direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C7×C4○D16, D16⋊3C14, Q32⋊3C14, C56.76D4, C28.71D8, SD32⋊3C14, C56.75C23, C112.23C22, (C2×C16)⋊6C14, C4○D8⋊1C14, (C7×D16)⋊7C2, (C7×Q32)⋊7C2, C8.13(C7×D4), C4.20(C7×D8), (C2×C112)⋊13C2, C16.6(C2×C14), (C7×SD32)⋊7C2, D8.2(C2×C14), (C2×C14).12D8, C4.10(D4×C14), C2.15(C14×D8), C14.87(C2×D8), C22.1(C7×D8), C28.317(C2×D4), (C2×C28).429D4, C8.6(C22×C14), Q16.2(C2×C14), (C7×D8).12C22, (C2×C56).434C22, (C7×Q16).14C22, (C7×C4○D8)⋊8C2, (C2×C4).85(C7×D4), (C2×C8).91(C2×C14), SmallGroup(448,916)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C4○D16
G = < a,b,c,d | a7=b4=d2=1, c8=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c7 >
Subgroups: 178 in 84 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C14, C14, C16, C2×C8, D8, SD16, Q16, C4○D4, C28, C28, C2×C14, C2×C14, C2×C16, D16, SD32, Q32, C4○D8, C56, C2×C28, C2×C28, C7×D4, C7×Q8, C4○D16, C112, C2×C56, C7×D8, C7×SD16, C7×Q16, C7×C4○D4, C2×C112, C7×D16, C7×SD32, C7×Q32, C7×C4○D8, C7×C4○D16
Quotients: C1, C2, C22, C7, D4, C23, C14, D8, C2×D4, C2×C14, C2×D8, C7×D4, C22×C14, C4○D16, C7×D8, D4×C14, C14×D8, C7×C4○D16
►On 224 points
Generators in S224
(1 195 185 89 112 137 119)(2 196 186 90 97 138 120)(3 197 187 91 98 139 121)(4 198 188 92 99 140 122)(5 199 189 93 100 141 123)(6 200 190 94 101 142 124)(7 201 191 95 102 143 125)(8 202 192 96 103 144 126)(9 203 177 81 104 129 127)(10 204 178 82 105 130 128)(11 205 179 83 106 131 113)(12 206 180 84 107 132 114)(13 207 181 85 108 133 115)(14 208 182 86 109 134 116)(15 193 183 87 110 135 117)(16 194 184 88 111 136 118)(17 80 172 53 213 150 40)(18 65 173 54 214 151 41)(19 66 174 55 215 152 42)(20 67 175 56 216 153 43)(21 68 176 57 217 154 44)(22 69 161 58 218 155 45)(23 70 162 59 219 156 46)(24 71 163 60 220 157 47)(25 72 164 61 221 158 48)(26 73 165 62 222 159 33)(27 74 166 63 223 160 34)(28 75 167 64 224 145 35)(29 76 168 49 209 146 36)(30 77 169 50 210 147 37)(31 78 170 51 211 148 38)(32 79 171 52 212 149 39)
(1 13 9 5)(2 14 10 6)(3 15 11 7)(4 16 12 8)(17 21 25 29)(18 22 26 30)(19 23 27 31)(20 24 28 32)(33 37 41 45)(34 38 42 46)(35 39 43 47)(36 40 44 48)(49 53 57 61)(50 54 58 62)(51 55 59 63)(52 56 60 64)(65 69 73 77)(66 70 74 78)(67 71 75 79)(68 72 76 80)(81 93 89 85)(82 94 90 86)(83 95 91 87)(84 96 92 88)(97 109 105 101)(98 110 106 102)(99 111 107 103)(100 112 108 104)(113 125 121 117)(114 126 122 118)(115 127 123 119)(116 128 124 120)(129 141 137 133)(130 142 138 134)(131 143 139 135)(132 144 140 136)(145 149 153 157)(146 150 154 158)(147 151 155 159)(148 152 156 160)(161 165 169 173)(162 166 170 174)(163 167 171 175)(164 168 172 176)(177 189 185 181)(178 190 186 182)(179 191 187 183)(180 192 188 184)(193 205 201 197)(194 206 202 198)(195 207 203 199)(196 208 204 200)(209 213 217 221)(210 214 218 222)(211 215 219 223)(212 216 220 224)
(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 53)(2 52)(3 51)(4 50)(5 49)(6 64)(7 63)(8 62)(9 61)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 54)(17 112)(18 111)(19 110)(20 109)(21 108)(22 107)(23 106)(24 105)(25 104)(26 103)(27 102)(28 101)(29 100)(30 99)(31 98)(32 97)(33 96)(34 95)(35 94)(36 93)(37 92)(38 91)(39 90)(40 89)(41 88)(42 87)(43 86)(44 85)(45 84)(46 83)(47 82)(48 81)(65 136)(66 135)(67 134)(68 133)(69 132)(70 131)(71 130)(72 129)(73 144)(74 143)(75 142)(76 141)(77 140)(78 139)(79 138)(80 137)(113 162)(114 161)(115 176)(116 175)(117 174)(118 173)(119 172)(120 171)(121 170)(122 169)(123 168)(124 167)(125 166)(126 165)(127 164)(128 163)(145 190)(146 189)(147 188)(148 187)(149 186)(150 185)(151 184)(152 183)(153 182)(154 181)(155 180)(156 179)(157 178)(158 177)(159 192)(160 191)(193 215)(194 214)(195 213)(196 212)(197 211)(198 210)(199 209)(200 224)(201 223)(202 222)(203 221)(204 220)(205 219)(206 218)(207 217)(208 216)
G:=sub<Sym(224)| (1,195,185,89,112,137,119)(2,196,186,90,97,138,120)(3,197,187,91,98,139,121)(4,198,188,92,99,140,122)(5,199,189,93,100,141,123)(6,200,190,94,101,142,124)(7,201,191,95,102,143,125)(8,202,192,96,103,144,126)(9,203,177,81,104,129,127)(10,204,178,82,105,130,128)(11,205,179,83,106,131,113)(12,206,180,84,107,132,114)(13,207,181,85,108,133,115)(14,208,182,86,109,134,116)(15,193,183,87,110,135,117)(16,194,184,88,111,136,118)(17,80,172,53,213,150,40)(18,65,173,54,214,151,41)(19,66,174,55,215,152,42)(20,67,175,56,216,153,43)(21,68,176,57,217,154,44)(22,69,161,58,218,155,45)(23,70,162,59,219,156,46)(24,71,163,60,220,157,47)(25,72,164,61,221,158,48)(26,73,165,62,222,159,33)(27,74,166,63,223,160,34)(28,75,167,64,224,145,35)(29,76,168,49,209,146,36)(30,77,169,50,210,147,37)(31,78,170,51,211,148,38)(32,79,171,52,212,149,39), (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32)(33,37,41,45)(34,38,42,46)(35,39,43,47)(36,40,44,48)(49,53,57,61)(50,54,58,62)(51,55,59,63)(52,56,60,64)(65,69,73,77)(66,70,74,78)(67,71,75,79)(68,72,76,80)(81,93,89,85)(82,94,90,86)(83,95,91,87)(84,96,92,88)(97,109,105,101)(98,110,106,102)(99,111,107,103)(100,112,108,104)(113,125,121,117)(114,126,122,118)(115,127,123,119)(116,128,124,120)(129,141,137,133)(130,142,138,134)(131,143,139,135)(132,144,140,136)(145,149,153,157)(146,150,154,158)(147,151,155,159)(148,152,156,160)(161,165,169,173)(162,166,170,174)(163,167,171,175)(164,168,172,176)(177,189,185,181)(178,190,186,182)(179,191,187,183)(180,192,188,184)(193,205,201,197)(194,206,202,198)(195,207,203,199)(196,208,204,200)(209,213,217,221)(210,214,218,222)(211,215,219,223)(212,216,220,224), (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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,112)(18,111)(19,110)(20,109)(21,108)(22,107)(23,106)(24,105)(25,104)(26,103)(27,102)(28,101)(29,100)(30,99)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,91)(39,90)(40,89)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(65,136)(66,135)(67,134)(68,133)(69,132)(70,131)(71,130)(72,129)(73,144)(74,143)(75,142)(76,141)(77,140)(78,139)(79,138)(80,137)(113,162)(114,161)(115,176)(116,175)(117,174)(118,173)(119,172)(120,171)(121,170)(122,169)(123,168)(124,167)(125,166)(126,165)(127,164)(128,163)(145,190)(146,189)(147,188)(148,187)(149,186)(150,185)(151,184)(152,183)(153,182)(154,181)(155,180)(156,179)(157,178)(158,177)(159,192)(160,191)(193,215)(194,214)(195,213)(196,212)(197,211)(198,210)(199,209)(200,224)(201,223)(202,222)(203,221)(204,220)(205,219)(206,218)(207,217)(208,216)>;
G:=Group( (1,195,185,89,112,137,119)(2,196,186,90,97,138,120)(3,197,187,91,98,139,121)(4,198,188,92,99,140,122)(5,199,189,93,100,141,123)(6,200,190,94,101,142,124)(7,201,191,95,102,143,125)(8,202,192,96,103,144,126)(9,203,177,81,104,129,127)(10,204,178,82,105,130,128)(11,205,179,83,106,131,113)(12,206,180,84,107,132,114)(13,207,181,85,108,133,115)(14,208,182,86,109,134,116)(15,193,183,87,110,135,117)(16,194,184,88,111,136,118)(17,80,172,53,213,150,40)(18,65,173,54,214,151,41)(19,66,174,55,215,152,42)(20,67,175,56,216,153,43)(21,68,176,57,217,154,44)(22,69,161,58,218,155,45)(23,70,162,59,219,156,46)(24,71,163,60,220,157,47)(25,72,164,61,221,158,48)(26,73,165,62,222,159,33)(27,74,166,63,223,160,34)(28,75,167,64,224,145,35)(29,76,168,49,209,146,36)(30,77,169,50,210,147,37)(31,78,170,51,211,148,38)(32,79,171,52,212,149,39), (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32)(33,37,41,45)(34,38,42,46)(35,39,43,47)(36,40,44,48)(49,53,57,61)(50,54,58,62)(51,55,59,63)(52,56,60,64)(65,69,73,77)(66,70,74,78)(67,71,75,79)(68,72,76,80)(81,93,89,85)(82,94,90,86)(83,95,91,87)(84,96,92,88)(97,109,105,101)(98,110,106,102)(99,111,107,103)(100,112,108,104)(113,125,121,117)(114,126,122,118)(115,127,123,119)(116,128,124,120)(129,141,137,133)(130,142,138,134)(131,143,139,135)(132,144,140,136)(145,149,153,157)(146,150,154,158)(147,151,155,159)(148,152,156,160)(161,165,169,173)(162,166,170,174)(163,167,171,175)(164,168,172,176)(177,189,185,181)(178,190,186,182)(179,191,187,183)(180,192,188,184)(193,205,201,197)(194,206,202,198)(195,207,203,199)(196,208,204,200)(209,213,217,221)(210,214,218,222)(211,215,219,223)(212,216,220,224), (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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,112)(18,111)(19,110)(20,109)(21,108)(22,107)(23,106)(24,105)(25,104)(26,103)(27,102)(28,101)(29,100)(30,99)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,91)(39,90)(40,89)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(65,136)(66,135)(67,134)(68,133)(69,132)(70,131)(71,130)(72,129)(73,144)(74,143)(75,142)(76,141)(77,140)(78,139)(79,138)(80,137)(113,162)(114,161)(115,176)(116,175)(117,174)(118,173)(119,172)(120,171)(121,170)(122,169)(123,168)(124,167)(125,166)(126,165)(127,164)(128,163)(145,190)(146,189)(147,188)(148,187)(149,186)(150,185)(151,184)(152,183)(153,182)(154,181)(155,180)(156,179)(157,178)(158,177)(159,192)(160,191)(193,215)(194,214)(195,213)(196,212)(197,211)(198,210)(199,209)(200,224)(201,223)(202,222)(203,221)(204,220)(205,219)(206,218)(207,217)(208,216) );
G=PermutationGroup([[(1,195,185,89,112,137,119),(2,196,186,90,97,138,120),(3,197,187,91,98,139,121),(4,198,188,92,99,140,122),(5,199,189,93,100,141,123),(6,200,190,94,101,142,124),(7,201,191,95,102,143,125),(8,202,192,96,103,144,126),(9,203,177,81,104,129,127),(10,204,178,82,105,130,128),(11,205,179,83,106,131,113),(12,206,180,84,107,132,114),(13,207,181,85,108,133,115),(14,208,182,86,109,134,116),(15,193,183,87,110,135,117),(16,194,184,88,111,136,118),(17,80,172,53,213,150,40),(18,65,173,54,214,151,41),(19,66,174,55,215,152,42),(20,67,175,56,216,153,43),(21,68,176,57,217,154,44),(22,69,161,58,218,155,45),(23,70,162,59,219,156,46),(24,71,163,60,220,157,47),(25,72,164,61,221,158,48),(26,73,165,62,222,159,33),(27,74,166,63,223,160,34),(28,75,167,64,224,145,35),(29,76,168,49,209,146,36),(30,77,169,50,210,147,37),(31,78,170,51,211,148,38),(32,79,171,52,212,149,39)], [(1,13,9,5),(2,14,10,6),(3,15,11,7),(4,16,12,8),(17,21,25,29),(18,22,26,30),(19,23,27,31),(20,24,28,32),(33,37,41,45),(34,38,42,46),(35,39,43,47),(36,40,44,48),(49,53,57,61),(50,54,58,62),(51,55,59,63),(52,56,60,64),(65,69,73,77),(66,70,74,78),(67,71,75,79),(68,72,76,80),(81,93,89,85),(82,94,90,86),(83,95,91,87),(84,96,92,88),(97,109,105,101),(98,110,106,102),(99,111,107,103),(100,112,108,104),(113,125,121,117),(114,126,122,118),(115,127,123,119),(116,128,124,120),(129,141,137,133),(130,142,138,134),(131,143,139,135),(132,144,140,136),(145,149,153,157),(146,150,154,158),(147,151,155,159),(148,152,156,160),(161,165,169,173),(162,166,170,174),(163,167,171,175),(164,168,172,176),(177,189,185,181),(178,190,186,182),(179,191,187,183),(180,192,188,184),(193,205,201,197),(194,206,202,198),(195,207,203,199),(196,208,204,200),(209,213,217,221),(210,214,218,222),(211,215,219,223),(212,216,220,224)], [(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)], [(1,53),(2,52),(3,51),(4,50),(5,49),(6,64),(7,63),(8,62),(9,61),(10,60),(11,59),(12,58),(13,57),(14,56),(15,55),(16,54),(17,112),(18,111),(19,110),(20,109),(21,108),(22,107),(23,106),(24,105),(25,104),(26,103),(27,102),(28,101),(29,100),(30,99),(31,98),(32,97),(33,96),(34,95),(35,94),(36,93),(37,92),(38,91),(39,90),(40,89),(41,88),(42,87),(43,86),(44,85),(45,84),(46,83),(47,82),(48,81),(65,136),(66,135),(67,134),(68,133),(69,132),(70,131),(71,130),(72,129),(73,144),(74,143),(75,142),(76,141),(77,140),(78,139),(79,138),(80,137),(113,162),(114,161),(115,176),(116,175),(117,174),(118,173),(119,172),(120,171),(121,170),(122,169),(123,168),(124,167),(125,166),(126,165),(127,164),(128,163),(145,190),(146,189),(147,188),(148,187),(149,186),(150,185),(151,184),(152,183),(153,182),(154,181),(155,180),(156,179),(157,178),(158,177),(159,192),(160,191),(193,215),(194,214),(195,213),(196,212),(197,211),(198,210),(199,209),(200,224),(201,223),(202,222),(203,221),(204,220),(205,219),(206,218),(207,217),(208,216)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14X | 16A | ··· | 16H | 28A | ··· | 28L | 28M | ··· | 28R | 28S | ··· | 28AD | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 8 | 8 | 1 | 1 | 2 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | D8 | D8 | C7×D4 | C7×D4 | C4○D16 | C7×D8 | C7×D8 | C7×C4○D16 |
| kernel | C7×C4○D16 | C2×C112 | C7×D16 | C7×SD32 | C7×Q32 | C7×C4○D8 | C4○D16 | C2×C16 | D16 | SD32 | Q32 | C4○D8 | C56 | C2×C28 | C28 | C2×C14 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 6 | 6 | 6 | 12 | 6 | 12 | 1 | 1 | 2 | 2 | 6 | 6 | 8 | 12 | 12 | 48 |
Matrix representation of C7×C4○D16 ►in GL2(𝔽113) generated by
| 109 | 0 |
| 0 | 109 |
| 98 | 0 |
| 0 | 98 |
| 65 | 0 |
| 0 | 40 |
| 0 | 40 |
| 65 | 0 |
G:=sub<GL(2,GF(113))| [109,0,0,109],[98,0,0,98],[65,0,0,40],[0,65,40,0] >;
C7×C4○D16 in GAP, Magma, Sage, TeX
C_7\times C_4\circ D_{16} % in TeX
G:=Group("C7xC4oD16"); // GroupNames label
G:=SmallGroup(448,916);
// by ID
G=gap.SmallGroup(448,916);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,1192,5884,2951,242,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^4=d^2=1,c^8=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^7>;
// generators/relations