(17 57 198 134)(18 86 199 163)(19 59 200 136)(20 88 201 165)(21 61 202 138)(22 90 203 167)(23 63 204 140)(24 92 205 113)(25 65 206 142)(26 94 207 115)(27 67 208 144)(28 96 209 117)(29 69 210 146)(30 98 211 119)(31 71 212 148)(32 100 213 121)(33 73 214 150)(34 102 215 123)(35 75 216 152)(36 104 217 125)(37 77 218 154)(38 106 219 127)(39 79 220 156)(40 108 221 129)(41 81 222 158)(42 110 223 131)(43 83 224 160)(44 112 169 133)(45 85 170 162)(46 58 171 135)(47 87 172 164)(48 60 173 137)(49 89 174 166)(50 62 175 139)(51 91 176 168)(52 64 177 141)(53 93 178 114)(54 66 179 143)(55 95 180 116)(56 68 181 145)
(1 43 29 15)(2 44 30 16)(3 45 31 17)(4 46 32 18)(5 47 33 19)(6 48 34 20)(7 49 35 21)(8 50 36 22)(9 51 37 23)(10 52 38 24)(11 53 39 25)(12 54 40 26)(13 55 41 27)(14 56 42 28)(57 99 85 71)(58 100 86 72)(59 101 87 73)(60 102 88 74)(61 103 89 75)(62 104 90 76)(63 105 91 77)(64 106 92 78)(65 107 93 79)(66 108 94 80)(67 109 95 81)(68 110 96 82)(69 111 97 83)(70 112 98 84)(113 155 141 127)(114 156 142 128)(115 157 143 129)(116 158 144 130)(117 159 145 131)(118 160 146 132)(119 161 147 133)(120 162 148 134)(121 163 149 135)(122 164 150 136)(123 165 151 137)(124 166 152 138)(125 167 153 139)(126 168 154 140)(169 211 197 183)(170 212 198 184)(171 213 199 185)(172 214 200 186)(173 215 201 187)(174 216 202 188)(175 217 203 189)(176 218 204 190)(177 219 205 191)(178 220 206 192)(179 221 207 193)(180 222 208 194)(181 223 209 195)(182 224 210 196)
(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)
(1 42 224 209 29 14 196 181)(2 208 169 13 30 180 197 41)(3 12 170 179 31 40 198 207)(4 178 171 39 32 206 199 11)(5 38 172 205 33 10 200 177)(6 204 173 9 34 176 201 37)(7 8 174 175 35 36 202 203)(15 56 182 223 43 28 210 195)(16 222 183 27 44 194 211 55)(17 26 184 193 45 54 212 221)(18 192 185 53 46 220 213 25)(19 52 186 219 47 24 214 191)(20 218 187 23 48 190 215 51)(21 22 188 189 49 50 216 217)(57 94 120 157 85 66 148 129)(58 156 121 65 86 128 149 93)(59 64 122 127 87 92 150 155)(60 126 123 91 88 154 151 63)(61 90 124 153 89 62 152 125)(67 112 130 119 95 84 158 147)(68 118 131 83 96 146 159 111)(69 82 132 145 97 110 160 117)(70 144 133 109 98 116 161 81)(71 108 134 115 99 80 162 143)(72 114 135 79 100 142 163 107)(73 78 136 141 101 106 164 113)(74 140 137 105 102 168 165 77)(75 104 138 167 103 76 166 139)
G:=sub<Sym(224)| (1,97,182,118)(2,70,183,147)(3,99,184,120)(4,72,185,149)(5,101,186,122)(6,74,187,151)(7,103,188,124)(8,76,189,153)(9,105,190,126)(10,78,191,155)(11,107,192,128)(12,80,193,157)(13,109,194,130)(14,82,195,159)(15,111,196,132)(16,84,197,161)(17,57,198,134)(18,86,199,163)(19,59,200,136)(20,88,201,165)(21,61,202,138)(22,90,203,167)(23,63,204,140)(24,92,205,113)(25,65,206,142)(26,94,207,115)(27,67,208,144)(28,96,209,117)(29,69,210,146)(30,98,211,119)(31,71,212,148)(32,100,213,121)(33,73,214,150)(34,102,215,123)(35,75,216,152)(36,104,217,125)(37,77,218,154)(38,106,219,127)(39,79,220,156)(40,108,221,129)(41,81,222,158)(42,110,223,131)(43,83,224,160)(44,112,169,133)(45,85,170,162)(46,58,171,135)(47,87,172,164)(48,60,173,137)(49,89,174,166)(50,62,175,139)(51,91,176,168)(52,64,177,141)(53,93,178,114)(54,66,179,143)(55,95,180,116)(56,68,181,145), (1,43,29,15)(2,44,30,16)(3,45,31,17)(4,46,32,18)(5,47,33,19)(6,48,34,20)(7,49,35,21)(8,50,36,22)(9,51,37,23)(10,52,38,24)(11,53,39,25)(12,54,40,26)(13,55,41,27)(14,56,42,28)(57,99,85,71)(58,100,86,72)(59,101,87,73)(60,102,88,74)(61,103,89,75)(62,104,90,76)(63,105,91,77)(64,106,92,78)(65,107,93,79)(66,108,94,80)(67,109,95,81)(68,110,96,82)(69,111,97,83)(70,112,98,84)(113,155,141,127)(114,156,142,128)(115,157,143,129)(116,158,144,130)(117,159,145,131)(118,160,146,132)(119,161,147,133)(120,162,148,134)(121,163,149,135)(122,164,150,136)(123,165,151,137)(124,166,152,138)(125,167,153,139)(126,168,154,140)(169,211,197,183)(170,212,198,184)(171,213,199,185)(172,214,200,186)(173,215,201,187)(174,216,202,188)(175,217,203,189)(176,218,204,190)(177,219,205,191)(178,220,206,192)(179,221,207,193)(180,222,208,194)(181,223,209,195)(182,224,210,196), (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), (1,42,224,209,29,14,196,181)(2,208,169,13,30,180,197,41)(3,12,170,179,31,40,198,207)(4,178,171,39,32,206,199,11)(5,38,172,205,33,10,200,177)(6,204,173,9,34,176,201,37)(7,8,174,175,35,36,202,203)(15,56,182,223,43,28,210,195)(16,222,183,27,44,194,211,55)(17,26,184,193,45,54,212,221)(18,192,185,53,46,220,213,25)(19,52,186,219,47,24,214,191)(20,218,187,23,48,190,215,51)(21,22,188,189,49,50,216,217)(57,94,120,157,85,66,148,129)(58,156,121,65,86,128,149,93)(59,64,122,127,87,92,150,155)(60,126,123,91,88,154,151,63)(61,90,124,153,89,62,152,125)(67,112,130,119,95,84,158,147)(68,118,131,83,96,146,159,111)(69,82,132,145,97,110,160,117)(70,144,133,109,98,116,161,81)(71,108,134,115,99,80,162,143)(72,114,135,79,100,142,163,107)(73,78,136,141,101,106,164,113)(74,140,137,105,102,168,165,77)(75,104,138,167,103,76,166,139)>;
G:=Group( (1,97,182,118)(2,70,183,147)(3,99,184,120)(4,72,185,149)(5,101,186,122)(6,74,187,151)(7,103,188,124)(8,76,189,153)(9,105,190,126)(10,78,191,155)(11,107,192,128)(12,80,193,157)(13,109,194,130)(14,82,195,159)(15,111,196,132)(16,84,197,161)(17,57,198,134)(18,86,199,163)(19,59,200,136)(20,88,201,165)(21,61,202,138)(22,90,203,167)(23,63,204,140)(24,92,205,113)(25,65,206,142)(26,94,207,115)(27,67,208,144)(28,96,209,117)(29,69,210,146)(30,98,211,119)(31,71,212,148)(32,100,213,121)(33,73,214,150)(34,102,215,123)(35,75,216,152)(36,104,217,125)(37,77,218,154)(38,106,219,127)(39,79,220,156)(40,108,221,129)(41,81,222,158)(42,110,223,131)(43,83,224,160)(44,112,169,133)(45,85,170,162)(46,58,171,135)(47,87,172,164)(48,60,173,137)(49,89,174,166)(50,62,175,139)(51,91,176,168)(52,64,177,141)(53,93,178,114)(54,66,179,143)(55,95,180,116)(56,68,181,145), (1,43,29,15)(2,44,30,16)(3,45,31,17)(4,46,32,18)(5,47,33,19)(6,48,34,20)(7,49,35,21)(8,50,36,22)(9,51,37,23)(10,52,38,24)(11,53,39,25)(12,54,40,26)(13,55,41,27)(14,56,42,28)(57,99,85,71)(58,100,86,72)(59,101,87,73)(60,102,88,74)(61,103,89,75)(62,104,90,76)(63,105,91,77)(64,106,92,78)(65,107,93,79)(66,108,94,80)(67,109,95,81)(68,110,96,82)(69,111,97,83)(70,112,98,84)(113,155,141,127)(114,156,142,128)(115,157,143,129)(116,158,144,130)(117,159,145,131)(118,160,146,132)(119,161,147,133)(120,162,148,134)(121,163,149,135)(122,164,150,136)(123,165,151,137)(124,166,152,138)(125,167,153,139)(126,168,154,140)(169,211,197,183)(170,212,198,184)(171,213,199,185)(172,214,200,186)(173,215,201,187)(174,216,202,188)(175,217,203,189)(176,218,204,190)(177,219,205,191)(178,220,206,192)(179,221,207,193)(180,222,208,194)(181,223,209,195)(182,224,210,196), (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), (1,42,224,209,29,14,196,181)(2,208,169,13,30,180,197,41)(3,12,170,179,31,40,198,207)(4,178,171,39,32,206,199,11)(5,38,172,205,33,10,200,177)(6,204,173,9,34,176,201,37)(7,8,174,175,35,36,202,203)(15,56,182,223,43,28,210,195)(16,222,183,27,44,194,211,55)(17,26,184,193,45,54,212,221)(18,192,185,53,46,220,213,25)(19,52,186,219,47,24,214,191)(20,218,187,23,48,190,215,51)(21,22,188,189,49,50,216,217)(57,94,120,157,85,66,148,129)(58,156,121,65,86,128,149,93)(59,64,122,127,87,92,150,155)(60,126,123,91,88,154,151,63)(61,90,124,153,89,62,152,125)(67,112,130,119,95,84,158,147)(68,118,131,83,96,146,159,111)(69,82,132,145,97,110,160,117)(70,144,133,109,98,116,161,81)(71,108,134,115,99,80,162,143)(72,114,135,79,100,142,163,107)(73,78,136,141,101,106,164,113)(74,140,137,105,102,168,165,77)(75,104,138,167,103,76,166,139) );
G=PermutationGroup([[(1,97,182,118),(2,70,183,147),(3,99,184,120),(4,72,185,149),(5,101,186,122),(6,74,187,151),(7,103,188,124),(8,76,189,153),(9,105,190,126),(10,78,191,155),(11,107,192,128),(12,80,193,157),(13,109,194,130),(14,82,195,159),(15,111,196,132),(16,84,197,161),(17,57,198,134),(18,86,199,163),(19,59,200,136),(20,88,201,165),(21,61,202,138),(22,90,203,167),(23,63,204,140),(24,92,205,113),(25,65,206,142),(26,94,207,115),(27,67,208,144),(28,96,209,117),(29,69,210,146),(30,98,211,119),(31,71,212,148),(32,100,213,121),(33,73,214,150),(34,102,215,123),(35,75,216,152),(36,104,217,125),(37,77,218,154),(38,106,219,127),(39,79,220,156),(40,108,221,129),(41,81,222,158),(42,110,223,131),(43,83,224,160),(44,112,169,133),(45,85,170,162),(46,58,171,135),(47,87,172,164),(48,60,173,137),(49,89,174,166),(50,62,175,139),(51,91,176,168),(52,64,177,141),(53,93,178,114),(54,66,179,143),(55,95,180,116),(56,68,181,145)], [(1,43,29,15),(2,44,30,16),(3,45,31,17),(4,46,32,18),(5,47,33,19),(6,48,34,20),(7,49,35,21),(8,50,36,22),(9,51,37,23),(10,52,38,24),(11,53,39,25),(12,54,40,26),(13,55,41,27),(14,56,42,28),(57,99,85,71),(58,100,86,72),(59,101,87,73),(60,102,88,74),(61,103,89,75),(62,104,90,76),(63,105,91,77),(64,106,92,78),(65,107,93,79),(66,108,94,80),(67,109,95,81),(68,110,96,82),(69,111,97,83),(70,112,98,84),(113,155,141,127),(114,156,142,128),(115,157,143,129),(116,158,144,130),(117,159,145,131),(118,160,146,132),(119,161,147,133),(120,162,148,134),(121,163,149,135),(122,164,150,136),(123,165,151,137),(124,166,152,138),(125,167,153,139),(126,168,154,140),(169,211,197,183),(170,212,198,184),(171,213,199,185),(172,214,200,186),(173,215,201,187),(174,216,202,188),(175,217,203,189),(176,218,204,190),(177,219,205,191),(178,220,206,192),(179,221,207,193),(180,222,208,194),(181,223,209,195),(182,224,210,196)], [(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)], [(1,42,224,209,29,14,196,181),(2,208,169,13,30,180,197,41),(3,12,170,179,31,40,198,207),(4,178,171,39,32,206,199,11),(5,38,172,205,33,10,200,177),(6,204,173,9,34,176,201,37),(7,8,174,175,35,36,202,203),(15,56,182,223,43,28,210,195),(16,222,183,27,44,194,211,55),(17,26,184,193,45,54,212,221),(18,192,185,53,46,220,213,25),(19,52,186,219,47,24,214,191),(20,218,187,23,48,190,215,51),(21,22,188,189,49,50,216,217),(57,94,120,157,85,66,148,129),(58,156,121,65,86,128,149,93),(59,64,122,127,87,92,150,155),(60,126,123,91,88,154,151,63),(61,90,124,153,89,62,152,125),(67,112,130,119,95,84,158,147),(68,118,131,83,96,146,159,111),(69,82,132,145,97,110,160,117),(70,144,133,109,98,116,161,81),(71,108,134,115,99,80,162,143),(72,114,135,79,100,142,163,107),(73,78,136,141,101,106,164,113),(74,140,137,105,102,168,165,77),(75,104,138,167,103,76,166,139)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D7 | C4○D4 | D14 | D14 | C8○D4 | C4×D7 | C4○D28 | D28.C4 |
| kernel | C42.185D14 | C4×C7⋊C8 | Dic7⋊C8 | D14⋊C8 | C7×C8⋊C4 | C42⋊D7 | Dic7⋊C4 | D14⋊C4 | C8⋊C4 | C28 | C42 | C2×C8 | C14 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 3 | 4 | 3 | 6 | 8 | 12 | 24 | 12 |
Matrix representation of C42.185D14 ►in GL4(𝔽113) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 69 | 49 |
| 0 | 0 | 32 | 44 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 81 | 109 | 0 | 0 |
| 4 | 4 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 60 | 95 |
| 75 | 55 | 0 | 0 |
| 58 | 38 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
G:=sub<GL(4,GF(113))| [15,0,0,0,0,15,0,0,0,0,69,32,0,0,49,44],[112,0,0,0,0,112,0,0,0,0,98,0,0,0,0,98],[81,4,0,0,109,4,0,0,0,0,18,60,0,0,0,95],[75,58,0,0,55,38,0,0,0,0,18,0,0,0,0,18] >;
C42.185D14 in GAP, Magma, Sage, TeX
C_4^2._{185}D_{14} % in TeX
G:=Group("C4^2.185D14"); // GroupNames label
G:=SmallGroup(448,243);
// by ID
G=gap.SmallGroup(448,243);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,120,422,387,58,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^14=b^-1,d^2=a^2*b,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c^13>;
// generators/relations