direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×C4⋊C8, C42.199D14, D14.5M4(2), C4⋊3(C8×D7), C28⋊1(C2×C8), (C4×D7)⋊1C8, C4.54(Q8×D7), Dic7⋊3(C2×C8), (C4×D7).53D4, D14.8(C2×C8), C28⋊C8⋊11C2, C4.204(D4×D7), (C4×D7).10Q8, Dic7⋊C8⋊22C2, D14.9(C4⋊C4), (C2×C8).214D14, C28.363(C2×D4), C14.9(C22×C8), C28.112(C2×Q8), (C4×Dic7).5C4, (D7×C42).1C2, C2.5(D7×M4(2)), (C4×C28).58C22, Dic7.10(C4⋊C4), (C2×C56).208C22, (C2×C28).829C23, C14.25(C2×M4(2)), (C4×Dic7).275C22, C7⋊1(C2×C4⋊C8), (C7×C4⋊C8)⋊13C2, C2.3(D7×C4⋊C4), (C2×C4×D7).5C4, C2.11(D7×C2×C8), C14.7(C2×C4⋊C4), (D7×C2×C8).13C2, C22.46(C2×C4×D7), (C2×C28).68(C2×C4), (C2×C4).144(C4×D7), (C2×C7⋊C8).304C22, (C2×C4×D7).307C22, (C2×C14).84(C22×C4), (C2×Dic7).87(C2×C4), (C22×D7).72(C2×C4), (C2×C4).771(C22×D7), SmallGroup(448,366)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×C4⋊C8
G = < a,b,c,d | a7=b2=c4=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 516 in 138 conjugacy classes, 73 normal (37 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, D7, C14, C42, C42, C2×C8, C2×C8, C22×C4, Dic7, Dic7, C28, C28, C28, D14, C2×C14, C4⋊C8, C4⋊C8, C2×C42, C22×C8, C7⋊C8, C56, C4×D7, C4×D7, C2×Dic7, C2×C28, C22×D7, C2×C4⋊C8, C8×D7, C2×C7⋊C8, C4×Dic7, C4×C28, C2×C56, C2×C4×D7, C28⋊C8, Dic7⋊C8, C7×C4⋊C8, D7×C42, D7×C2×C8, D7×C4⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C23, D7, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, D14, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), C4×D7, C22×D7, C2×C4⋊C8, C8×D7, C2×C4×D7, D4×D7, Q8×D7, D7×C4⋊C4, D7×C2×C8, D7×M4(2), D7×C4⋊C8
►On 224 points
Generators in S224
(1 13 45 179 176 99 74)(2 14 46 180 169 100 75)(3 15 47 181 170 101 76)(4 16 48 182 171 102 77)(5 9 41 183 172 103 78)(6 10 42 184 173 104 79)(7 11 43 177 174 97 80)(8 12 44 178 175 98 73)(17 54 144 106 70 166 135)(18 55 137 107 71 167 136)(19 56 138 108 72 168 129)(20 49 139 109 65 161 130)(21 50 140 110 66 162 131)(22 51 141 111 67 163 132)(23 52 142 112 68 164 133)(24 53 143 105 69 165 134)(25 152 209 160 118 199 36)(26 145 210 153 119 200 37)(27 146 211 154 120 193 38)(28 147 212 155 113 194 39)(29 148 213 156 114 195 40)(30 149 214 157 115 196 33)(31 150 215 158 116 197 34)(32 151 216 159 117 198 35)(57 192 125 87 222 201 96)(58 185 126 88 223 202 89)(59 186 127 81 224 203 90)(60 187 128 82 217 204 91)(61 188 121 83 218 205 92)(62 189 122 84 219 206 93)(63 190 123 85 220 207 94)(64 191 124 86 221 208 95)
(1 109)(2 110)(3 111)(4 112)(5 105)(6 106)(7 107)(8 108)(9 143)(10 144)(11 137)(12 138)(13 139)(14 140)(15 141)(16 142)(17 184)(18 177)(19 178)(20 179)(21 180)(22 181)(23 182)(24 183)(25 57)(26 58)(27 59)(28 60)(29 61)(30 62)(31 63)(32 64)(33 189)(34 190)(35 191)(36 192)(37 185)(38 186)(39 187)(40 188)(41 53)(42 54)(43 55)(44 56)(45 49)(46 50)(47 51)(48 52)(65 74)(66 75)(67 76)(68 77)(69 78)(70 79)(71 80)(72 73)(81 120)(82 113)(83 114)(84 115)(85 116)(86 117)(87 118)(88 119)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 167)(98 168)(99 161)(100 162)(101 163)(102 164)(103 165)(104 166)(121 195)(122 196)(123 197)(124 198)(125 199)(126 200)(127 193)(128 194)(129 175)(130 176)(131 169)(132 170)(133 171)(134 172)(135 173)(136 174)(153 223)(154 224)(155 217)(156 218)(157 219)(158 220)(159 221)(160 222)(201 209)(202 210)(203 211)(204 212)(205 213)(206 214)(207 215)(208 216)
(1 155 69 208)(2 201 70 156)(3 157 71 202)(4 203 72 158)(5 159 65 204)(6 205 66 160)(7 153 67 206)(8 207 68 154)(9 117 161 91)(10 92 162 118)(11 119 163 93)(12 94 164 120)(13 113 165 95)(14 96 166 114)(15 115 167 89)(16 90 168 116)(17 40 180 192)(18 185 181 33)(19 34 182 186)(20 187 183 35)(21 36 184 188)(22 189 177 37)(23 38 178 190)(24 191 179 39)(25 173 121 50)(26 51 122 174)(27 175 123 52)(28 53 124 176)(29 169 125 54)(30 55 126 170)(31 171 127 56)(32 49 128 172)(41 198 130 60)(42 61 131 199)(43 200 132 62)(44 63 133 193)(45 194 134 64)(46 57 135 195)(47 196 136 58)(48 59 129 197)(73 220 112 211)(74 212 105 221)(75 222 106 213)(76 214 107 223)(77 224 108 215)(78 216 109 217)(79 218 110 209)(80 210 111 219)(81 138 150 102)(82 103 151 139)(83 140 152 104)(84 97 145 141)(85 142 146 98)(86 99 147 143)(87 144 148 100)(88 101 149 137)
(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)
G:=sub<Sym(224)| (1,13,45,179,176,99,74)(2,14,46,180,169,100,75)(3,15,47,181,170,101,76)(4,16,48,182,171,102,77)(5,9,41,183,172,103,78)(6,10,42,184,173,104,79)(7,11,43,177,174,97,80)(8,12,44,178,175,98,73)(17,54,144,106,70,166,135)(18,55,137,107,71,167,136)(19,56,138,108,72,168,129)(20,49,139,109,65,161,130)(21,50,140,110,66,162,131)(22,51,141,111,67,163,132)(23,52,142,112,68,164,133)(24,53,143,105,69,165,134)(25,152,209,160,118,199,36)(26,145,210,153,119,200,37)(27,146,211,154,120,193,38)(28,147,212,155,113,194,39)(29,148,213,156,114,195,40)(30,149,214,157,115,196,33)(31,150,215,158,116,197,34)(32,151,216,159,117,198,35)(57,192,125,87,222,201,96)(58,185,126,88,223,202,89)(59,186,127,81,224,203,90)(60,187,128,82,217,204,91)(61,188,121,83,218,205,92)(62,189,122,84,219,206,93)(63,190,123,85,220,207,94)(64,191,124,86,221,208,95), (1,109)(2,110)(3,111)(4,112)(5,105)(6,106)(7,107)(8,108)(9,143)(10,144)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,184)(18,177)(19,178)(20,179)(21,180)(22,181)(23,182)(24,183)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,189)(34,190)(35,191)(36,192)(37,185)(38,186)(39,187)(40,188)(41,53)(42,54)(43,55)(44,56)(45,49)(46,50)(47,51)(48,52)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(71,80)(72,73)(81,120)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,167)(98,168)(99,161)(100,162)(101,163)(102,164)(103,165)(104,166)(121,195)(122,196)(123,197)(124,198)(125,199)(126,200)(127,193)(128,194)(129,175)(130,176)(131,169)(132,170)(133,171)(134,172)(135,173)(136,174)(153,223)(154,224)(155,217)(156,218)(157,219)(158,220)(159,221)(160,222)(201,209)(202,210)(203,211)(204,212)(205,213)(206,214)(207,215)(208,216), (1,155,69,208)(2,201,70,156)(3,157,71,202)(4,203,72,158)(5,159,65,204)(6,205,66,160)(7,153,67,206)(8,207,68,154)(9,117,161,91)(10,92,162,118)(11,119,163,93)(12,94,164,120)(13,113,165,95)(14,96,166,114)(15,115,167,89)(16,90,168,116)(17,40,180,192)(18,185,181,33)(19,34,182,186)(20,187,183,35)(21,36,184,188)(22,189,177,37)(23,38,178,190)(24,191,179,39)(25,173,121,50)(26,51,122,174)(27,175,123,52)(28,53,124,176)(29,169,125,54)(30,55,126,170)(31,171,127,56)(32,49,128,172)(41,198,130,60)(42,61,131,199)(43,200,132,62)(44,63,133,193)(45,194,134,64)(46,57,135,195)(47,196,136,58)(48,59,129,197)(73,220,112,211)(74,212,105,221)(75,222,106,213)(76,214,107,223)(77,224,108,215)(78,216,109,217)(79,218,110,209)(80,210,111,219)(81,138,150,102)(82,103,151,139)(83,140,152,104)(84,97,145,141)(85,142,146,98)(86,99,147,143)(87,144,148,100)(88,101,149,137), (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)>;
G:=Group( (1,13,45,179,176,99,74)(2,14,46,180,169,100,75)(3,15,47,181,170,101,76)(4,16,48,182,171,102,77)(5,9,41,183,172,103,78)(6,10,42,184,173,104,79)(7,11,43,177,174,97,80)(8,12,44,178,175,98,73)(17,54,144,106,70,166,135)(18,55,137,107,71,167,136)(19,56,138,108,72,168,129)(20,49,139,109,65,161,130)(21,50,140,110,66,162,131)(22,51,141,111,67,163,132)(23,52,142,112,68,164,133)(24,53,143,105,69,165,134)(25,152,209,160,118,199,36)(26,145,210,153,119,200,37)(27,146,211,154,120,193,38)(28,147,212,155,113,194,39)(29,148,213,156,114,195,40)(30,149,214,157,115,196,33)(31,150,215,158,116,197,34)(32,151,216,159,117,198,35)(57,192,125,87,222,201,96)(58,185,126,88,223,202,89)(59,186,127,81,224,203,90)(60,187,128,82,217,204,91)(61,188,121,83,218,205,92)(62,189,122,84,219,206,93)(63,190,123,85,220,207,94)(64,191,124,86,221,208,95), (1,109)(2,110)(3,111)(4,112)(5,105)(6,106)(7,107)(8,108)(9,143)(10,144)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,184)(18,177)(19,178)(20,179)(21,180)(22,181)(23,182)(24,183)(25,57)(26,58)(27,59)(28,60)(29,61)(30,62)(31,63)(32,64)(33,189)(34,190)(35,191)(36,192)(37,185)(38,186)(39,187)(40,188)(41,53)(42,54)(43,55)(44,56)(45,49)(46,50)(47,51)(48,52)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(71,80)(72,73)(81,120)(82,113)(83,114)(84,115)(85,116)(86,117)(87,118)(88,119)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,167)(98,168)(99,161)(100,162)(101,163)(102,164)(103,165)(104,166)(121,195)(122,196)(123,197)(124,198)(125,199)(126,200)(127,193)(128,194)(129,175)(130,176)(131,169)(132,170)(133,171)(134,172)(135,173)(136,174)(153,223)(154,224)(155,217)(156,218)(157,219)(158,220)(159,221)(160,222)(201,209)(202,210)(203,211)(204,212)(205,213)(206,214)(207,215)(208,216), (1,155,69,208)(2,201,70,156)(3,157,71,202)(4,203,72,158)(5,159,65,204)(6,205,66,160)(7,153,67,206)(8,207,68,154)(9,117,161,91)(10,92,162,118)(11,119,163,93)(12,94,164,120)(13,113,165,95)(14,96,166,114)(15,115,167,89)(16,90,168,116)(17,40,180,192)(18,185,181,33)(19,34,182,186)(20,187,183,35)(21,36,184,188)(22,189,177,37)(23,38,178,190)(24,191,179,39)(25,173,121,50)(26,51,122,174)(27,175,123,52)(28,53,124,176)(29,169,125,54)(30,55,126,170)(31,171,127,56)(32,49,128,172)(41,198,130,60)(42,61,131,199)(43,200,132,62)(44,63,133,193)(45,194,134,64)(46,57,135,195)(47,196,136,58)(48,59,129,197)(73,220,112,211)(74,212,105,221)(75,222,106,213)(76,214,107,223)(77,224,108,215)(78,216,109,217)(79,218,110,209)(80,210,111,219)(81,138,150,102)(82,103,151,139)(83,140,152,104)(84,97,145,141)(85,142,146,98)(86,99,147,143)(87,144,148,100)(88,101,149,137), (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) );
G=PermutationGroup([[(1,13,45,179,176,99,74),(2,14,46,180,169,100,75),(3,15,47,181,170,101,76),(4,16,48,182,171,102,77),(5,9,41,183,172,103,78),(6,10,42,184,173,104,79),(7,11,43,177,174,97,80),(8,12,44,178,175,98,73),(17,54,144,106,70,166,135),(18,55,137,107,71,167,136),(19,56,138,108,72,168,129),(20,49,139,109,65,161,130),(21,50,140,110,66,162,131),(22,51,141,111,67,163,132),(23,52,142,112,68,164,133),(24,53,143,105,69,165,134),(25,152,209,160,118,199,36),(26,145,210,153,119,200,37),(27,146,211,154,120,193,38),(28,147,212,155,113,194,39),(29,148,213,156,114,195,40),(30,149,214,157,115,196,33),(31,150,215,158,116,197,34),(32,151,216,159,117,198,35),(57,192,125,87,222,201,96),(58,185,126,88,223,202,89),(59,186,127,81,224,203,90),(60,187,128,82,217,204,91),(61,188,121,83,218,205,92),(62,189,122,84,219,206,93),(63,190,123,85,220,207,94),(64,191,124,86,221,208,95)], [(1,109),(2,110),(3,111),(4,112),(5,105),(6,106),(7,107),(8,108),(9,143),(10,144),(11,137),(12,138),(13,139),(14,140),(15,141),(16,142),(17,184),(18,177),(19,178),(20,179),(21,180),(22,181),(23,182),(24,183),(25,57),(26,58),(27,59),(28,60),(29,61),(30,62),(31,63),(32,64),(33,189),(34,190),(35,191),(36,192),(37,185),(38,186),(39,187),(40,188),(41,53),(42,54),(43,55),(44,56),(45,49),(46,50),(47,51),(48,52),(65,74),(66,75),(67,76),(68,77),(69,78),(70,79),(71,80),(72,73),(81,120),(82,113),(83,114),(84,115),(85,116),(86,117),(87,118),(88,119),(89,145),(90,146),(91,147),(92,148),(93,149),(94,150),(95,151),(96,152),(97,167),(98,168),(99,161),(100,162),(101,163),(102,164),(103,165),(104,166),(121,195),(122,196),(123,197),(124,198),(125,199),(126,200),(127,193),(128,194),(129,175),(130,176),(131,169),(132,170),(133,171),(134,172),(135,173),(136,174),(153,223),(154,224),(155,217),(156,218),(157,219),(158,220),(159,221),(160,222),(201,209),(202,210),(203,211),(204,212),(205,213),(206,214),(207,215),(208,216)], [(1,155,69,208),(2,201,70,156),(3,157,71,202),(4,203,72,158),(5,159,65,204),(6,205,66,160),(7,153,67,206),(8,207,68,154),(9,117,161,91),(10,92,162,118),(11,119,163,93),(12,94,164,120),(13,113,165,95),(14,96,166,114),(15,115,167,89),(16,90,168,116),(17,40,180,192),(18,185,181,33),(19,34,182,186),(20,187,183,35),(21,36,184,188),(22,189,177,37),(23,38,178,190),(24,191,179,39),(25,173,121,50),(26,51,122,174),(27,175,123,52),(28,53,124,176),(29,169,125,54),(30,55,126,170),(31,171,127,56),(32,49,128,172),(41,198,130,60),(42,61,131,199),(43,200,132,62),(44,63,133,193),(45,194,134,64),(46,57,135,195),(47,196,136,58),(48,59,129,197),(73,220,112,211),(74,212,105,221),(75,222,106,213),(76,214,107,223),(77,224,108,215),(78,216,109,217),(79,218,110,209),(80,210,111,219),(81,138,150,102),(82,103,151,139),(83,140,152,104),(84,97,145,141),(85,142,146,98),(86,99,147,143),(87,144,148,100),(88,101,149,137)], [(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)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 14A | ··· | 14I | 28A | ··· | 28L | 28M | ··· | 28X | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 7 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D7 | M4(2) | D14 | D14 | C4×D7 | C8×D7 | D4×D7 | Q8×D7 | D7×M4(2) |
| kernel | D7×C4⋊C8 | C28⋊C8 | Dic7⋊C8 | C7×C4⋊C8 | D7×C42 | D7×C2×C8 | C4×Dic7 | C2×C4×D7 | C4×D7 | C4×D7 | C4×D7 | C4⋊C8 | D14 | C42 | C2×C8 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 16 | 2 | 2 | 3 | 4 | 3 | 6 | 12 | 24 | 3 | 3 | 6 |
Matrix representation of D7×C4⋊C8 ►in GL4(𝔽113) generated by
| 24 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 112 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 18 |
| 0 | 0 | 67 | 107 |
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 15 | 112 |
| 0 | 0 | 0 | 98 |
G:=sub<GL(4,GF(113))| [24,112,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,112,0,0,112,0,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,6,67,0,0,18,107],[18,0,0,0,0,18,0,0,0,0,15,0,0,0,112,98] >;
D7×C4⋊C8 in GAP, Magma, Sage, TeX
D_7\times C_4\rtimes C_8
% in TeX
G:=Group("D7xC4:C8"); // GroupNames label
G:=SmallGroup(448,366);
// by ID
G=gap.SmallGroup(448,366);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,58,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^4=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations