direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D8.D7, D8.7D14, C14⋊2SD32, C56.21D4, C28.22D8, C56.25C23, Dic28⋊12C22, C7⋊3(C2×SD32), C7⋊C16⋊8C22, (C2×D8).2D7, C4.9(D4⋊D7), (C14×D8).3C2, (C2×C14).43D8, C14.64(C2×D8), C28.161(C2×D4), (C2×C28).181D4, (C2×C8).234D14, C8.14(C7⋊D4), (C7×D8).7C22, C8.31(C22×D7), (C2×Dic28)⋊17C2, (C2×C56).86C22, C22.22(D4⋊D7), (C2×C7⋊C16)⋊7C2, C4.3(C2×C7⋊D4), C2.19(C2×D4⋊D7), (C2×C4).143(C7⋊D4), SmallGroup(448,682)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D8.D7
G = < a,b,c,d,e | a2=b8=c2=d7=1, e2=b4, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=b5c, ede-1=d-1 >
Subgroups: 420 in 90 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C16, C2×C8, D8, D8, Q16, C2×D4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C2×C16, SD32, C2×D8, C2×Q16, C56, Dic14, C2×Dic7, C2×C28, C7×D4, C22×C14, C2×SD32, C7⋊C16, Dic28, Dic28, C2×C56, C7×D8, C7×D8, C2×Dic14, D4×C14, C2×C7⋊C16, D8.D7, C2×Dic28, C14×D8, C2×D8.D7
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, SD32, C2×D8, C7⋊D4, C22×D7, C2×SD32, D4⋊D7, C2×C7⋊D4, D8.D7, C2×D4⋊D7, C2×D8.D7
►On 224 points
Generators in S224
(1 108)(2 109)(3 110)(4 111)(5 112)(6 105)(7 106)(8 107)(9 212)(10 213)(11 214)(12 215)(13 216)(14 209)(15 210)(16 211)(17 206)(18 207)(19 208)(20 201)(21 202)(22 203)(23 204)(24 205)(25 200)(26 193)(27 194)(28 195)(29 196)(30 197)(31 198)(32 199)(33 186)(34 187)(35 188)(36 189)(37 190)(38 191)(39 192)(40 185)(41 180)(42 181)(43 182)(44 183)(45 184)(46 177)(47 178)(48 179)(49 174)(50 175)(51 176)(52 169)(53 170)(54 171)(55 172)(56 173)(57 135)(58 136)(59 129)(60 130)(61 131)(62 132)(63 133)(64 134)(65 143)(66 144)(67 137)(68 138)(69 139)(70 140)(71 141)(72 142)(73 115)(74 116)(75 117)(76 118)(77 119)(78 120)(79 113)(80 114)(81 123)(82 124)(83 125)(84 126)(85 127)(86 128)(87 121)(88 122)(89 160)(90 153)(91 154)(92 155)(93 156)(94 157)(95 158)(96 159)(97 146)(98 147)(99 148)(100 149)(101 150)(102 151)(103 152)(104 145)(161 217)(162 218)(163 219)(164 220)(165 221)(166 222)(167 223)(168 224)
(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)
(1 107)(2 106)(3 105)(4 112)(5 111)(6 110)(7 109)(8 108)(9 212)(10 211)(11 210)(12 209)(13 216)(14 215)(15 214)(16 213)(17 208)(18 207)(19 206)(20 205)(21 204)(22 203)(23 202)(24 201)(25 196)(26 195)(27 194)(28 193)(29 200)(30 199)(31 198)(32 197)(33 186)(34 185)(35 192)(36 191)(37 190)(38 189)(39 188)(40 187)(41 184)(42 183)(43 182)(44 181)(45 180)(46 179)(47 178)(48 177)(49 170)(50 169)(51 176)(52 175)(53 174)(54 173)(55 172)(56 171)(57 134)(58 133)(59 132)(60 131)(61 130)(62 129)(63 136)(64 135)(65 138)(66 137)(67 144)(68 143)(69 142)(70 141)(71 140)(72 139)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 120)(80 119)(81 126)(82 125)(83 124)(84 123)(85 122)(86 121)(87 128)(88 127)(89 157)(90 156)(91 155)(92 154)(93 153)(94 160)(95 159)(96 158)(97 145)(98 152)(99 151)(100 150)(101 149)(102 148)(103 147)(104 146)(161 221)(162 220)(163 219)(164 218)(165 217)(166 224)(167 223)(168 222)
(1 75 57 71 96 97 87)(2 76 58 72 89 98 88)(3 77 59 65 90 99 81)(4 78 60 66 91 100 82)(5 79 61 67 92 101 83)(6 80 62 68 93 102 84)(7 73 63 69 94 103 85)(8 74 64 70 95 104 86)(9 27 33 51 219 47 22)(10 28 34 52 220 48 23)(11 29 35 53 221 41 24)(12 30 36 54 222 42 17)(13 31 37 55 223 43 18)(14 32 38 56 224 44 19)(15 25 39 49 217 45 20)(16 26 40 50 218 46 21)(105 114 132 138 156 151 126)(106 115 133 139 157 152 127)(107 116 134 140 158 145 128)(108 117 135 141 159 146 121)(109 118 136 142 160 147 122)(110 119 129 143 153 148 123)(111 120 130 144 154 149 124)(112 113 131 137 155 150 125)(161 184 201 210 200 192 174)(162 177 202 211 193 185 175)(163 178 203 212 194 186 176)(164 179 204 213 195 187 169)(165 180 205 214 196 188 170)(166 181 206 215 197 189 171)(167 182 207 216 198 190 172)(168 183 208 209 199 191 173)
(1 173 5 169)(2 172 6 176)(3 171 7 175)(4 170 8 174)(9 142 13 138)(10 141 14 137)(11 140 15 144)(12 139 16 143)(17 157 21 153)(18 156 22 160)(19 155 23 159)(20 154 24 158)(25 130 29 134)(26 129 30 133)(27 136 31 132)(28 135 32 131)(33 118 37 114)(34 117 38 113)(35 116 39 120)(36 115 40 119)(41 145 45 149)(42 152 46 148)(43 151 47 147)(44 150 48 146)(49 111 53 107)(50 110 54 106)(51 109 55 105)(52 108 56 112)(57 199 61 195)(58 198 62 194)(59 197 63 193)(60 196 64 200)(65 215 69 211)(66 214 70 210)(67 213 71 209)(68 212 72 216)(73 185 77 189)(74 192 78 188)(75 191 79 187)(76 190 80 186)(81 166 85 162)(82 165 86 161)(83 164 87 168)(84 163 88 167)(89 207 93 203)(90 206 94 202)(91 205 95 201)(92 204 96 208)(97 183 101 179)(98 182 102 178)(99 181 103 177)(100 180 104 184)(121 224 125 220)(122 223 126 219)(123 222 127 218)(124 221 128 217)
G:=sub<Sym(224)| (1,108)(2,109)(3,110)(4,111)(5,112)(6,105)(7,106)(8,107)(9,212)(10,213)(11,214)(12,215)(13,216)(14,209)(15,210)(16,211)(17,206)(18,207)(19,208)(20,201)(21,202)(22,203)(23,204)(24,205)(25,200)(26,193)(27,194)(28,195)(29,196)(30,197)(31,198)(32,199)(33,186)(34,187)(35,188)(36,189)(37,190)(38,191)(39,192)(40,185)(41,180)(42,181)(43,182)(44,183)(45,184)(46,177)(47,178)(48,179)(49,174)(50,175)(51,176)(52,169)(53,170)(54,171)(55,172)(56,173)(57,135)(58,136)(59,129)(60,130)(61,131)(62,132)(63,133)(64,134)(65,143)(66,144)(67,137)(68,138)(69,139)(70,140)(71,141)(72,142)(73,115)(74,116)(75,117)(76,118)(77,119)(78,120)(79,113)(80,114)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,121)(88,122)(89,160)(90,153)(91,154)(92,155)(93,156)(94,157)(95,158)(96,159)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(103,152)(104,145)(161,217)(162,218)(163,219)(164,220)(165,221)(166,222)(167,223)(168,224), (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), (1,107)(2,106)(3,105)(4,112)(5,111)(6,110)(7,109)(8,108)(9,212)(10,211)(11,210)(12,209)(13,216)(14,215)(15,214)(16,213)(17,208)(18,207)(19,206)(20,205)(21,204)(22,203)(23,202)(24,201)(25,196)(26,195)(27,194)(28,193)(29,200)(30,199)(31,198)(32,197)(33,186)(34,185)(35,192)(36,191)(37,190)(38,189)(39,188)(40,187)(41,184)(42,183)(43,182)(44,181)(45,180)(46,179)(47,178)(48,177)(49,170)(50,169)(51,176)(52,175)(53,174)(54,173)(55,172)(56,171)(57,134)(58,133)(59,132)(60,131)(61,130)(62,129)(63,136)(64,135)(65,138)(66,137)(67,144)(68,143)(69,142)(70,141)(71,140)(72,139)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,120)(80,119)(81,126)(82,125)(83,124)(84,123)(85,122)(86,121)(87,128)(88,127)(89,157)(90,156)(91,155)(92,154)(93,153)(94,160)(95,159)(96,158)(97,145)(98,152)(99,151)(100,150)(101,149)(102,148)(103,147)(104,146)(161,221)(162,220)(163,219)(164,218)(165,217)(166,224)(167,223)(168,222), (1,75,57,71,96,97,87)(2,76,58,72,89,98,88)(3,77,59,65,90,99,81)(4,78,60,66,91,100,82)(5,79,61,67,92,101,83)(6,80,62,68,93,102,84)(7,73,63,69,94,103,85)(8,74,64,70,95,104,86)(9,27,33,51,219,47,22)(10,28,34,52,220,48,23)(11,29,35,53,221,41,24)(12,30,36,54,222,42,17)(13,31,37,55,223,43,18)(14,32,38,56,224,44,19)(15,25,39,49,217,45,20)(16,26,40,50,218,46,21)(105,114,132,138,156,151,126)(106,115,133,139,157,152,127)(107,116,134,140,158,145,128)(108,117,135,141,159,146,121)(109,118,136,142,160,147,122)(110,119,129,143,153,148,123)(111,120,130,144,154,149,124)(112,113,131,137,155,150,125)(161,184,201,210,200,192,174)(162,177,202,211,193,185,175)(163,178,203,212,194,186,176)(164,179,204,213,195,187,169)(165,180,205,214,196,188,170)(166,181,206,215,197,189,171)(167,182,207,216,198,190,172)(168,183,208,209,199,191,173), (1,173,5,169)(2,172,6,176)(3,171,7,175)(4,170,8,174)(9,142,13,138)(10,141,14,137)(11,140,15,144)(12,139,16,143)(17,157,21,153)(18,156,22,160)(19,155,23,159)(20,154,24,158)(25,130,29,134)(26,129,30,133)(27,136,31,132)(28,135,32,131)(33,118,37,114)(34,117,38,113)(35,116,39,120)(36,115,40,119)(41,145,45,149)(42,152,46,148)(43,151,47,147)(44,150,48,146)(49,111,53,107)(50,110,54,106)(51,109,55,105)(52,108,56,112)(57,199,61,195)(58,198,62,194)(59,197,63,193)(60,196,64,200)(65,215,69,211)(66,214,70,210)(67,213,71,209)(68,212,72,216)(73,185,77,189)(74,192,78,188)(75,191,79,187)(76,190,80,186)(81,166,85,162)(82,165,86,161)(83,164,87,168)(84,163,88,167)(89,207,93,203)(90,206,94,202)(91,205,95,201)(92,204,96,208)(97,183,101,179)(98,182,102,178)(99,181,103,177)(100,180,104,184)(121,224,125,220)(122,223,126,219)(123,222,127,218)(124,221,128,217)>;
G:=Group( (1,108)(2,109)(3,110)(4,111)(5,112)(6,105)(7,106)(8,107)(9,212)(10,213)(11,214)(12,215)(13,216)(14,209)(15,210)(16,211)(17,206)(18,207)(19,208)(20,201)(21,202)(22,203)(23,204)(24,205)(25,200)(26,193)(27,194)(28,195)(29,196)(30,197)(31,198)(32,199)(33,186)(34,187)(35,188)(36,189)(37,190)(38,191)(39,192)(40,185)(41,180)(42,181)(43,182)(44,183)(45,184)(46,177)(47,178)(48,179)(49,174)(50,175)(51,176)(52,169)(53,170)(54,171)(55,172)(56,173)(57,135)(58,136)(59,129)(60,130)(61,131)(62,132)(63,133)(64,134)(65,143)(66,144)(67,137)(68,138)(69,139)(70,140)(71,141)(72,142)(73,115)(74,116)(75,117)(76,118)(77,119)(78,120)(79,113)(80,114)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,121)(88,122)(89,160)(90,153)(91,154)(92,155)(93,156)(94,157)(95,158)(96,159)(97,146)(98,147)(99,148)(100,149)(101,150)(102,151)(103,152)(104,145)(161,217)(162,218)(163,219)(164,220)(165,221)(166,222)(167,223)(168,224), (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), (1,107)(2,106)(3,105)(4,112)(5,111)(6,110)(7,109)(8,108)(9,212)(10,211)(11,210)(12,209)(13,216)(14,215)(15,214)(16,213)(17,208)(18,207)(19,206)(20,205)(21,204)(22,203)(23,202)(24,201)(25,196)(26,195)(27,194)(28,193)(29,200)(30,199)(31,198)(32,197)(33,186)(34,185)(35,192)(36,191)(37,190)(38,189)(39,188)(40,187)(41,184)(42,183)(43,182)(44,181)(45,180)(46,179)(47,178)(48,177)(49,170)(50,169)(51,176)(52,175)(53,174)(54,173)(55,172)(56,171)(57,134)(58,133)(59,132)(60,131)(61,130)(62,129)(63,136)(64,135)(65,138)(66,137)(67,144)(68,143)(69,142)(70,141)(71,140)(72,139)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,120)(80,119)(81,126)(82,125)(83,124)(84,123)(85,122)(86,121)(87,128)(88,127)(89,157)(90,156)(91,155)(92,154)(93,153)(94,160)(95,159)(96,158)(97,145)(98,152)(99,151)(100,150)(101,149)(102,148)(103,147)(104,146)(161,221)(162,220)(163,219)(164,218)(165,217)(166,224)(167,223)(168,222), (1,75,57,71,96,97,87)(2,76,58,72,89,98,88)(3,77,59,65,90,99,81)(4,78,60,66,91,100,82)(5,79,61,67,92,101,83)(6,80,62,68,93,102,84)(7,73,63,69,94,103,85)(8,74,64,70,95,104,86)(9,27,33,51,219,47,22)(10,28,34,52,220,48,23)(11,29,35,53,221,41,24)(12,30,36,54,222,42,17)(13,31,37,55,223,43,18)(14,32,38,56,224,44,19)(15,25,39,49,217,45,20)(16,26,40,50,218,46,21)(105,114,132,138,156,151,126)(106,115,133,139,157,152,127)(107,116,134,140,158,145,128)(108,117,135,141,159,146,121)(109,118,136,142,160,147,122)(110,119,129,143,153,148,123)(111,120,130,144,154,149,124)(112,113,131,137,155,150,125)(161,184,201,210,200,192,174)(162,177,202,211,193,185,175)(163,178,203,212,194,186,176)(164,179,204,213,195,187,169)(165,180,205,214,196,188,170)(166,181,206,215,197,189,171)(167,182,207,216,198,190,172)(168,183,208,209,199,191,173), (1,173,5,169)(2,172,6,176)(3,171,7,175)(4,170,8,174)(9,142,13,138)(10,141,14,137)(11,140,15,144)(12,139,16,143)(17,157,21,153)(18,156,22,160)(19,155,23,159)(20,154,24,158)(25,130,29,134)(26,129,30,133)(27,136,31,132)(28,135,32,131)(33,118,37,114)(34,117,38,113)(35,116,39,120)(36,115,40,119)(41,145,45,149)(42,152,46,148)(43,151,47,147)(44,150,48,146)(49,111,53,107)(50,110,54,106)(51,109,55,105)(52,108,56,112)(57,199,61,195)(58,198,62,194)(59,197,63,193)(60,196,64,200)(65,215,69,211)(66,214,70,210)(67,213,71,209)(68,212,72,216)(73,185,77,189)(74,192,78,188)(75,191,79,187)(76,190,80,186)(81,166,85,162)(82,165,86,161)(83,164,87,168)(84,163,88,167)(89,207,93,203)(90,206,94,202)(91,205,95,201)(92,204,96,208)(97,183,101,179)(98,182,102,178)(99,181,103,177)(100,180,104,184)(121,224,125,220)(122,223,126,219)(123,222,127,218)(124,221,128,217) );
G=PermutationGroup([[(1,108),(2,109),(3,110),(4,111),(5,112),(6,105),(7,106),(8,107),(9,212),(10,213),(11,214),(12,215),(13,216),(14,209),(15,210),(16,211),(17,206),(18,207),(19,208),(20,201),(21,202),(22,203),(23,204),(24,205),(25,200),(26,193),(27,194),(28,195),(29,196),(30,197),(31,198),(32,199),(33,186),(34,187),(35,188),(36,189),(37,190),(38,191),(39,192),(40,185),(41,180),(42,181),(43,182),(44,183),(45,184),(46,177),(47,178),(48,179),(49,174),(50,175),(51,176),(52,169),(53,170),(54,171),(55,172),(56,173),(57,135),(58,136),(59,129),(60,130),(61,131),(62,132),(63,133),(64,134),(65,143),(66,144),(67,137),(68,138),(69,139),(70,140),(71,141),(72,142),(73,115),(74,116),(75,117),(76,118),(77,119),(78,120),(79,113),(80,114),(81,123),(82,124),(83,125),(84,126),(85,127),(86,128),(87,121),(88,122),(89,160),(90,153),(91,154),(92,155),(93,156),(94,157),(95,158),(96,159),(97,146),(98,147),(99,148),(100,149),(101,150),(102,151),(103,152),(104,145),(161,217),(162,218),(163,219),(164,220),(165,221),(166,222),(167,223),(168,224)], [(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)], [(1,107),(2,106),(3,105),(4,112),(5,111),(6,110),(7,109),(8,108),(9,212),(10,211),(11,210),(12,209),(13,216),(14,215),(15,214),(16,213),(17,208),(18,207),(19,206),(20,205),(21,204),(22,203),(23,202),(24,201),(25,196),(26,195),(27,194),(28,193),(29,200),(30,199),(31,198),(32,197),(33,186),(34,185),(35,192),(36,191),(37,190),(38,189),(39,188),(40,187),(41,184),(42,183),(43,182),(44,181),(45,180),(46,179),(47,178),(48,177),(49,170),(50,169),(51,176),(52,175),(53,174),(54,173),(55,172),(56,171),(57,134),(58,133),(59,132),(60,131),(61,130),(62,129),(63,136),(64,135),(65,138),(66,137),(67,144),(68,143),(69,142),(70,141),(71,140),(72,139),(73,118),(74,117),(75,116),(76,115),(77,114),(78,113),(79,120),(80,119),(81,126),(82,125),(83,124),(84,123),(85,122),(86,121),(87,128),(88,127),(89,157),(90,156),(91,155),(92,154),(93,153),(94,160),(95,159),(96,158),(97,145),(98,152),(99,151),(100,150),(101,149),(102,148),(103,147),(104,146),(161,221),(162,220),(163,219),(164,218),(165,217),(166,224),(167,223),(168,222)], [(1,75,57,71,96,97,87),(2,76,58,72,89,98,88),(3,77,59,65,90,99,81),(4,78,60,66,91,100,82),(5,79,61,67,92,101,83),(6,80,62,68,93,102,84),(7,73,63,69,94,103,85),(8,74,64,70,95,104,86),(9,27,33,51,219,47,22),(10,28,34,52,220,48,23),(11,29,35,53,221,41,24),(12,30,36,54,222,42,17),(13,31,37,55,223,43,18),(14,32,38,56,224,44,19),(15,25,39,49,217,45,20),(16,26,40,50,218,46,21),(105,114,132,138,156,151,126),(106,115,133,139,157,152,127),(107,116,134,140,158,145,128),(108,117,135,141,159,146,121),(109,118,136,142,160,147,122),(110,119,129,143,153,148,123),(111,120,130,144,154,149,124),(112,113,131,137,155,150,125),(161,184,201,210,200,192,174),(162,177,202,211,193,185,175),(163,178,203,212,194,186,176),(164,179,204,213,195,187,169),(165,180,205,214,196,188,170),(166,181,206,215,197,189,171),(167,182,207,216,198,190,172),(168,183,208,209,199,191,173)], [(1,173,5,169),(2,172,6,176),(3,171,7,175),(4,170,8,174),(9,142,13,138),(10,141,14,137),(11,140,15,144),(12,139,16,143),(17,157,21,153),(18,156,22,160),(19,155,23,159),(20,154,24,158),(25,130,29,134),(26,129,30,133),(27,136,31,132),(28,135,32,131),(33,118,37,114),(34,117,38,113),(35,116,39,120),(36,115,40,119),(41,145,45,149),(42,152,46,148),(43,151,47,147),(44,150,48,146),(49,111,53,107),(50,110,54,106),(51,109,55,105),(52,108,56,112),(57,199,61,195),(58,198,62,194),(59,197,63,193),(60,196,64,200),(65,215,69,211),(66,214,70,210),(67,213,71,209),(68,212,72,216),(73,185,77,189),(74,192,78,188),(75,191,79,187),(76,190,80,186),(81,166,85,162),(82,165,86,161),(83,164,87,168),(84,163,88,167),(89,207,93,203),(90,206,94,202),(91,205,95,201),(92,204,96,208),(97,183,101,179),(98,182,102,178),(99,181,103,177),(100,180,104,184),(121,224,125,220),(122,223,126,219),(123,222,127,218),(124,221,128,217)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14U | 16A | ··· | 16H | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 14 | ··· | 14 | 4 | ··· | 4 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D8 | D8 | D14 | D14 | SD32 | C7⋊D4 | C7⋊D4 | D4⋊D7 | D4⋊D7 | D8.D7 |
| kernel | C2×D8.D7 | C2×C7⋊C16 | D8.D7 | C2×Dic28 | C14×D8 | C56 | C2×C28 | C2×D8 | C28 | C2×C14 | C2×C8 | D8 | C14 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 8 | 6 | 6 | 3 | 3 | 12 |
Matrix representation of C2×D8.D7 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 62 | 62 | 0 | 0 |
| 82 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 51 | 51 | 0 | 0 |
| 31 | 62 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 79 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 97 | 49 |
| 97 | 88 | 0 | 0 |
| 60 | 16 | 0 | 0 |
| 0 | 0 | 77 | 71 |
| 0 | 0 | 12 | 36 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[62,82,0,0,62,0,0,0,0,0,112,0,0,0,0,112],[51,31,0,0,51,62,0,0,0,0,1,79,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,30,97,0,0,0,49],[97,60,0,0,88,16,0,0,0,0,77,12,0,0,71,36] >;
C2×D8.D7 in GAP, Magma, Sage, TeX
C_2\times D_8.D_7
% in TeX
G:=Group("C2xD8.D7"); // GroupNames label
G:=SmallGroup(448,682);
// by ID
G=gap.SmallGroup(448,682);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,254,675,185,192,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^7=1,e^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^5*c,e*d*e^-1=d^-1>;
// generators/relations