direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C8⋊D7, C28⋊5M4(2), D14.1C42, C42.283D14, Dic7.1C42, C8⋊12(C4×D7), (C4×C8)⋊13D7, C56⋊25(C2×C4), (C4×C56)⋊20C2, C7⋊1(C4×M4(2)), C56⋊C4⋊30C2, C2.7(D7×C42), (C2×C8).323D14, C14.6(C2×C42), (C4×Dic7).13C4, (D7×C42).12C2, C14.3(C2×M4(2)), (C4×C28).339C22, (C2×C56).405C22, (C2×C28).805C23, C28.123(C22×C4), (C4×Dic7).264C22, (C4×C7⋊C8)⋊20C2, C7⋊C8⋊17(C2×C4), C4.97(C2×C4×D7), (C2×C4×D7).13C4, C2.2(C2×C8⋊D7), C22.37(C2×C4×D7), (C4×D7).19(C2×C4), (C2×C4).174(C4×D7), (C2×C28).247(C2×C4), (C2×C8⋊D7).13C2, (C2×C7⋊C8).291C22, (C2×C4×D7).267C22, (C2×C14).60(C22×C4), (C2×Dic7).80(C2×C4), (C22×D7).51(C2×C4), (C2×C4).747(C22×D7), SmallGroup(448,221)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C8⋊D7
G = < a,b,c,d | a4=b8=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Subgroups: 516 in 142 conjugacy classes, 83 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, M4(2), C22×C4, Dic7, Dic7, C28, D14, D14, C2×C14, C4×C8, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C7⋊C8, C56, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C4×M4(2), C8⋊D7, C2×C7⋊C8, C4×Dic7, C4×Dic7, C4×C28, C2×C56, C2×C4×D7, C2×C4×D7, C4×C7⋊C8, C56⋊C4, C4×C56, D7×C42, C2×C8⋊D7, C4×C8⋊D7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C42, M4(2), C22×C4, D14, C2×C42, C2×M4(2), C4×D7, C22×D7, C4×M4(2), C8⋊D7, C2×C4×D7, D7×C42, C2×C8⋊D7, C4×C8⋊D7
►On 224 points
Generators in S224
(1 137 18 165)(2 138 19 166)(3 139 20 167)(4 140 21 168)(5 141 22 161)(6 142 23 162)(7 143 24 163)(8 144 17 164)(9 95 147 157)(10 96 148 158)(11 89 149 159)(12 90 150 160)(13 91 151 153)(14 92 152 154)(15 93 145 155)(16 94 146 156)(25 113 191 98)(26 114 192 99)(27 115 185 100)(28 116 186 101)(29 117 187 102)(30 118 188 103)(31 119 189 104)(32 120 190 97)(33 177 136 43)(34 178 129 44)(35 179 130 45)(36 180 131 46)(37 181 132 47)(38 182 133 48)(39 183 134 41)(40 184 135 42)(49 57 174 202)(50 58 175 203)(51 59 176 204)(52 60 169 205)(53 61 170 206)(54 62 171 207)(55 63 172 208)(56 64 173 201)(65 224 112 213)(66 217 105 214)(67 218 106 215)(68 219 107 216)(69 220 108 209)(70 221 109 210)(71 222 110 211)(72 223 111 212)(73 85 124 200)(74 86 125 193)(75 87 126 194)(76 88 127 195)(77 81 128 196)(78 82 121 197)(79 83 122 198)(80 84 123 199)
(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 66 99 176 123 41 13)(2 67 100 169 124 42 14)(3 68 101 170 125 43 15)(4 69 102 171 126 44 16)(5 70 103 172 127 45 9)(6 71 104 173 128 46 10)(7 72 97 174 121 47 11)(8 65 98 175 122 48 12)(17 112 113 50 79 182 150)(18 105 114 51 80 183 151)(19 106 115 52 73 184 152)(20 107 116 53 74 177 145)(21 108 117 54 75 178 146)(22 109 118 55 76 179 147)(23 110 119 56 77 180 148)(24 111 120 49 78 181 149)(25 203 198 38 90 144 224)(26 204 199 39 91 137 217)(27 205 200 40 92 138 218)(28 206 193 33 93 139 219)(29 207 194 34 94 140 220)(30 208 195 35 95 141 221)(31 201 196 36 96 142 222)(32 202 197 37 89 143 223)(57 82 132 159 163 212 190)(58 83 133 160 164 213 191)(59 84 134 153 165 214 192)(60 85 135 154 166 215 185)(61 86 136 155 167 216 186)(62 87 129 156 168 209 187)(63 88 130 157 161 210 188)(64 81 131 158 162 211 189)
(1 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)(17 146)(18 151)(19 148)(20 145)(21 150)(22 147)(23 152)(24 149)(25 194)(26 199)(27 196)(28 193)(29 198)(30 195)(31 200)(32 197)(33 219)(34 224)(35 221)(36 218)(37 223)(38 220)(39 217)(40 222)(41 66)(42 71)(43 68)(44 65)(45 70)(46 67)(47 72)(48 69)(50 54)(52 56)(58 62)(60 64)(73 119)(74 116)(75 113)(76 118)(77 115)(78 120)(79 117)(80 114)(81 185)(82 190)(83 187)(84 192)(85 189)(86 186)(87 191)(88 188)(89 143)(90 140)(91 137)(92 142)(93 139)(94 144)(95 141)(96 138)(97 121)(98 126)(99 123)(100 128)(101 125)(102 122)(103 127)(104 124)(105 183)(106 180)(107 177)(108 182)(109 179)(110 184)(111 181)(112 178)(129 213)(130 210)(131 215)(132 212)(133 209)(134 214)(135 211)(136 216)(153 165)(154 162)(155 167)(156 164)(157 161)(158 166)(159 163)(160 168)(169 173)(171 175)(201 205)(203 207)
G:=sub<Sym(224)| (1,137,18,165)(2,138,19,166)(3,139,20,167)(4,140,21,168)(5,141,22,161)(6,142,23,162)(7,143,24,163)(8,144,17,164)(9,95,147,157)(10,96,148,158)(11,89,149,159)(12,90,150,160)(13,91,151,153)(14,92,152,154)(15,93,145,155)(16,94,146,156)(25,113,191,98)(26,114,192,99)(27,115,185,100)(28,116,186,101)(29,117,187,102)(30,118,188,103)(31,119,189,104)(32,120,190,97)(33,177,136,43)(34,178,129,44)(35,179,130,45)(36,180,131,46)(37,181,132,47)(38,182,133,48)(39,183,134,41)(40,184,135,42)(49,57,174,202)(50,58,175,203)(51,59,176,204)(52,60,169,205)(53,61,170,206)(54,62,171,207)(55,63,172,208)(56,64,173,201)(65,224,112,213)(66,217,105,214)(67,218,106,215)(68,219,107,216)(69,220,108,209)(70,221,109,210)(71,222,110,211)(72,223,111,212)(73,85,124,200)(74,86,125,193)(75,87,126,194)(76,88,127,195)(77,81,128,196)(78,82,121,197)(79,83,122,198)(80,84,123,199), (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), (1,66,99,176,123,41,13)(2,67,100,169,124,42,14)(3,68,101,170,125,43,15)(4,69,102,171,126,44,16)(5,70,103,172,127,45,9)(6,71,104,173,128,46,10)(7,72,97,174,121,47,11)(8,65,98,175,122,48,12)(17,112,113,50,79,182,150)(18,105,114,51,80,183,151)(19,106,115,52,73,184,152)(20,107,116,53,74,177,145)(21,108,117,54,75,178,146)(22,109,118,55,76,179,147)(23,110,119,56,77,180,148)(24,111,120,49,78,181,149)(25,203,198,38,90,144,224)(26,204,199,39,91,137,217)(27,205,200,40,92,138,218)(28,206,193,33,93,139,219)(29,207,194,34,94,140,220)(30,208,195,35,95,141,221)(31,201,196,36,96,142,222)(32,202,197,37,89,143,223)(57,82,132,159,163,212,190)(58,83,133,160,164,213,191)(59,84,134,153,165,214,192)(60,85,135,154,166,215,185)(61,86,136,155,167,216,186)(62,87,129,156,168,209,187)(63,88,130,157,161,210,188)(64,81,131,158,162,211,189), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16)(17,146)(18,151)(19,148)(20,145)(21,150)(22,147)(23,152)(24,149)(25,194)(26,199)(27,196)(28,193)(29,198)(30,195)(31,200)(32,197)(33,219)(34,224)(35,221)(36,218)(37,223)(38,220)(39,217)(40,222)(41,66)(42,71)(43,68)(44,65)(45,70)(46,67)(47,72)(48,69)(50,54)(52,56)(58,62)(60,64)(73,119)(74,116)(75,113)(76,118)(77,115)(78,120)(79,117)(80,114)(81,185)(82,190)(83,187)(84,192)(85,189)(86,186)(87,191)(88,188)(89,143)(90,140)(91,137)(92,142)(93,139)(94,144)(95,141)(96,138)(97,121)(98,126)(99,123)(100,128)(101,125)(102,122)(103,127)(104,124)(105,183)(106,180)(107,177)(108,182)(109,179)(110,184)(111,181)(112,178)(129,213)(130,210)(131,215)(132,212)(133,209)(134,214)(135,211)(136,216)(153,165)(154,162)(155,167)(156,164)(157,161)(158,166)(159,163)(160,168)(169,173)(171,175)(201,205)(203,207)>;
G:=Group( (1,137,18,165)(2,138,19,166)(3,139,20,167)(4,140,21,168)(5,141,22,161)(6,142,23,162)(7,143,24,163)(8,144,17,164)(9,95,147,157)(10,96,148,158)(11,89,149,159)(12,90,150,160)(13,91,151,153)(14,92,152,154)(15,93,145,155)(16,94,146,156)(25,113,191,98)(26,114,192,99)(27,115,185,100)(28,116,186,101)(29,117,187,102)(30,118,188,103)(31,119,189,104)(32,120,190,97)(33,177,136,43)(34,178,129,44)(35,179,130,45)(36,180,131,46)(37,181,132,47)(38,182,133,48)(39,183,134,41)(40,184,135,42)(49,57,174,202)(50,58,175,203)(51,59,176,204)(52,60,169,205)(53,61,170,206)(54,62,171,207)(55,63,172,208)(56,64,173,201)(65,224,112,213)(66,217,105,214)(67,218,106,215)(68,219,107,216)(69,220,108,209)(70,221,109,210)(71,222,110,211)(72,223,111,212)(73,85,124,200)(74,86,125,193)(75,87,126,194)(76,88,127,195)(77,81,128,196)(78,82,121,197)(79,83,122,198)(80,84,123,199), (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), (1,66,99,176,123,41,13)(2,67,100,169,124,42,14)(3,68,101,170,125,43,15)(4,69,102,171,126,44,16)(5,70,103,172,127,45,9)(6,71,104,173,128,46,10)(7,72,97,174,121,47,11)(8,65,98,175,122,48,12)(17,112,113,50,79,182,150)(18,105,114,51,80,183,151)(19,106,115,52,73,184,152)(20,107,116,53,74,177,145)(21,108,117,54,75,178,146)(22,109,118,55,76,179,147)(23,110,119,56,77,180,148)(24,111,120,49,78,181,149)(25,203,198,38,90,144,224)(26,204,199,39,91,137,217)(27,205,200,40,92,138,218)(28,206,193,33,93,139,219)(29,207,194,34,94,140,220)(30,208,195,35,95,141,221)(31,201,196,36,96,142,222)(32,202,197,37,89,143,223)(57,82,132,159,163,212,190)(58,83,133,160,164,213,191)(59,84,134,153,165,214,192)(60,85,135,154,166,215,185)(61,86,136,155,167,216,186)(62,87,129,156,168,209,187)(63,88,130,157,161,210,188)(64,81,131,158,162,211,189), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16)(17,146)(18,151)(19,148)(20,145)(21,150)(22,147)(23,152)(24,149)(25,194)(26,199)(27,196)(28,193)(29,198)(30,195)(31,200)(32,197)(33,219)(34,224)(35,221)(36,218)(37,223)(38,220)(39,217)(40,222)(41,66)(42,71)(43,68)(44,65)(45,70)(46,67)(47,72)(48,69)(50,54)(52,56)(58,62)(60,64)(73,119)(74,116)(75,113)(76,118)(77,115)(78,120)(79,117)(80,114)(81,185)(82,190)(83,187)(84,192)(85,189)(86,186)(87,191)(88,188)(89,143)(90,140)(91,137)(92,142)(93,139)(94,144)(95,141)(96,138)(97,121)(98,126)(99,123)(100,128)(101,125)(102,122)(103,127)(104,124)(105,183)(106,180)(107,177)(108,182)(109,179)(110,184)(111,181)(112,178)(129,213)(130,210)(131,215)(132,212)(133,209)(134,214)(135,211)(136,216)(153,165)(154,162)(155,167)(156,164)(157,161)(158,166)(159,163)(160,168)(169,173)(171,175)(201,205)(203,207) );
G=PermutationGroup([[(1,137,18,165),(2,138,19,166),(3,139,20,167),(4,140,21,168),(5,141,22,161),(6,142,23,162),(7,143,24,163),(8,144,17,164),(9,95,147,157),(10,96,148,158),(11,89,149,159),(12,90,150,160),(13,91,151,153),(14,92,152,154),(15,93,145,155),(16,94,146,156),(25,113,191,98),(26,114,192,99),(27,115,185,100),(28,116,186,101),(29,117,187,102),(30,118,188,103),(31,119,189,104),(32,120,190,97),(33,177,136,43),(34,178,129,44),(35,179,130,45),(36,180,131,46),(37,181,132,47),(38,182,133,48),(39,183,134,41),(40,184,135,42),(49,57,174,202),(50,58,175,203),(51,59,176,204),(52,60,169,205),(53,61,170,206),(54,62,171,207),(55,63,172,208),(56,64,173,201),(65,224,112,213),(66,217,105,214),(67,218,106,215),(68,219,107,216),(69,220,108,209),(70,221,109,210),(71,222,110,211),(72,223,111,212),(73,85,124,200),(74,86,125,193),(75,87,126,194),(76,88,127,195),(77,81,128,196),(78,82,121,197),(79,83,122,198),(80,84,123,199)], [(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)], [(1,66,99,176,123,41,13),(2,67,100,169,124,42,14),(3,68,101,170,125,43,15),(4,69,102,171,126,44,16),(5,70,103,172,127,45,9),(6,71,104,173,128,46,10),(7,72,97,174,121,47,11),(8,65,98,175,122,48,12),(17,112,113,50,79,182,150),(18,105,114,51,80,183,151),(19,106,115,52,73,184,152),(20,107,116,53,74,177,145),(21,108,117,54,75,178,146),(22,109,118,55,76,179,147),(23,110,119,56,77,180,148),(24,111,120,49,78,181,149),(25,203,198,38,90,144,224),(26,204,199,39,91,137,217),(27,205,200,40,92,138,218),(28,206,193,33,93,139,219),(29,207,194,34,94,140,220),(30,208,195,35,95,141,221),(31,201,196,36,96,142,222),(32,202,197,37,89,143,223),(57,82,132,159,163,212,190),(58,83,133,160,164,213,191),(59,84,134,153,165,214,192),(60,85,135,154,166,215,185),(61,86,136,155,167,216,186),(62,87,129,156,168,209,187),(63,88,130,157,161,210,188),(64,81,131,158,162,211,189)], [(1,13),(2,10),(3,15),(4,12),(5,9),(6,14),(7,11),(8,16),(17,146),(18,151),(19,148),(20,145),(21,150),(22,147),(23,152),(24,149),(25,194),(26,199),(27,196),(28,193),(29,198),(30,195),(31,200),(32,197),(33,219),(34,224),(35,221),(36,218),(37,223),(38,220),(39,217),(40,222),(41,66),(42,71),(43,68),(44,65),(45,70),(46,67),(47,72),(48,69),(50,54),(52,56),(58,62),(60,64),(73,119),(74,116),(75,113),(76,118),(77,115),(78,120),(79,117),(80,114),(81,185),(82,190),(83,187),(84,192),(85,189),(86,186),(87,191),(88,188),(89,143),(90,140),(91,137),(92,142),(93,139),(94,144),(95,141),(96,138),(97,121),(98,126),(99,123),(100,128),(101,125),(102,122),(103,127),(104,124),(105,183),(106,180),(107,177),(108,182),(109,179),(110,184),(111,181),(112,178),(129,213),(130,210),(131,215),(132,212),(133,209),(134,214),(135,211),(136,216),(153,165),(154,162),(155,167),(156,164),(157,161),(158,166),(159,163),(160,168),(169,173),(171,175),(201,205),(203,207)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4R | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 14A | ··· | 14I | 28A | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 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 | 14 | 14 | 1 | ··· | 1 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D7 | M4(2) | D14 | D14 | C4×D7 | C4×D7 | C8⋊D7 |
| kernel | C4×C8⋊D7 | C4×C7⋊C8 | C56⋊C4 | C4×C56 | D7×C42 | C2×C8⋊D7 | C8⋊D7 | C4×Dic7 | C2×C4×D7 | C4×C8 | C28 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 16 | 4 | 4 | 3 | 8 | 3 | 6 | 24 | 12 | 48 |
Matrix representation of C4×C8⋊D7 ►in GL4(𝔽113) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 66 | 100 |
| 0 | 0 | 32 | 47 |
| 0 | 112 | 0 | 0 |
| 1 | 79 | 0 | 0 |
| 0 | 0 | 104 | 112 |
| 0 | 0 | 72 | 33 |
| 79 | 25 | 0 | 0 |
| 112 | 34 | 0 | 0 |
| 0 | 0 | 33 | 103 |
| 0 | 0 | 41 | 80 |
G:=sub<GL(4,GF(113))| [15,0,0,0,0,15,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,66,32,0,0,100,47],[0,1,0,0,112,79,0,0,0,0,104,72,0,0,112,33],[79,112,0,0,25,34,0,0,0,0,33,41,0,0,103,80] >;
C4×C8⋊D7 in GAP, Magma, Sage, TeX
C_4\times C_8\rtimes D_7
% in TeX
G:=Group("C4xC8:D7"); // GroupNames label
G:=SmallGroup(448,221);
// by ID
G=gap.SmallGroup(448,221);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,758,58,136,18822]);
// 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,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations