metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q32⋊2D7, D14.9D8, C16.3D14, Q16.4D14, C56.21C23, Dic7.11D8, D56.4C22, C112.10C22, Dic28.6C22, C7⋊C8.5D4, C4.9(D4×D7), (D7×Q16)⋊5C2, (C7×Q32)⋊4C2, C7⋊Q32⋊4C2, C2.24(D7×D8), C16⋊D7⋊4C2, C112⋊C2⋊4C2, Q8.D14.C2, (C4×D7).10D4, C14.40(C2×D8), C28.15(C2×D4), C7⋊SD32⋊3C2, C7⋊3(Q32⋊C2), C7⋊C16.2C22, (C8×D7).6C22, C8.27(C22×D7), (C7×Q16).5C22, SmallGroup(448,452)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q32⋊D7
G = < a,b,c,d | a16=c7=d2=1, b2=a8, bab-1=a-1, ac=ca, dad=a9, bc=cb, dbd=a8b, dcd=c-1 >
Subgroups: 528 in 82 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C8, C2×C4, D4, Q8, D7, C14, C16, C16, C2×C8, D8, SD16, Q16, Q16, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, M5(2), SD32, Q32, Q32, C2×Q16, C4○D8, C7⋊C8, C56, Dic14, C4×D7, C4×D7, D28, C7×Q8, Q32⋊C2, C7⋊C16, C112, C8×D7, D56, Dic28, Q8⋊D7, C7⋊Q16, C7×Q16, Q8×D7, Q8⋊2D7, C16⋊D7, C112⋊C2, C7⋊SD32, C7⋊Q32, C7×Q32, D7×Q16, Q8.D14, Q32⋊D7
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, C22×D7, Q32⋊C2, D4×D7, D7×D8, Q32⋊D7
►On 224 points
Generators in S224
(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 84 9 92)(2 83 10 91)(3 82 11 90)(4 81 12 89)(5 96 13 88)(6 95 14 87)(7 94 15 86)(8 93 16 85)(17 69 25 77)(18 68 26 76)(19 67 27 75)(20 66 28 74)(21 65 29 73)(22 80 30 72)(23 79 31 71)(24 78 32 70)(33 201 41 193)(34 200 42 208)(35 199 43 207)(36 198 44 206)(37 197 45 205)(38 196 46 204)(39 195 47 203)(40 194 48 202)(49 117 57 125)(50 116 58 124)(51 115 59 123)(52 114 60 122)(53 113 61 121)(54 128 62 120)(55 127 63 119)(56 126 64 118)(97 215 105 223)(98 214 106 222)(99 213 107 221)(100 212 108 220)(101 211 109 219)(102 210 110 218)(103 209 111 217)(104 224 112 216)(129 152 137 160)(130 151 138 159)(131 150 139 158)(132 149 140 157)(133 148 141 156)(134 147 142 155)(135 146 143 154)(136 145 144 153)(161 185 169 177)(162 184 170 192)(163 183 171 191)(164 182 172 190)(165 181 173 189)(166 180 174 188)(167 179 175 187)(168 178 176 186)
(1 44 148 172 119 111 79)(2 45 149 173 120 112 80)(3 46 150 174 121 97 65)(4 47 151 175 122 98 66)(5 48 152 176 123 99 67)(6 33 153 161 124 100 68)(7 34 154 162 125 101 69)(8 35 155 163 126 102 70)(9 36 156 164 127 103 71)(10 37 157 165 128 104 72)(11 38 158 166 113 105 73)(12 39 159 167 114 106 74)(13 40 160 168 115 107 75)(14 41 145 169 116 108 76)(15 42 146 170 117 109 77)(16 43 147 171 118 110 78)(17 86 208 143 192 57 219)(18 87 193 144 177 58 220)(19 88 194 129 178 59 221)(20 89 195 130 179 60 222)(21 90 196 131 180 61 223)(22 91 197 132 181 62 224)(23 92 198 133 182 63 209)(24 93 199 134 183 64 210)(25 94 200 135 184 49 211)(26 95 201 136 185 50 212)(27 96 202 137 186 51 213)(28 81 203 138 187 52 214)(29 82 204 139 188 53 215)(30 83 205 140 189 54 216)(31 84 206 141 190 55 217)(32 85 207 142 191 56 218)
(1 79)(2 72)(3 65)(4 74)(5 67)(6 76)(7 69)(8 78)(9 71)(10 80)(11 73)(12 66)(13 75)(14 68)(15 77)(16 70)(17 94)(18 87)(19 96)(20 89)(21 82)(22 91)(23 84)(24 93)(25 86)(26 95)(27 88)(28 81)(29 90)(30 83)(31 92)(32 85)(33 108)(34 101)(35 110)(36 103)(37 112)(38 105)(39 98)(40 107)(41 100)(42 109)(43 102)(44 111)(45 104)(46 97)(47 106)(48 99)(49 143)(50 136)(51 129)(52 138)(53 131)(54 140)(55 133)(56 142)(57 135)(58 144)(59 137)(60 130)(61 139)(62 132)(63 141)(64 134)(113 158)(114 151)(115 160)(116 153)(117 146)(118 155)(119 148)(120 157)(121 150)(122 159)(123 152)(124 145)(125 154)(126 147)(127 156)(128 149)(161 169)(163 171)(165 173)(167 175)(178 186)(180 188)(182 190)(184 192)(193 220)(194 213)(195 222)(196 215)(197 224)(198 217)(199 210)(200 219)(201 212)(202 221)(203 214)(204 223)(205 216)(206 209)(207 218)(208 211)
G:=sub<Sym(224)| (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), (1,84,9,92)(2,83,10,91)(3,82,11,90)(4,81,12,89)(5,96,13,88)(6,95,14,87)(7,94,15,86)(8,93,16,85)(17,69,25,77)(18,68,26,76)(19,67,27,75)(20,66,28,74)(21,65,29,73)(22,80,30,72)(23,79,31,71)(24,78,32,70)(33,201,41,193)(34,200,42,208)(35,199,43,207)(36,198,44,206)(37,197,45,205)(38,196,46,204)(39,195,47,203)(40,194,48,202)(49,117,57,125)(50,116,58,124)(51,115,59,123)(52,114,60,122)(53,113,61,121)(54,128,62,120)(55,127,63,119)(56,126,64,118)(97,215,105,223)(98,214,106,222)(99,213,107,221)(100,212,108,220)(101,211,109,219)(102,210,110,218)(103,209,111,217)(104,224,112,216)(129,152,137,160)(130,151,138,159)(131,150,139,158)(132,149,140,157)(133,148,141,156)(134,147,142,155)(135,146,143,154)(136,145,144,153)(161,185,169,177)(162,184,170,192)(163,183,171,191)(164,182,172,190)(165,181,173,189)(166,180,174,188)(167,179,175,187)(168,178,176,186), (1,44,148,172,119,111,79)(2,45,149,173,120,112,80)(3,46,150,174,121,97,65)(4,47,151,175,122,98,66)(5,48,152,176,123,99,67)(6,33,153,161,124,100,68)(7,34,154,162,125,101,69)(8,35,155,163,126,102,70)(9,36,156,164,127,103,71)(10,37,157,165,128,104,72)(11,38,158,166,113,105,73)(12,39,159,167,114,106,74)(13,40,160,168,115,107,75)(14,41,145,169,116,108,76)(15,42,146,170,117,109,77)(16,43,147,171,118,110,78)(17,86,208,143,192,57,219)(18,87,193,144,177,58,220)(19,88,194,129,178,59,221)(20,89,195,130,179,60,222)(21,90,196,131,180,61,223)(22,91,197,132,181,62,224)(23,92,198,133,182,63,209)(24,93,199,134,183,64,210)(25,94,200,135,184,49,211)(26,95,201,136,185,50,212)(27,96,202,137,186,51,213)(28,81,203,138,187,52,214)(29,82,204,139,188,53,215)(30,83,205,140,189,54,216)(31,84,206,141,190,55,217)(32,85,207,142,191,56,218), (1,79)(2,72)(3,65)(4,74)(5,67)(6,76)(7,69)(8,78)(9,71)(10,80)(11,73)(12,66)(13,75)(14,68)(15,77)(16,70)(17,94)(18,87)(19,96)(20,89)(21,82)(22,91)(23,84)(24,93)(25,86)(26,95)(27,88)(28,81)(29,90)(30,83)(31,92)(32,85)(33,108)(34,101)(35,110)(36,103)(37,112)(38,105)(39,98)(40,107)(41,100)(42,109)(43,102)(44,111)(45,104)(46,97)(47,106)(48,99)(49,143)(50,136)(51,129)(52,138)(53,131)(54,140)(55,133)(56,142)(57,135)(58,144)(59,137)(60,130)(61,139)(62,132)(63,141)(64,134)(113,158)(114,151)(115,160)(116,153)(117,146)(118,155)(119,148)(120,157)(121,150)(122,159)(123,152)(124,145)(125,154)(126,147)(127,156)(128,149)(161,169)(163,171)(165,173)(167,175)(178,186)(180,188)(182,190)(184,192)(193,220)(194,213)(195,222)(196,215)(197,224)(198,217)(199,210)(200,219)(201,212)(202,221)(203,214)(204,223)(205,216)(206,209)(207,218)(208,211)>;
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), (1,84,9,92)(2,83,10,91)(3,82,11,90)(4,81,12,89)(5,96,13,88)(6,95,14,87)(7,94,15,86)(8,93,16,85)(17,69,25,77)(18,68,26,76)(19,67,27,75)(20,66,28,74)(21,65,29,73)(22,80,30,72)(23,79,31,71)(24,78,32,70)(33,201,41,193)(34,200,42,208)(35,199,43,207)(36,198,44,206)(37,197,45,205)(38,196,46,204)(39,195,47,203)(40,194,48,202)(49,117,57,125)(50,116,58,124)(51,115,59,123)(52,114,60,122)(53,113,61,121)(54,128,62,120)(55,127,63,119)(56,126,64,118)(97,215,105,223)(98,214,106,222)(99,213,107,221)(100,212,108,220)(101,211,109,219)(102,210,110,218)(103,209,111,217)(104,224,112,216)(129,152,137,160)(130,151,138,159)(131,150,139,158)(132,149,140,157)(133,148,141,156)(134,147,142,155)(135,146,143,154)(136,145,144,153)(161,185,169,177)(162,184,170,192)(163,183,171,191)(164,182,172,190)(165,181,173,189)(166,180,174,188)(167,179,175,187)(168,178,176,186), (1,44,148,172,119,111,79)(2,45,149,173,120,112,80)(3,46,150,174,121,97,65)(4,47,151,175,122,98,66)(5,48,152,176,123,99,67)(6,33,153,161,124,100,68)(7,34,154,162,125,101,69)(8,35,155,163,126,102,70)(9,36,156,164,127,103,71)(10,37,157,165,128,104,72)(11,38,158,166,113,105,73)(12,39,159,167,114,106,74)(13,40,160,168,115,107,75)(14,41,145,169,116,108,76)(15,42,146,170,117,109,77)(16,43,147,171,118,110,78)(17,86,208,143,192,57,219)(18,87,193,144,177,58,220)(19,88,194,129,178,59,221)(20,89,195,130,179,60,222)(21,90,196,131,180,61,223)(22,91,197,132,181,62,224)(23,92,198,133,182,63,209)(24,93,199,134,183,64,210)(25,94,200,135,184,49,211)(26,95,201,136,185,50,212)(27,96,202,137,186,51,213)(28,81,203,138,187,52,214)(29,82,204,139,188,53,215)(30,83,205,140,189,54,216)(31,84,206,141,190,55,217)(32,85,207,142,191,56,218), (1,79)(2,72)(3,65)(4,74)(5,67)(6,76)(7,69)(8,78)(9,71)(10,80)(11,73)(12,66)(13,75)(14,68)(15,77)(16,70)(17,94)(18,87)(19,96)(20,89)(21,82)(22,91)(23,84)(24,93)(25,86)(26,95)(27,88)(28,81)(29,90)(30,83)(31,92)(32,85)(33,108)(34,101)(35,110)(36,103)(37,112)(38,105)(39,98)(40,107)(41,100)(42,109)(43,102)(44,111)(45,104)(46,97)(47,106)(48,99)(49,143)(50,136)(51,129)(52,138)(53,131)(54,140)(55,133)(56,142)(57,135)(58,144)(59,137)(60,130)(61,139)(62,132)(63,141)(64,134)(113,158)(114,151)(115,160)(116,153)(117,146)(118,155)(119,148)(120,157)(121,150)(122,159)(123,152)(124,145)(125,154)(126,147)(127,156)(128,149)(161,169)(163,171)(165,173)(167,175)(178,186)(180,188)(182,190)(184,192)(193,220)(194,213)(195,222)(196,215)(197,224)(198,217)(199,210)(200,219)(201,212)(202,221)(203,214)(204,223)(205,216)(206,209)(207,218)(208,211) );
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)], [(1,84,9,92),(2,83,10,91),(3,82,11,90),(4,81,12,89),(5,96,13,88),(6,95,14,87),(7,94,15,86),(8,93,16,85),(17,69,25,77),(18,68,26,76),(19,67,27,75),(20,66,28,74),(21,65,29,73),(22,80,30,72),(23,79,31,71),(24,78,32,70),(33,201,41,193),(34,200,42,208),(35,199,43,207),(36,198,44,206),(37,197,45,205),(38,196,46,204),(39,195,47,203),(40,194,48,202),(49,117,57,125),(50,116,58,124),(51,115,59,123),(52,114,60,122),(53,113,61,121),(54,128,62,120),(55,127,63,119),(56,126,64,118),(97,215,105,223),(98,214,106,222),(99,213,107,221),(100,212,108,220),(101,211,109,219),(102,210,110,218),(103,209,111,217),(104,224,112,216),(129,152,137,160),(130,151,138,159),(131,150,139,158),(132,149,140,157),(133,148,141,156),(134,147,142,155),(135,146,143,154),(136,145,144,153),(161,185,169,177),(162,184,170,192),(163,183,171,191),(164,182,172,190),(165,181,173,189),(166,180,174,188),(167,179,175,187),(168,178,176,186)], [(1,44,148,172,119,111,79),(2,45,149,173,120,112,80),(3,46,150,174,121,97,65),(4,47,151,175,122,98,66),(5,48,152,176,123,99,67),(6,33,153,161,124,100,68),(7,34,154,162,125,101,69),(8,35,155,163,126,102,70),(9,36,156,164,127,103,71),(10,37,157,165,128,104,72),(11,38,158,166,113,105,73),(12,39,159,167,114,106,74),(13,40,160,168,115,107,75),(14,41,145,169,116,108,76),(15,42,146,170,117,109,77),(16,43,147,171,118,110,78),(17,86,208,143,192,57,219),(18,87,193,144,177,58,220),(19,88,194,129,178,59,221),(20,89,195,130,179,60,222),(21,90,196,131,180,61,223),(22,91,197,132,181,62,224),(23,92,198,133,182,63,209),(24,93,199,134,183,64,210),(25,94,200,135,184,49,211),(26,95,201,136,185,50,212),(27,96,202,137,186,51,213),(28,81,203,138,187,52,214),(29,82,204,139,188,53,215),(30,83,205,140,189,54,216),(31,84,206,141,190,55,217),(32,85,207,142,191,56,218)], [(1,79),(2,72),(3,65),(4,74),(5,67),(6,76),(7,69),(8,78),(9,71),(10,80),(11,73),(12,66),(13,75),(14,68),(15,77),(16,70),(17,94),(18,87),(19,96),(20,89),(21,82),(22,91),(23,84),(24,93),(25,86),(26,95),(27,88),(28,81),(29,90),(30,83),(31,92),(32,85),(33,108),(34,101),(35,110),(36,103),(37,112),(38,105),(39,98),(40,107),(41,100),(42,109),(43,102),(44,111),(45,104),(46,97),(47,106),(48,99),(49,143),(50,136),(51,129),(52,138),(53,131),(54,140),(55,133),(56,142),(57,135),(58,144),(59,137),(60,130),(61,139),(62,132),(63,141),(64,134),(113,158),(114,151),(115,160),(116,153),(117,146),(118,155),(119,148),(120,157),(121,150),(122,159),(123,152),(124,145),(125,154),(126,147),(127,156),(128,149),(161,169),(163,171),(165,173),(167,175),(178,186),(180,188),(182,190),(184,192),(193,220),(194,213),(195,222),(196,215),(197,224),(198,217),(199,210),(200,219),(201,212),(202,221),(203,214),(204,223),(205,216),(206,209),(207,218),(208,211)]])
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 14A | 14B | 14C | 16A | 16B | 16C | 16D | 28A | 28B | 28C | 28D | ··· | 28I | 56A | ··· | 56F | 112A | ··· | 112L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | 28 | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 14 | 56 | 2 | 8 | 8 | 14 | 56 | 2 | 2 | 2 | 2 | 2 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 4 | 4 | 4 | 16 | ··· | 16 | 4 | ··· | 4 | 4 | ··· | 4 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | D8 | D14 | D14 | Q32⋊C2 | D4×D7 | D7×D8 | Q32⋊D7 |
| kernel | Q32⋊D7 | C16⋊D7 | C112⋊C2 | C7⋊SD32 | C7⋊Q32 | C7×Q32 | D7×Q16 | Q8.D14 | C7⋊C8 | C4×D7 | Q32 | Dic7 | D14 | C16 | Q16 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 2 | 3 | 6 | 12 |
Matrix representation of Q32⋊D7 ►in GL4(𝔽113) generated by
| 110 | 85 | 43 | 100 |
| 28 | 3 | 13 | 70 |
| 70 | 13 | 110 | 85 |
| 100 | 43 | 28 | 3 |
| 74 | 88 | 89 | 2 |
| 25 | 39 | 111 | 24 |
| 89 | 2 | 39 | 25 |
| 111 | 24 | 88 | 74 |
| 0 | 1 | 0 | 0 |
| 112 | 24 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 24 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(113))| [110,28,70,100,85,3,13,43,43,13,110,28,100,70,85,3],[74,25,89,111,88,39,2,24,89,111,39,88,2,24,25,74],[0,112,0,0,1,24,0,0,0,0,0,112,0,0,1,24],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
Q32⋊D7 in GAP, Magma, Sage, TeX
Q_{32}\rtimes D_7 % in TeX
G:=Group("Q32:D7"); // GroupNames label
G:=SmallGroup(448,452);
// by ID
G=gap.SmallGroup(448,452);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,135,184,346,185,192,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^16=c^7=d^2=1,b^2=a^8,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^9,b*c=c*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations