direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D8⋊C4, D8⋊3C28, C8⋊3(C2×C28), (C7×D8)⋊9C4, C56⋊21(C2×C4), D4⋊2(C2×C28), (C4×D4)⋊2C14, C4.Q8⋊3C14, C8⋊C4⋊2C14, (D4×C28)⋊31C2, (C2×D8).6C14, C2.17(D4×C28), (C14×D8).13C2, C14.119(C4×D4), (C2×C28).458D4, D4⋊C4⋊16C14, C4.14(C22×C28), C42.10(C2×C14), C22.56(D4×C14), C28.261(C4○D4), (C2×C28).909C23, (C4×C28).251C22, (C2×C56).331C22, C28.159(C22×C4), C14.130(C8⋊C22), (D4×C14).293C22, C4.6(C7×C4○D4), (C7×D4)⋊15(C2×C4), (C7×C8⋊C4)⋊11C2, (C7×C4.Q8)⋊12C2, C2.5(C7×C8⋊C22), C4⋊C4.50(C2×C14), (C2×C8).20(C2×C14), (C2×C4).104(C7×D4), (C7×D4⋊C4)⋊39C2, (C2×D4).51(C2×C14), (C2×C14).632(C2×D4), (C7×C4⋊C4).371C22, (C2×C4).84(C22×C14), SmallGroup(448,850)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D8⋊C4
G = < a,b,c,d | a7=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b5, dcd-1=b4c >
Subgroups: 250 in 132 conjugacy classes, 70 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C28, C28, C2×C14, C2×C14, C8⋊C4, D4⋊C4, C4.Q8, C4×D4, C2×D8, C56, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, D8⋊C4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C7×D8, C22×C28, D4×C14, C7×C8⋊C4, C7×D4⋊C4, C7×C4.Q8, D4×C28, C14×D8, C7×D8⋊C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22×C4, C2×D4, C4○D4, C28, C2×C14, C4×D4, C8⋊C22, C2×C28, C7×D4, C22×C14, D8⋊C4, C22×C28, D4×C14, C7×C4○D4, D4×C28, C7×C8⋊C22, C7×D8⋊C4
►On 224 points
Generators in S224
(1 115 60 133 75 125 67)(2 116 61 134 76 126 68)(3 117 62 135 77 127 69)(4 118 63 136 78 128 70)(5 119 64 129 79 121 71)(6 120 57 130 80 122 72)(7 113 58 131 73 123 65)(8 114 59 132 74 124 66)(9 162 194 25 186 17 178)(10 163 195 26 187 18 179)(11 164 196 27 188 19 180)(12 165 197 28 189 20 181)(13 166 198 29 190 21 182)(14 167 199 30 191 22 183)(15 168 200 31 192 23 184)(16 161 193 32 185 24 177)(33 176 218 49 210 41 202)(34 169 219 50 211 42 203)(35 170 220 51 212 43 204)(36 171 221 52 213 44 205)(37 172 222 53 214 45 206)(38 173 223 54 215 46 207)(39 174 224 55 216 47 208)(40 175 217 56 209 48 201)(81 105 155 97 147 89 139)(82 106 156 98 148 90 140)(83 107 157 99 149 91 141)(84 108 158 100 150 92 142)(85 109 159 101 151 93 143)(86 110 160 102 152 94 144)(87 111 153 103 145 95 137)(88 112 154 104 146 96 138)
(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 37)(2 36)(3 35)(4 34)(5 33)(6 40)(7 39)(8 38)(9 138)(10 137)(11 144)(12 143)(13 142)(14 141)(15 140)(16 139)(17 146)(18 145)(19 152)(20 151)(21 150)(22 149)(23 148)(24 147)(25 154)(26 153)(27 160)(28 159)(29 158)(30 157)(31 156)(32 155)(41 121)(42 128)(43 127)(44 126)(45 125)(46 124)(47 123)(48 122)(49 129)(50 136)(51 135)(52 134)(53 133)(54 132)(55 131)(56 130)(57 217)(58 224)(59 223)(60 222)(61 221)(62 220)(63 219)(64 218)(65 208)(66 207)(67 206)(68 205)(69 204)(70 203)(71 202)(72 201)(73 216)(74 215)(75 214)(76 213)(77 212)(78 211)(79 210)(80 209)(81 161)(82 168)(83 167)(84 166)(85 165)(86 164)(87 163)(88 162)(89 177)(90 184)(91 183)(92 182)(93 181)(94 180)(95 179)(96 178)(97 185)(98 192)(99 191)(100 190)(101 189)(102 188)(103 187)(104 186)(105 193)(106 200)(107 199)(108 198)(109 197)(110 196)(111 195)(112 194)(113 174)(114 173)(115 172)(116 171)(117 170)(118 169)(119 176)(120 175)
(1 84 34 163)(2 81 35 168)(3 86 36 165)(4 83 37 162)(5 88 38 167)(6 85 39 164)(7 82 40 161)(8 87 33 166)(9 70 141 206)(10 67 142 203)(11 72 143 208)(12 69 144 205)(13 66 137 202)(14 71 138 207)(15 68 139 204)(16 65 140 201)(17 78 149 214)(18 75 150 211)(19 80 151 216)(20 77 152 213)(21 74 145 210)(22 79 146 215)(23 76 147 212)(24 73 148 209)(25 63 157 222)(26 60 158 219)(27 57 159 224)(28 62 160 221)(29 59 153 218)(30 64 154 223)(31 61 155 220)(32 58 156 217)(41 182 124 95)(42 179 125 92)(43 184 126 89)(44 181 127 94)(45 178 128 91)(46 183 121 96)(47 180 122 93)(48 177 123 90)(49 190 132 103)(50 187 133 100)(51 192 134 97)(52 189 135 102)(53 186 136 99)(54 191 129 104)(55 188 130 101)(56 185 131 98)(105 170 200 116)(106 175 193 113)(107 172 194 118)(108 169 195 115)(109 174 196 120)(110 171 197 117)(111 176 198 114)(112 173 199 119)
G:=sub<Sym(224)| (1,115,60,133,75,125,67)(2,116,61,134,76,126,68)(3,117,62,135,77,127,69)(4,118,63,136,78,128,70)(5,119,64,129,79,121,71)(6,120,57,130,80,122,72)(7,113,58,131,73,123,65)(8,114,59,132,74,124,66)(9,162,194,25,186,17,178)(10,163,195,26,187,18,179)(11,164,196,27,188,19,180)(12,165,197,28,189,20,181)(13,166,198,29,190,21,182)(14,167,199,30,191,22,183)(15,168,200,31,192,23,184)(16,161,193,32,185,24,177)(33,176,218,49,210,41,202)(34,169,219,50,211,42,203)(35,170,220,51,212,43,204)(36,171,221,52,213,44,205)(37,172,222,53,214,45,206)(38,173,223,54,215,46,207)(39,174,224,55,216,47,208)(40,175,217,56,209,48,201)(81,105,155,97,147,89,139)(82,106,156,98,148,90,140)(83,107,157,99,149,91,141)(84,108,158,100,150,92,142)(85,109,159,101,151,93,143)(86,110,160,102,152,94,144)(87,111,153,103,145,95,137)(88,112,154,104,146,96,138), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,40)(7,39)(8,38)(9,138)(10,137)(11,144)(12,143)(13,142)(14,141)(15,140)(16,139)(17,146)(18,145)(19,152)(20,151)(21,150)(22,149)(23,148)(24,147)(25,154)(26,153)(27,160)(28,159)(29,158)(30,157)(31,156)(32,155)(41,121)(42,128)(43,127)(44,126)(45,125)(46,124)(47,123)(48,122)(49,129)(50,136)(51,135)(52,134)(53,133)(54,132)(55,131)(56,130)(57,217)(58,224)(59,223)(60,222)(61,221)(62,220)(63,219)(64,218)(65,208)(66,207)(67,206)(68,205)(69,204)(70,203)(71,202)(72,201)(73,216)(74,215)(75,214)(76,213)(77,212)(78,211)(79,210)(80,209)(81,161)(82,168)(83,167)(84,166)(85,165)(86,164)(87,163)(88,162)(89,177)(90,184)(91,183)(92,182)(93,181)(94,180)(95,179)(96,178)(97,185)(98,192)(99,191)(100,190)(101,189)(102,188)(103,187)(104,186)(105,193)(106,200)(107,199)(108,198)(109,197)(110,196)(111,195)(112,194)(113,174)(114,173)(115,172)(116,171)(117,170)(118,169)(119,176)(120,175), (1,84,34,163)(2,81,35,168)(3,86,36,165)(4,83,37,162)(5,88,38,167)(6,85,39,164)(7,82,40,161)(8,87,33,166)(9,70,141,206)(10,67,142,203)(11,72,143,208)(12,69,144,205)(13,66,137,202)(14,71,138,207)(15,68,139,204)(16,65,140,201)(17,78,149,214)(18,75,150,211)(19,80,151,216)(20,77,152,213)(21,74,145,210)(22,79,146,215)(23,76,147,212)(24,73,148,209)(25,63,157,222)(26,60,158,219)(27,57,159,224)(28,62,160,221)(29,59,153,218)(30,64,154,223)(31,61,155,220)(32,58,156,217)(41,182,124,95)(42,179,125,92)(43,184,126,89)(44,181,127,94)(45,178,128,91)(46,183,121,96)(47,180,122,93)(48,177,123,90)(49,190,132,103)(50,187,133,100)(51,192,134,97)(52,189,135,102)(53,186,136,99)(54,191,129,104)(55,188,130,101)(56,185,131,98)(105,170,200,116)(106,175,193,113)(107,172,194,118)(108,169,195,115)(109,174,196,120)(110,171,197,117)(111,176,198,114)(112,173,199,119)>;
G:=Group( (1,115,60,133,75,125,67)(2,116,61,134,76,126,68)(3,117,62,135,77,127,69)(4,118,63,136,78,128,70)(5,119,64,129,79,121,71)(6,120,57,130,80,122,72)(7,113,58,131,73,123,65)(8,114,59,132,74,124,66)(9,162,194,25,186,17,178)(10,163,195,26,187,18,179)(11,164,196,27,188,19,180)(12,165,197,28,189,20,181)(13,166,198,29,190,21,182)(14,167,199,30,191,22,183)(15,168,200,31,192,23,184)(16,161,193,32,185,24,177)(33,176,218,49,210,41,202)(34,169,219,50,211,42,203)(35,170,220,51,212,43,204)(36,171,221,52,213,44,205)(37,172,222,53,214,45,206)(38,173,223,54,215,46,207)(39,174,224,55,216,47,208)(40,175,217,56,209,48,201)(81,105,155,97,147,89,139)(82,106,156,98,148,90,140)(83,107,157,99,149,91,141)(84,108,158,100,150,92,142)(85,109,159,101,151,93,143)(86,110,160,102,152,94,144)(87,111,153,103,145,95,137)(88,112,154,104,146,96,138), (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,40)(7,39)(8,38)(9,138)(10,137)(11,144)(12,143)(13,142)(14,141)(15,140)(16,139)(17,146)(18,145)(19,152)(20,151)(21,150)(22,149)(23,148)(24,147)(25,154)(26,153)(27,160)(28,159)(29,158)(30,157)(31,156)(32,155)(41,121)(42,128)(43,127)(44,126)(45,125)(46,124)(47,123)(48,122)(49,129)(50,136)(51,135)(52,134)(53,133)(54,132)(55,131)(56,130)(57,217)(58,224)(59,223)(60,222)(61,221)(62,220)(63,219)(64,218)(65,208)(66,207)(67,206)(68,205)(69,204)(70,203)(71,202)(72,201)(73,216)(74,215)(75,214)(76,213)(77,212)(78,211)(79,210)(80,209)(81,161)(82,168)(83,167)(84,166)(85,165)(86,164)(87,163)(88,162)(89,177)(90,184)(91,183)(92,182)(93,181)(94,180)(95,179)(96,178)(97,185)(98,192)(99,191)(100,190)(101,189)(102,188)(103,187)(104,186)(105,193)(106,200)(107,199)(108,198)(109,197)(110,196)(111,195)(112,194)(113,174)(114,173)(115,172)(116,171)(117,170)(118,169)(119,176)(120,175), (1,84,34,163)(2,81,35,168)(3,86,36,165)(4,83,37,162)(5,88,38,167)(6,85,39,164)(7,82,40,161)(8,87,33,166)(9,70,141,206)(10,67,142,203)(11,72,143,208)(12,69,144,205)(13,66,137,202)(14,71,138,207)(15,68,139,204)(16,65,140,201)(17,78,149,214)(18,75,150,211)(19,80,151,216)(20,77,152,213)(21,74,145,210)(22,79,146,215)(23,76,147,212)(24,73,148,209)(25,63,157,222)(26,60,158,219)(27,57,159,224)(28,62,160,221)(29,59,153,218)(30,64,154,223)(31,61,155,220)(32,58,156,217)(41,182,124,95)(42,179,125,92)(43,184,126,89)(44,181,127,94)(45,178,128,91)(46,183,121,96)(47,180,122,93)(48,177,123,90)(49,190,132,103)(50,187,133,100)(51,192,134,97)(52,189,135,102)(53,186,136,99)(54,191,129,104)(55,188,130,101)(56,185,131,98)(105,170,200,116)(106,175,193,113)(107,172,194,118)(108,169,195,115)(109,174,196,120)(110,171,197,117)(111,176,198,114)(112,173,199,119) );
G=PermutationGroup([[(1,115,60,133,75,125,67),(2,116,61,134,76,126,68),(3,117,62,135,77,127,69),(4,118,63,136,78,128,70),(5,119,64,129,79,121,71),(6,120,57,130,80,122,72),(7,113,58,131,73,123,65),(8,114,59,132,74,124,66),(9,162,194,25,186,17,178),(10,163,195,26,187,18,179),(11,164,196,27,188,19,180),(12,165,197,28,189,20,181),(13,166,198,29,190,21,182),(14,167,199,30,191,22,183),(15,168,200,31,192,23,184),(16,161,193,32,185,24,177),(33,176,218,49,210,41,202),(34,169,219,50,211,42,203),(35,170,220,51,212,43,204),(36,171,221,52,213,44,205),(37,172,222,53,214,45,206),(38,173,223,54,215,46,207),(39,174,224,55,216,47,208),(40,175,217,56,209,48,201),(81,105,155,97,147,89,139),(82,106,156,98,148,90,140),(83,107,157,99,149,91,141),(84,108,158,100,150,92,142),(85,109,159,101,151,93,143),(86,110,160,102,152,94,144),(87,111,153,103,145,95,137),(88,112,154,104,146,96,138)], [(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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,40),(7,39),(8,38),(9,138),(10,137),(11,144),(12,143),(13,142),(14,141),(15,140),(16,139),(17,146),(18,145),(19,152),(20,151),(21,150),(22,149),(23,148),(24,147),(25,154),(26,153),(27,160),(28,159),(29,158),(30,157),(31,156),(32,155),(41,121),(42,128),(43,127),(44,126),(45,125),(46,124),(47,123),(48,122),(49,129),(50,136),(51,135),(52,134),(53,133),(54,132),(55,131),(56,130),(57,217),(58,224),(59,223),(60,222),(61,221),(62,220),(63,219),(64,218),(65,208),(66,207),(67,206),(68,205),(69,204),(70,203),(71,202),(72,201),(73,216),(74,215),(75,214),(76,213),(77,212),(78,211),(79,210),(80,209),(81,161),(82,168),(83,167),(84,166),(85,165),(86,164),(87,163),(88,162),(89,177),(90,184),(91,183),(92,182),(93,181),(94,180),(95,179),(96,178),(97,185),(98,192),(99,191),(100,190),(101,189),(102,188),(103,187),(104,186),(105,193),(106,200),(107,199),(108,198),(109,197),(110,196),(111,195),(112,194),(113,174),(114,173),(115,172),(116,171),(117,170),(118,169),(119,176),(120,175)], [(1,84,34,163),(2,81,35,168),(3,86,36,165),(4,83,37,162),(5,88,38,167),(6,85,39,164),(7,82,40,161),(8,87,33,166),(9,70,141,206),(10,67,142,203),(11,72,143,208),(12,69,144,205),(13,66,137,202),(14,71,138,207),(15,68,139,204),(16,65,140,201),(17,78,149,214),(18,75,150,211),(19,80,151,216),(20,77,152,213),(21,74,145,210),(22,79,146,215),(23,76,147,212),(24,73,148,209),(25,63,157,222),(26,60,158,219),(27,57,159,224),(28,62,160,221),(29,59,153,218),(30,64,154,223),(31,61,155,220),(32,58,156,217),(41,182,124,95),(42,179,125,92),(43,184,126,89),(44,181,127,94),(45,178,128,91),(46,183,121,96),(47,180,122,93),(48,177,123,90),(49,190,132,103),(50,187,133,100),(51,192,134,97),(52,189,135,102),(53,186,136,99),(54,191,129,104),(55,188,130,101),(56,185,131,98),(105,170,200,116),(106,175,193,113),(107,172,194,118),(108,169,195,115),(109,174,196,120),(110,171,197,117),(111,176,198,114),(112,173,199,119)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AP | 28A | ··· | 28AJ | 28AK | ··· | 28BH | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C14 | C28 | D4 | C4○D4 | C7×D4 | C7×C4○D4 | C8⋊C22 | C7×C8⋊C22 |
| kernel | C7×D8⋊C4 | C7×C8⋊C4 | C7×D4⋊C4 | C7×C4.Q8 | D4×C28 | C14×D8 | C7×D8 | D8⋊C4 | C8⋊C4 | D4⋊C4 | C4.Q8 | C4×D4 | C2×D8 | D8 | C2×C28 | C28 | C2×C4 | C4 | C14 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 6 | 6 | 12 | 6 | 12 | 6 | 48 | 2 | 2 | 12 | 12 | 2 | 12 |
Matrix representation of C7×D8⋊C4 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 106 | 0 | 0 | 0 |
| 0 | 0 | 0 | 106 | 0 | 0 |
| 0 | 0 | 0 | 0 | 106 | 0 |
| 0 | 0 | 0 | 0 | 0 | 106 |
| 107 | 53 | 0 | 0 | 0 | 0 |
| 59 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 106 | 7 | 112 | 1 |
| 0 | 0 | 9 | 106 | 112 | 112 |
| 0 | 0 | 49 | 64 | 112 | 1 |
| 0 | 0 | 50 | 48 | 112 | 15 |
| 6 | 60 | 0 | 0 | 0 | 0 |
| 103 | 107 | 0 | 0 | 0 | 0 |
| 0 | 0 | 106 | 7 | 112 | 1 |
| 0 | 0 | 104 | 7 | 1 | 1 |
| 0 | 0 | 49 | 64 | 112 | 1 |
| 0 | 0 | 64 | 64 | 98 | 1 |
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 111 | 0 |
| 0 | 0 | 16 | 98 | 0 | 111 |
| 0 | 0 | 1 | 0 | 112 | 0 |
| 0 | 0 | 1 | 0 | 97 | 15 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,106,0,0,0,0,0,0,106,0,0,0,0,0,0,106,0,0,0,0,0,0,106],[107,59,0,0,0,0,53,6,0,0,0,0,0,0,106,9,49,50,0,0,7,106,64,48,0,0,112,112,112,112,0,0,1,112,1,15],[6,103,0,0,0,0,60,107,0,0,0,0,0,0,106,104,49,64,0,0,7,7,64,64,0,0,112,1,112,98,0,0,1,1,1,1],[15,0,0,0,0,0,0,15,0,0,0,0,0,0,1,16,1,1,0,0,0,98,0,0,0,0,111,0,112,97,0,0,0,111,0,15] >;
C7×D8⋊C4 in GAP, Magma, Sage, TeX
C_7\times D_8\rtimes C_4
% in TeX
G:=Group("C7xD8:C4"); // GroupNames label
G:=SmallGroup(448,850);
// by ID
G=gap.SmallGroup(448,850);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,4790,1780,9804,4911,172]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^5,d*c*d^-1=b^4*c>;
// generators/relations