direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×C4×C8, D14.6C42, C42.281D14, Dic7.6C42, C28⋊5(C2×C8), (C4×C56)⋊18C2, C56⋊24(C2×C4), Dic7⋊5(C2×C8), D14.7(C2×C8), C2.1(D7×C42), (C8×Dic7)⋊33C2, (C2×C8).338D14, C14.5(C2×C42), C14.2(C22×C8), (C4×Dic7).23C4, (D7×C42).16C2, (C2×C56).404C22, (C4×C28).337C22, C28.122(C22×C4), (C2×C28).802C23, (C4×Dic7).298C22, C7⋊1(C2×C4×C8), (C4×C7⋊C8)⋊27C2, C2.1(D7×C2×C8), C7⋊C8⋊28(C2×C4), C4.96(C2×C4×D7), (C2×C4×D7).23C4, (D7×C2×C8).19C2, C22.34(C2×C4×D7), (C4×D7).36(C2×C4), (C2×C4).172(C4×D7), (C2×C28).245(C2×C4), (C2×C7⋊C8).328C22, (C2×C4×D7).304C22, (C2×C14).57(C22×C4), (C22×D7).70(C2×C4), (C2×C4).744(C22×D7), (C2×Dic7).106(C2×C4), SmallGroup(448,218)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C4×C8 |
Generators and relations for D7×C4×C8
G = < a,b,c,d | a4=b8=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 516 in 162 conjugacy classes, 103 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, C23, D7, C14, C14, C42, C42, C2×C8, C2×C8, C22×C4, Dic7, C28, D14, C2×C14, C4×C8, C4×C8, C2×C42, C22×C8, C7⋊C8, C56, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C2×C4×C8, C8×D7, C2×C7⋊C8, C4×Dic7, C4×Dic7, C4×C28, C2×C56, C2×C4×D7, C2×C4×D7, C4×C7⋊C8, C8×Dic7, C4×C56, D7×C42, D7×C2×C8, D7×C4×C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C42, C2×C8, C22×C4, D14, C4×C8, C2×C42, C22×C8, C4×D7, C22×D7, C2×C4×C8, C8×D7, C2×C4×D7, D7×C42, D7×C2×C8, D7×C4×C8
►On 224 points
Generators in S224
(1 221 47 212)(2 222 48 213)(3 223 41 214)(4 224 42 215)(5 217 43 216)(6 218 44 209)(7 219 45 210)(8 220 46 211)(9 40 162 199)(10 33 163 200)(11 34 164 193)(12 35 165 194)(13 36 166 195)(14 37 167 196)(15 38 168 197)(16 39 161 198)(17 70 130 117)(18 71 131 118)(19 72 132 119)(20 65 133 120)(21 66 134 113)(22 67 135 114)(23 68 136 115)(24 69 129 116)(25 169 192 64)(26 170 185 57)(27 171 186 58)(28 172 187 59)(29 173 188 60)(30 174 189 61)(31 175 190 62)(32 176 191 63)(49 103 142 150)(50 104 143 151)(51 97 144 152)(52 98 137 145)(53 99 138 146)(54 100 139 147)(55 101 140 148)(56 102 141 149)(73 183 128 90)(74 184 121 91)(75 177 122 92)(76 178 123 93)(77 179 124 94)(78 180 125 95)(79 181 126 96)(80 182 127 89)(81 112 206 159)(82 105 207 160)(83 106 208 153)(84 107 201 154)(85 108 202 155)(86 109 203 156)(87 110 204 157)(88 111 205 158)
(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)
(1 207 161 94 56 170 132)(2 208 162 95 49 171 133)(3 201 163 96 50 172 134)(4 202 164 89 51 173 135)(5 203 165 90 52 174 136)(6 204 166 91 53 175 129)(7 205 167 92 54 176 130)(8 206 168 93 55 169 131)(9 180 142 58 20 48 83)(10 181 143 59 21 41 84)(11 182 144 60 22 42 85)(12 183 137 61 23 43 86)(13 184 138 62 24 44 87)(14 177 139 63 17 45 88)(15 178 140 64 18 46 81)(16 179 141 57 19 47 82)(25 71 211 112 38 123 148)(26 72 212 105 39 124 149)(27 65 213 106 40 125 150)(28 66 214 107 33 126 151)(29 67 215 108 34 127 152)(30 68 216 109 35 128 145)(31 69 209 110 36 121 146)(32 70 210 111 37 122 147)(73 98 189 115 217 156 194)(74 99 190 116 218 157 195)(75 100 191 117 219 158 196)(76 101 192 118 220 159 197)(77 102 185 119 221 160 198)(78 103 186 120 222 153 199)(79 104 187 113 223 154 200)(80 97 188 114 224 155 193)
(1 136)(2 129)(3 130)(4 131)(5 132)(6 133)(7 134)(8 135)(9 138)(10 139)(11 140)(12 141)(13 142)(14 143)(15 144)(16 137)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 108)(26 109)(27 110)(28 111)(29 112)(30 105)(31 106)(32 107)(33 147)(34 148)(35 149)(36 150)(37 151)(38 152)(39 145)(40 146)(49 166)(50 167)(51 168)(52 161)(53 162)(54 163)(55 164)(56 165)(57 86)(58 87)(59 88)(60 81)(61 82)(62 83)(63 84)(64 85)(65 209)(66 210)(67 211)(68 212)(69 213)(70 214)(71 215)(72 216)(73 77)(74 78)(75 79)(76 80)(89 93)(90 94)(91 95)(92 96)(97 197)(98 198)(99 199)(100 200)(101 193)(102 194)(103 195)(104 196)(113 219)(114 220)(115 221)(116 222)(117 223)(118 224)(119 217)(120 218)(121 125)(122 126)(123 127)(124 128)(153 190)(154 191)(155 192)(156 185)(157 186)(158 187)(159 188)(160 189)(169 202)(170 203)(171 204)(172 205)(173 206)(174 207)(175 208)(176 201)(177 181)(178 182)(179 183)(180 184)
G:=sub<Sym(224)| (1,221,47,212)(2,222,48,213)(3,223,41,214)(4,224,42,215)(5,217,43,216)(6,218,44,209)(7,219,45,210)(8,220,46,211)(9,40,162,199)(10,33,163,200)(11,34,164,193)(12,35,165,194)(13,36,166,195)(14,37,167,196)(15,38,168,197)(16,39,161,198)(17,70,130,117)(18,71,131,118)(19,72,132,119)(20,65,133,120)(21,66,134,113)(22,67,135,114)(23,68,136,115)(24,69,129,116)(25,169,192,64)(26,170,185,57)(27,171,186,58)(28,172,187,59)(29,173,188,60)(30,174,189,61)(31,175,190,62)(32,176,191,63)(49,103,142,150)(50,104,143,151)(51,97,144,152)(52,98,137,145)(53,99,138,146)(54,100,139,147)(55,101,140,148)(56,102,141,149)(73,183,128,90)(74,184,121,91)(75,177,122,92)(76,178,123,93)(77,179,124,94)(78,180,125,95)(79,181,126,96)(80,182,127,89)(81,112,206,159)(82,105,207,160)(83,106,208,153)(84,107,201,154)(85,108,202,155)(86,109,203,156)(87,110,204,157)(88,111,205,158), (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), (1,207,161,94,56,170,132)(2,208,162,95,49,171,133)(3,201,163,96,50,172,134)(4,202,164,89,51,173,135)(5,203,165,90,52,174,136)(6,204,166,91,53,175,129)(7,205,167,92,54,176,130)(8,206,168,93,55,169,131)(9,180,142,58,20,48,83)(10,181,143,59,21,41,84)(11,182,144,60,22,42,85)(12,183,137,61,23,43,86)(13,184,138,62,24,44,87)(14,177,139,63,17,45,88)(15,178,140,64,18,46,81)(16,179,141,57,19,47,82)(25,71,211,112,38,123,148)(26,72,212,105,39,124,149)(27,65,213,106,40,125,150)(28,66,214,107,33,126,151)(29,67,215,108,34,127,152)(30,68,216,109,35,128,145)(31,69,209,110,36,121,146)(32,70,210,111,37,122,147)(73,98,189,115,217,156,194)(74,99,190,116,218,157,195)(75,100,191,117,219,158,196)(76,101,192,118,220,159,197)(77,102,185,119,221,160,198)(78,103,186,120,222,153,199)(79,104,187,113,223,154,200)(80,97,188,114,224,155,193), (1,136)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,138)(10,139)(11,140)(12,141)(13,142)(14,143)(15,144)(16,137)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,108)(26,109)(27,110)(28,111)(29,112)(30,105)(31,106)(32,107)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,145)(40,146)(49,166)(50,167)(51,168)(52,161)(53,162)(54,163)(55,164)(56,165)(57,86)(58,87)(59,88)(60,81)(61,82)(62,83)(63,84)(64,85)(65,209)(66,210)(67,211)(68,212)(69,213)(70,214)(71,215)(72,216)(73,77)(74,78)(75,79)(76,80)(89,93)(90,94)(91,95)(92,96)(97,197)(98,198)(99,199)(100,200)(101,193)(102,194)(103,195)(104,196)(113,219)(114,220)(115,221)(116,222)(117,223)(118,224)(119,217)(120,218)(121,125)(122,126)(123,127)(124,128)(153,190)(154,191)(155,192)(156,185)(157,186)(158,187)(159,188)(160,189)(169,202)(170,203)(171,204)(172,205)(173,206)(174,207)(175,208)(176,201)(177,181)(178,182)(179,183)(180,184)>;
G:=Group( (1,221,47,212)(2,222,48,213)(3,223,41,214)(4,224,42,215)(5,217,43,216)(6,218,44,209)(7,219,45,210)(8,220,46,211)(9,40,162,199)(10,33,163,200)(11,34,164,193)(12,35,165,194)(13,36,166,195)(14,37,167,196)(15,38,168,197)(16,39,161,198)(17,70,130,117)(18,71,131,118)(19,72,132,119)(20,65,133,120)(21,66,134,113)(22,67,135,114)(23,68,136,115)(24,69,129,116)(25,169,192,64)(26,170,185,57)(27,171,186,58)(28,172,187,59)(29,173,188,60)(30,174,189,61)(31,175,190,62)(32,176,191,63)(49,103,142,150)(50,104,143,151)(51,97,144,152)(52,98,137,145)(53,99,138,146)(54,100,139,147)(55,101,140,148)(56,102,141,149)(73,183,128,90)(74,184,121,91)(75,177,122,92)(76,178,123,93)(77,179,124,94)(78,180,125,95)(79,181,126,96)(80,182,127,89)(81,112,206,159)(82,105,207,160)(83,106,208,153)(84,107,201,154)(85,108,202,155)(86,109,203,156)(87,110,204,157)(88,111,205,158), (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), (1,207,161,94,56,170,132)(2,208,162,95,49,171,133)(3,201,163,96,50,172,134)(4,202,164,89,51,173,135)(5,203,165,90,52,174,136)(6,204,166,91,53,175,129)(7,205,167,92,54,176,130)(8,206,168,93,55,169,131)(9,180,142,58,20,48,83)(10,181,143,59,21,41,84)(11,182,144,60,22,42,85)(12,183,137,61,23,43,86)(13,184,138,62,24,44,87)(14,177,139,63,17,45,88)(15,178,140,64,18,46,81)(16,179,141,57,19,47,82)(25,71,211,112,38,123,148)(26,72,212,105,39,124,149)(27,65,213,106,40,125,150)(28,66,214,107,33,126,151)(29,67,215,108,34,127,152)(30,68,216,109,35,128,145)(31,69,209,110,36,121,146)(32,70,210,111,37,122,147)(73,98,189,115,217,156,194)(74,99,190,116,218,157,195)(75,100,191,117,219,158,196)(76,101,192,118,220,159,197)(77,102,185,119,221,160,198)(78,103,186,120,222,153,199)(79,104,187,113,223,154,200)(80,97,188,114,224,155,193), (1,136)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,138)(10,139)(11,140)(12,141)(13,142)(14,143)(15,144)(16,137)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,108)(26,109)(27,110)(28,111)(29,112)(30,105)(31,106)(32,107)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,145)(40,146)(49,166)(50,167)(51,168)(52,161)(53,162)(54,163)(55,164)(56,165)(57,86)(58,87)(59,88)(60,81)(61,82)(62,83)(63,84)(64,85)(65,209)(66,210)(67,211)(68,212)(69,213)(70,214)(71,215)(72,216)(73,77)(74,78)(75,79)(76,80)(89,93)(90,94)(91,95)(92,96)(97,197)(98,198)(99,199)(100,200)(101,193)(102,194)(103,195)(104,196)(113,219)(114,220)(115,221)(116,222)(117,223)(118,224)(119,217)(120,218)(121,125)(122,126)(123,127)(124,128)(153,190)(154,191)(155,192)(156,185)(157,186)(158,187)(159,188)(160,189)(169,202)(170,203)(171,204)(172,205)(173,206)(174,207)(175,208)(176,201)(177,181)(178,182)(179,183)(180,184) );
G=PermutationGroup([[(1,221,47,212),(2,222,48,213),(3,223,41,214),(4,224,42,215),(5,217,43,216),(6,218,44,209),(7,219,45,210),(8,220,46,211),(9,40,162,199),(10,33,163,200),(11,34,164,193),(12,35,165,194),(13,36,166,195),(14,37,167,196),(15,38,168,197),(16,39,161,198),(17,70,130,117),(18,71,131,118),(19,72,132,119),(20,65,133,120),(21,66,134,113),(22,67,135,114),(23,68,136,115),(24,69,129,116),(25,169,192,64),(26,170,185,57),(27,171,186,58),(28,172,187,59),(29,173,188,60),(30,174,189,61),(31,175,190,62),(32,176,191,63),(49,103,142,150),(50,104,143,151),(51,97,144,152),(52,98,137,145),(53,99,138,146),(54,100,139,147),(55,101,140,148),(56,102,141,149),(73,183,128,90),(74,184,121,91),(75,177,122,92),(76,178,123,93),(77,179,124,94),(78,180,125,95),(79,181,126,96),(80,182,127,89),(81,112,206,159),(82,105,207,160),(83,106,208,153),(84,107,201,154),(85,108,202,155),(86,109,203,156),(87,110,204,157),(88,111,205,158)], [(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)], [(1,207,161,94,56,170,132),(2,208,162,95,49,171,133),(3,201,163,96,50,172,134),(4,202,164,89,51,173,135),(5,203,165,90,52,174,136),(6,204,166,91,53,175,129),(7,205,167,92,54,176,130),(8,206,168,93,55,169,131),(9,180,142,58,20,48,83),(10,181,143,59,21,41,84),(11,182,144,60,22,42,85),(12,183,137,61,23,43,86),(13,184,138,62,24,44,87),(14,177,139,63,17,45,88),(15,178,140,64,18,46,81),(16,179,141,57,19,47,82),(25,71,211,112,38,123,148),(26,72,212,105,39,124,149),(27,65,213,106,40,125,150),(28,66,214,107,33,126,151),(29,67,215,108,34,127,152),(30,68,216,109,35,128,145),(31,69,209,110,36,121,146),(32,70,210,111,37,122,147),(73,98,189,115,217,156,194),(74,99,190,116,218,157,195),(75,100,191,117,219,158,196),(76,101,192,118,220,159,197),(77,102,185,119,221,160,198),(78,103,186,120,222,153,199),(79,104,187,113,223,154,200),(80,97,188,114,224,155,193)], [(1,136),(2,129),(3,130),(4,131),(5,132),(6,133),(7,134),(8,135),(9,138),(10,139),(11,140),(12,141),(13,142),(14,143),(15,144),(16,137),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,108),(26,109),(27,110),(28,111),(29,112),(30,105),(31,106),(32,107),(33,147),(34,148),(35,149),(36,150),(37,151),(38,152),(39,145),(40,146),(49,166),(50,167),(51,168),(52,161),(53,162),(54,163),(55,164),(56,165),(57,86),(58,87),(59,88),(60,81),(61,82),(62,83),(63,84),(64,85),(65,209),(66,210),(67,211),(68,212),(69,213),(70,214),(71,215),(72,216),(73,77),(74,78),(75,79),(76,80),(89,93),(90,94),(91,95),(92,96),(97,197),(98,198),(99,199),(100,200),(101,193),(102,194),(103,195),(104,196),(113,219),(114,220),(115,221),(116,222),(117,223),(118,224),(119,217),(120,218),(121,125),(122,126),(123,127),(124,128),(153,190),(154,191),(155,192),(156,185),(157,186),(158,187),(159,188),(160,189),(169,202),(170,203),(171,204),(172,205),(173,206),(174,207),(175,208),(176,201),(177,181),(178,182),(179,183),(180,184)]])
160 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4L | 4M | ··· | 4X | 7A | 7B | 7C | 8A | ··· | 8P | 8Q | ··· | 8AF | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
160 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 | C2 | C4 | C4 | C4 | C8 | D7 | D14 | D14 | C4×D7 | C4×D7 | C8×D7 |
| kernel | D7×C4×C8 | C4×C7⋊C8 | C8×Dic7 | C4×C56 | D7×C42 | D7×C2×C8 | C8×D7 | C4×Dic7 | C2×C4×D7 | C4×D7 | C4×C8 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 16 | 4 | 4 | 32 | 3 | 3 | 6 | 24 | 12 | 48 |
Matrix representation of D7×C4×C8 ►in GL3(𝔽113) generated by
| 15 | 0 | 0 |
| 0 | 98 | 0 |
| 0 | 0 | 98 |
| 15 | 0 | 0 |
| 0 | 95 | 0 |
| 0 | 0 | 95 |
| 1 | 0 | 0 |
| 0 | 88 | 2 |
| 0 | 112 | 104 |
| 112 | 0 | 0 |
| 0 | 25 | 54 |
| 0 | 1 | 88 |
G:=sub<GL(3,GF(113))| [15,0,0,0,98,0,0,0,98],[15,0,0,0,95,0,0,0,95],[1,0,0,0,88,112,0,2,104],[112,0,0,0,25,1,0,54,88] >;
D7×C4×C8 in GAP, Magma, Sage, TeX
D_7\times C_4\times C_8
% in TeX
G:=Group("D7xC4xC8"); // GroupNames label
G:=SmallGroup(448,218);
// by ID
G=gap.SmallGroup(448,218);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,58,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^8=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations