direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C8⋊5D4, C56⋊27D4, C28⋊9SD16, C8⋊5(C7×D4), C4⋊Q8⋊7C14, (C4×C56)⋊28C2, (C4×C8)⋊12C14, C4⋊1(C7×SD16), C4.1(D4×C14), (C2×C28).421D4, C4⋊1D4.6C14, C28.308(C2×D4), (C2×SD16)⋊14C14, (C14×SD16)⋊31C2, C42.79(C2×C14), C14.96(C2×SD16), C2.16(C14×SD16), C14.42(C4⋊1D4), (C2×C28).948C23, (C2×C56).438C22, (C4×C28).363C22, C22.113(D4×C14), (D4×C14).202C22, (Q8×C14).176C22, (C7×C4⋊Q8)⋊28C2, (C2×C4).77(C7×D4), C2.5(C7×C4⋊1D4), (C2×C8).94(C2×C14), (C2×D4).25(C2×C14), (C7×C4⋊1D4).13C2, (C2×C14).669(C2×D4), (C2×Q8).20(C2×C14), (C2×C4).123(C22×C14), SmallGroup(448,900)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8⋊5D4
G = < a,b,c,d | a7=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b3, dcd=c-1 >
Subgroups: 290 in 142 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, C28, C28, C2×C14, C2×C14, C4×C8, C4⋊1D4, C4⋊Q8, C2×SD16, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C8⋊5D4, C4×C28, C7×C4⋊C4, C2×C56, C7×SD16, D4×C14, D4×C14, Q8×C14, C4×C56, C7×C4⋊1D4, C7×C4⋊Q8, C14×SD16, C7×C8⋊5D4
Quotients: C1, C2, C22, C7, D4, C23, C14, SD16, C2×D4, C2×C14, C4⋊1D4, C2×SD16, C7×D4, C22×C14, C8⋊5D4, C7×SD16, D4×C14, C7×C4⋊1D4, C14×SD16, C7×C8⋊5D4
►On 224 points
Generators in S224
(1 134 78 126 70 118 62)(2 135 79 127 71 119 63)(3 136 80 128 72 120 64)(4 129 73 121 65 113 57)(5 130 74 122 66 114 58)(6 131 75 123 67 115 59)(7 132 76 124 68 116 60)(8 133 77 125 69 117 61)(9 163 195 25 187 17 179)(10 164 196 26 188 18 180)(11 165 197 27 189 19 181)(12 166 198 28 190 20 182)(13 167 199 29 191 21 183)(14 168 200 30 192 22 184)(15 161 193 31 185 23 177)(16 162 194 32 186 24 178)(33 169 219 49 211 41 203)(34 170 220 50 212 42 204)(35 171 221 51 213 43 205)(36 172 222 52 214 44 206)(37 173 223 53 215 45 207)(38 174 224 54 216 46 208)(39 175 217 55 209 47 201)(40 176 218 56 210 48 202)(81 107 155 99 147 90 141)(82 108 156 100 148 91 142)(83 109 157 101 149 92 143)(84 110 158 102 150 93 144)(85 111 159 103 151 94 137)(86 112 160 104 152 95 138)(87 105 153 97 145 96 139)(88 106 154 98 146 89 140)
(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)
(1 149 37 167)(2 150 38 168)(3 151 39 161)(4 152 40 162)(5 145 33 163)(6 146 34 164)(7 147 35 165)(8 148 36 166)(9 58 97 203)(10 59 98 204)(11 60 99 205)(12 61 100 206)(13 62 101 207)(14 63 102 208)(15 64 103 201)(16 57 104 202)(17 66 105 211)(18 67 106 212)(19 68 107 213)(20 69 108 214)(21 70 109 215)(22 71 110 216)(23 72 111 209)(24 65 112 210)(25 74 139 219)(26 75 140 220)(27 76 141 221)(28 77 142 222)(29 78 143 223)(30 79 144 224)(31 80 137 217)(32 73 138 218)(41 179 114 153)(42 180 115 154)(43 181 116 155)(44 182 117 156)(45 183 118 157)(46 184 119 158)(47 177 120 159)(48 178 113 160)(49 187 122 87)(50 188 123 88)(51 189 124 81)(52 190 125 82)(53 191 126 83)(54 192 127 84)(55 185 128 85)(56 186 121 86)(89 170 196 131)(90 171 197 132)(91 172 198 133)(92 173 199 134)(93 174 200 135)(94 175 193 136)(95 176 194 129)(96 169 195 130)
(1 37)(2 40)(3 35)(4 38)(5 33)(6 36)(7 39)(8 34)(10 12)(11 15)(14 16)(18 20)(19 23)(22 24)(26 28)(27 31)(30 32)(41 114)(42 117)(43 120)(44 115)(45 118)(46 113)(47 116)(48 119)(49 122)(50 125)(51 128)(52 123)(53 126)(54 121)(55 124)(56 127)(57 208)(58 203)(59 206)(60 201)(61 204)(62 207)(63 202)(64 205)(65 216)(66 211)(67 214)(68 209)(69 212)(70 215)(71 210)(72 213)(73 224)(74 219)(75 222)(76 217)(77 220)(78 223)(79 218)(80 221)(81 85)(82 88)(84 86)(89 91)(90 94)(93 95)(98 100)(99 103)(102 104)(106 108)(107 111)(110 112)(129 174)(130 169)(131 172)(132 175)(133 170)(134 173)(135 176)(136 171)(137 141)(138 144)(140 142)(146 148)(147 151)(150 152)(154 156)(155 159)(158 160)(161 165)(162 168)(164 166)(177 181)(178 184)(180 182)(185 189)(186 192)(188 190)(193 197)(194 200)(196 198)
G:=sub<Sym(224)| (1,134,78,126,70,118,62)(2,135,79,127,71,119,63)(3,136,80,128,72,120,64)(4,129,73,121,65,113,57)(5,130,74,122,66,114,58)(6,131,75,123,67,115,59)(7,132,76,124,68,116,60)(8,133,77,125,69,117,61)(9,163,195,25,187,17,179)(10,164,196,26,188,18,180)(11,165,197,27,189,19,181)(12,166,198,28,190,20,182)(13,167,199,29,191,21,183)(14,168,200,30,192,22,184)(15,161,193,31,185,23,177)(16,162,194,32,186,24,178)(33,169,219,49,211,41,203)(34,170,220,50,212,42,204)(35,171,221,51,213,43,205)(36,172,222,52,214,44,206)(37,173,223,53,215,45,207)(38,174,224,54,216,46,208)(39,175,217,55,209,47,201)(40,176,218,56,210,48,202)(81,107,155,99,147,90,141)(82,108,156,100,148,91,142)(83,109,157,101,149,92,143)(84,110,158,102,150,93,144)(85,111,159,103,151,94,137)(86,112,160,104,152,95,138)(87,105,153,97,145,96,139)(88,106,154,98,146,89,140), (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), (1,149,37,167)(2,150,38,168)(3,151,39,161)(4,152,40,162)(5,145,33,163)(6,146,34,164)(7,147,35,165)(8,148,36,166)(9,58,97,203)(10,59,98,204)(11,60,99,205)(12,61,100,206)(13,62,101,207)(14,63,102,208)(15,64,103,201)(16,57,104,202)(17,66,105,211)(18,67,106,212)(19,68,107,213)(20,69,108,214)(21,70,109,215)(22,71,110,216)(23,72,111,209)(24,65,112,210)(25,74,139,219)(26,75,140,220)(27,76,141,221)(28,77,142,222)(29,78,143,223)(30,79,144,224)(31,80,137,217)(32,73,138,218)(41,179,114,153)(42,180,115,154)(43,181,116,155)(44,182,117,156)(45,183,118,157)(46,184,119,158)(47,177,120,159)(48,178,113,160)(49,187,122,87)(50,188,123,88)(51,189,124,81)(52,190,125,82)(53,191,126,83)(54,192,127,84)(55,185,128,85)(56,186,121,86)(89,170,196,131)(90,171,197,132)(91,172,198,133)(92,173,199,134)(93,174,200,135)(94,175,193,136)(95,176,194,129)(96,169,195,130), (1,37)(2,40)(3,35)(4,38)(5,33)(6,36)(7,39)(8,34)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24)(26,28)(27,31)(30,32)(41,114)(42,117)(43,120)(44,115)(45,118)(46,113)(47,116)(48,119)(49,122)(50,125)(51,128)(52,123)(53,126)(54,121)(55,124)(56,127)(57,208)(58,203)(59,206)(60,201)(61,204)(62,207)(63,202)(64,205)(65,216)(66,211)(67,214)(68,209)(69,212)(70,215)(71,210)(72,213)(73,224)(74,219)(75,222)(76,217)(77,220)(78,223)(79,218)(80,221)(81,85)(82,88)(84,86)(89,91)(90,94)(93,95)(98,100)(99,103)(102,104)(106,108)(107,111)(110,112)(129,174)(130,169)(131,172)(132,175)(133,170)(134,173)(135,176)(136,171)(137,141)(138,144)(140,142)(146,148)(147,151)(150,152)(154,156)(155,159)(158,160)(161,165)(162,168)(164,166)(177,181)(178,184)(180,182)(185,189)(186,192)(188,190)(193,197)(194,200)(196,198)>;
G:=Group( (1,134,78,126,70,118,62)(2,135,79,127,71,119,63)(3,136,80,128,72,120,64)(4,129,73,121,65,113,57)(5,130,74,122,66,114,58)(6,131,75,123,67,115,59)(7,132,76,124,68,116,60)(8,133,77,125,69,117,61)(9,163,195,25,187,17,179)(10,164,196,26,188,18,180)(11,165,197,27,189,19,181)(12,166,198,28,190,20,182)(13,167,199,29,191,21,183)(14,168,200,30,192,22,184)(15,161,193,31,185,23,177)(16,162,194,32,186,24,178)(33,169,219,49,211,41,203)(34,170,220,50,212,42,204)(35,171,221,51,213,43,205)(36,172,222,52,214,44,206)(37,173,223,53,215,45,207)(38,174,224,54,216,46,208)(39,175,217,55,209,47,201)(40,176,218,56,210,48,202)(81,107,155,99,147,90,141)(82,108,156,100,148,91,142)(83,109,157,101,149,92,143)(84,110,158,102,150,93,144)(85,111,159,103,151,94,137)(86,112,160,104,152,95,138)(87,105,153,97,145,96,139)(88,106,154,98,146,89,140), (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), (1,149,37,167)(2,150,38,168)(3,151,39,161)(4,152,40,162)(5,145,33,163)(6,146,34,164)(7,147,35,165)(8,148,36,166)(9,58,97,203)(10,59,98,204)(11,60,99,205)(12,61,100,206)(13,62,101,207)(14,63,102,208)(15,64,103,201)(16,57,104,202)(17,66,105,211)(18,67,106,212)(19,68,107,213)(20,69,108,214)(21,70,109,215)(22,71,110,216)(23,72,111,209)(24,65,112,210)(25,74,139,219)(26,75,140,220)(27,76,141,221)(28,77,142,222)(29,78,143,223)(30,79,144,224)(31,80,137,217)(32,73,138,218)(41,179,114,153)(42,180,115,154)(43,181,116,155)(44,182,117,156)(45,183,118,157)(46,184,119,158)(47,177,120,159)(48,178,113,160)(49,187,122,87)(50,188,123,88)(51,189,124,81)(52,190,125,82)(53,191,126,83)(54,192,127,84)(55,185,128,85)(56,186,121,86)(89,170,196,131)(90,171,197,132)(91,172,198,133)(92,173,199,134)(93,174,200,135)(94,175,193,136)(95,176,194,129)(96,169,195,130), (1,37)(2,40)(3,35)(4,38)(5,33)(6,36)(7,39)(8,34)(10,12)(11,15)(14,16)(18,20)(19,23)(22,24)(26,28)(27,31)(30,32)(41,114)(42,117)(43,120)(44,115)(45,118)(46,113)(47,116)(48,119)(49,122)(50,125)(51,128)(52,123)(53,126)(54,121)(55,124)(56,127)(57,208)(58,203)(59,206)(60,201)(61,204)(62,207)(63,202)(64,205)(65,216)(66,211)(67,214)(68,209)(69,212)(70,215)(71,210)(72,213)(73,224)(74,219)(75,222)(76,217)(77,220)(78,223)(79,218)(80,221)(81,85)(82,88)(84,86)(89,91)(90,94)(93,95)(98,100)(99,103)(102,104)(106,108)(107,111)(110,112)(129,174)(130,169)(131,172)(132,175)(133,170)(134,173)(135,176)(136,171)(137,141)(138,144)(140,142)(146,148)(147,151)(150,152)(154,156)(155,159)(158,160)(161,165)(162,168)(164,166)(177,181)(178,184)(180,182)(185,189)(186,192)(188,190)(193,197)(194,200)(196,198) );
G=PermutationGroup([[(1,134,78,126,70,118,62),(2,135,79,127,71,119,63),(3,136,80,128,72,120,64),(4,129,73,121,65,113,57),(5,130,74,122,66,114,58),(6,131,75,123,67,115,59),(7,132,76,124,68,116,60),(8,133,77,125,69,117,61),(9,163,195,25,187,17,179),(10,164,196,26,188,18,180),(11,165,197,27,189,19,181),(12,166,198,28,190,20,182),(13,167,199,29,191,21,183),(14,168,200,30,192,22,184),(15,161,193,31,185,23,177),(16,162,194,32,186,24,178),(33,169,219,49,211,41,203),(34,170,220,50,212,42,204),(35,171,221,51,213,43,205),(36,172,222,52,214,44,206),(37,173,223,53,215,45,207),(38,174,224,54,216,46,208),(39,175,217,55,209,47,201),(40,176,218,56,210,48,202),(81,107,155,99,147,90,141),(82,108,156,100,148,91,142),(83,109,157,101,149,92,143),(84,110,158,102,150,93,144),(85,111,159,103,151,94,137),(86,112,160,104,152,95,138),(87,105,153,97,145,96,139),(88,106,154,98,146,89,140)], [(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)], [(1,149,37,167),(2,150,38,168),(3,151,39,161),(4,152,40,162),(5,145,33,163),(6,146,34,164),(7,147,35,165),(8,148,36,166),(9,58,97,203),(10,59,98,204),(11,60,99,205),(12,61,100,206),(13,62,101,207),(14,63,102,208),(15,64,103,201),(16,57,104,202),(17,66,105,211),(18,67,106,212),(19,68,107,213),(20,69,108,214),(21,70,109,215),(22,71,110,216),(23,72,111,209),(24,65,112,210),(25,74,139,219),(26,75,140,220),(27,76,141,221),(28,77,142,222),(29,78,143,223),(30,79,144,224),(31,80,137,217),(32,73,138,218),(41,179,114,153),(42,180,115,154),(43,181,116,155),(44,182,117,156),(45,183,118,157),(46,184,119,158),(47,177,120,159),(48,178,113,160),(49,187,122,87),(50,188,123,88),(51,189,124,81),(52,190,125,82),(53,191,126,83),(54,192,127,84),(55,185,128,85),(56,186,121,86),(89,170,196,131),(90,171,197,132),(91,172,198,133),(92,173,199,134),(93,174,200,135),(94,175,193,136),(95,176,194,129),(96,169,195,130)], [(1,37),(2,40),(3,35),(4,38),(5,33),(6,36),(7,39),(8,34),(10,12),(11,15),(14,16),(18,20),(19,23),(22,24),(26,28),(27,31),(30,32),(41,114),(42,117),(43,120),(44,115),(45,118),(46,113),(47,116),(48,119),(49,122),(50,125),(51,128),(52,123),(53,126),(54,121),(55,124),(56,127),(57,208),(58,203),(59,206),(60,201),(61,204),(62,207),(63,202),(64,205),(65,216),(66,211),(67,214),(68,209),(69,212),(70,215),(71,210),(72,213),(73,224),(74,219),(75,222),(76,217),(77,220),(78,223),(79,218),(80,221),(81,85),(82,88),(84,86),(89,91),(90,94),(93,95),(98,100),(99,103),(102,104),(106,108),(107,111),(110,112),(129,174),(130,169),(131,172),(132,175),(133,170),(134,173),(135,176),(136,171),(137,141),(138,144),(140,142),(146,148),(147,151),(150,152),(154,156),(155,159),(158,160),(161,165),(162,168),(164,166),(177,181),(178,184),(180,182),(185,189),(186,192),(188,190),(193,197),(194,200),(196,198)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28AJ | 28AK | ··· | 28AV | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | ··· | 2 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | D4 | D4 | SD16 | C7×D4 | C7×D4 | C7×SD16 |
| kernel | C7×C8⋊5D4 | C4×C56 | C7×C4⋊1D4 | C7×C4⋊Q8 | C14×SD16 | C8⋊5D4 | C4×C8 | C4⋊1D4 | C4⋊Q8 | C2×SD16 | C56 | C2×C28 | C28 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 6 | 6 | 24 | 4 | 2 | 8 | 24 | 12 | 48 |
Matrix representation of C7×C8⋊5D4 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 26 | 13 | 0 | 0 |
| 87 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 54 | 2 |
| 0 | 0 | 67 | 59 |
| 1 | 1 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 54 | 1 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,28,0,0,0,0,28],[26,87,0,0,13,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,54,67,0,0,2,59],[1,0,0,0,1,112,0,0,0,0,112,54,0,0,0,1] >;
C7×C8⋊5D4 in GAP, Magma, Sage, TeX
C_7\times C_8\rtimes_5D_4
% in TeX
G:=Group("C7xC8:5D4"); // GroupNames label
G:=SmallGroup(448,900);
// by ID
G=gap.SmallGroup(448,900);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,400,2438,604,9804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^3,d*c*d=c^-1>;
// generators/relations