metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2.D8⋊5D7, D14⋊C8⋊26C2, C4⋊C4.50D14, (C2×C8).29D14, C4⋊D28.8C2, C14.D8⋊22C2, C2.D56⋊27C2, C4.82(C4○D28), C28.40(C4○D4), C14.76(C4○D8), C4.Dic14⋊21C2, (C22×D7).28D4, C22.231(D4×D7), C2.23(D8⋊D7), C14.42(C8⋊C22), (C2×C56).243C22, (C2×C28).301C23, C4.30(Q8⋊2D7), (C2×Dic7).168D4, (C2×D28).83C22, C7⋊5(C23.19D4), C2.14(Q8.D14), C4⋊Dic7.126C22, C2.17(D14.5D4), C14.47(C22.D4), C4⋊C4⋊7D7⋊7C2, (C7×C2.D8)⋊13C2, (C2×C7⋊C8).71C22, (C2×C4×D7).40C22, (C2×C14).306(C2×D4), (C7×C4⋊C4).94C22, (C2×C4).404(C22×D7), SmallGroup(448,419)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.D8⋊D7
G = < a,b,c,d,e | a2=b8=d7=e2=1, c2=a, ab=ba, ece=ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=ab5, cd=dc, ede=d-1 >
Subgroups: 684 in 106 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C7⋊C8, C56, C4×D7, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C23.19D4, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×D28, C2×D28, C4.Dic14, C14.D8, D14⋊C8, C2.D56, C7×C2.D8, C4⋊C4⋊7D7, C4⋊D28, C2.D8⋊D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22.D4, C4○D8, C8⋊C22, C22×D7, C23.19D4, C4○D28, D4×D7, Q8⋊2D7, D14.5D4, D8⋊D7, Q8.D14, C2.D8⋊D7
►On 224 points
Generators in S224
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 194)(10 195)(11 196)(12 197)(13 198)(14 199)(15 200)(16 193)(17 150)(18 151)(19 152)(20 145)(21 146)(22 147)(23 148)(24 149)(25 125)(26 126)(27 127)(28 128)(29 121)(30 122)(31 123)(32 124)(33 171)(34 172)(35 173)(36 174)(37 175)(38 176)(39 169)(40 170)(41 116)(42 117)(43 118)(44 119)(45 120)(46 113)(47 114)(48 115)(57 144)(58 137)(59 138)(60 139)(61 140)(62 141)(63 142)(64 143)(65 162)(66 163)(67 164)(68 165)(69 166)(70 167)(71 168)(72 161)(73 188)(74 189)(75 190)(76 191)(77 192)(78 185)(79 186)(80 187)(81 135)(82 136)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 182)(90 183)(91 184)(92 177)(93 178)(94 179)(95 180)(96 181)(97 109)(98 110)(99 111)(100 112)(101 105)(102 106)(103 107)(104 108)(153 209)(154 210)(155 211)(156 212)(157 213)(158 214)(159 215)(160 216)(201 222)(202 223)(203 224)(204 217)(205 218)(206 219)(207 220)(208 221)
(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)
(1 212 51 156)(2 211 52 155)(3 210 53 154)(4 209 54 153)(5 216 55 160)(6 215 56 159)(7 214 49 158)(8 213 50 157)(9 183 194 90)(10 182 195 89)(11 181 196 96)(12 180 197 95)(13 179 198 94)(14 178 199 93)(15 177 200 92)(16 184 193 91)(17 175 150 37)(18 174 151 36)(19 173 152 35)(20 172 145 34)(21 171 146 33)(22 170 147 40)(23 169 148 39)(24 176 149 38)(25 67 125 164)(26 66 126 163)(27 65 127 162)(28 72 128 161)(29 71 121 168)(30 70 122 167)(31 69 123 166)(32 68 124 165)(41 76 116 191)(42 75 117 190)(43 74 118 189)(44 73 119 188)(45 80 120 187)(46 79 113 186)(47 78 114 185)(48 77 115 192)(57 135 144 81)(58 134 137 88)(59 133 138 87)(60 132 139 86)(61 131 140 85)(62 130 141 84)(63 129 142 83)(64 136 143 82)(97 218 109 205)(98 217 110 204)(99 224 111 203)(100 223 112 202)(101 222 105 201)(102 221 106 208)(103 220 107 207)(104 219 108 206)
(1 25 40 90 83 118 111)(2 26 33 91 84 119 112)(3 27 34 92 85 120 105)(4 28 35 93 86 113 106)(5 29 36 94 87 114 107)(6 30 37 95 88 115 108)(7 31 38 96 81 116 109)(8 32 39 89 82 117 110)(9 63 189 203 212 67 22)(10 64 190 204 213 68 23)(11 57 191 205 214 69 24)(12 58 192 206 215 70 17)(13 59 185 207 216 71 18)(14 60 186 208 209 72 19)(15 61 187 201 210 65 20)(16 62 188 202 211 66 21)(41 97 49 123 176 181 135)(42 98 50 124 169 182 136)(43 99 51 125 170 183 129)(44 100 52 126 171 184 130)(45 101 53 127 172 177 131)(46 102 54 128 173 178 132)(47 103 55 121 174 179 133)(48 104 56 122 175 180 134)(73 223 155 163 146 193 141)(74 224 156 164 147 194 142)(75 217 157 165 148 195 143)(76 218 158 166 149 196 144)(77 219 159 167 150 197 137)(78 220 160 168 151 198 138)(79 221 153 161 152 199 139)(80 222 154 162 145 200 140)
(1 111)(2 104)(3 105)(4 98)(5 107)(6 100)(7 109)(8 102)(9 194)(10 14)(11 196)(12 16)(13 198)(15 200)(17 62)(18 138)(19 64)(20 140)(21 58)(22 142)(23 60)(24 144)(25 118)(26 48)(27 120)(28 42)(29 114)(30 44)(31 116)(32 46)(33 134)(34 85)(35 136)(36 87)(37 130)(38 81)(39 132)(40 83)(41 123)(43 125)(45 127)(47 121)(49 97)(50 106)(51 99)(52 108)(53 101)(54 110)(55 103)(56 112)(57 149)(59 151)(61 145)(63 147)(65 80)(66 192)(67 74)(68 186)(69 76)(70 188)(71 78)(72 190)(73 167)(75 161)(77 163)(79 165)(82 173)(84 175)(86 169)(88 171)(89 178)(91 180)(93 182)(95 184)(113 124)(115 126)(117 128)(119 122)(129 170)(131 172)(133 174)(135 176)(137 146)(139 148)(141 150)(143 152)(153 217)(154 201)(155 219)(156 203)(157 221)(158 205)(159 223)(160 207)(162 187)(164 189)(166 191)(168 185)(193 197)(195 199)(202 215)(204 209)(206 211)(208 213)(210 222)(212 224)(214 218)(216 220)
G:=sub<Sym(224)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,194)(10,195)(11,196)(12,197)(13,198)(14,199)(15,200)(16,193)(17,150)(18,151)(19,152)(20,145)(21,146)(22,147)(23,148)(24,149)(25,125)(26,126)(27,127)(28,128)(29,121)(30,122)(31,123)(32,124)(33,171)(34,172)(35,173)(36,174)(37,175)(38,176)(39,169)(40,170)(41,116)(42,117)(43,118)(44,119)(45,120)(46,113)(47,114)(48,115)(57,144)(58,137)(59,138)(60,139)(61,140)(62,141)(63,142)(64,143)(65,162)(66,163)(67,164)(68,165)(69,166)(70,167)(71,168)(72,161)(73,188)(74,189)(75,190)(76,191)(77,192)(78,185)(79,186)(80,187)(81,135)(82,136)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,182)(90,183)(91,184)(92,177)(93,178)(94,179)(95,180)(96,181)(97,109)(98,110)(99,111)(100,112)(101,105)(102,106)(103,107)(104,108)(153,209)(154,210)(155,211)(156,212)(157,213)(158,214)(159,215)(160,216)(201,222)(202,223)(203,224)(204,217)(205,218)(206,219)(207,220)(208,221), (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), (1,212,51,156)(2,211,52,155)(3,210,53,154)(4,209,54,153)(5,216,55,160)(6,215,56,159)(7,214,49,158)(8,213,50,157)(9,183,194,90)(10,182,195,89)(11,181,196,96)(12,180,197,95)(13,179,198,94)(14,178,199,93)(15,177,200,92)(16,184,193,91)(17,175,150,37)(18,174,151,36)(19,173,152,35)(20,172,145,34)(21,171,146,33)(22,170,147,40)(23,169,148,39)(24,176,149,38)(25,67,125,164)(26,66,126,163)(27,65,127,162)(28,72,128,161)(29,71,121,168)(30,70,122,167)(31,69,123,166)(32,68,124,165)(41,76,116,191)(42,75,117,190)(43,74,118,189)(44,73,119,188)(45,80,120,187)(46,79,113,186)(47,78,114,185)(48,77,115,192)(57,135,144,81)(58,134,137,88)(59,133,138,87)(60,132,139,86)(61,131,140,85)(62,130,141,84)(63,129,142,83)(64,136,143,82)(97,218,109,205)(98,217,110,204)(99,224,111,203)(100,223,112,202)(101,222,105,201)(102,221,106,208)(103,220,107,207)(104,219,108,206), (1,25,40,90,83,118,111)(2,26,33,91,84,119,112)(3,27,34,92,85,120,105)(4,28,35,93,86,113,106)(5,29,36,94,87,114,107)(6,30,37,95,88,115,108)(7,31,38,96,81,116,109)(8,32,39,89,82,117,110)(9,63,189,203,212,67,22)(10,64,190,204,213,68,23)(11,57,191,205,214,69,24)(12,58,192,206,215,70,17)(13,59,185,207,216,71,18)(14,60,186,208,209,72,19)(15,61,187,201,210,65,20)(16,62,188,202,211,66,21)(41,97,49,123,176,181,135)(42,98,50,124,169,182,136)(43,99,51,125,170,183,129)(44,100,52,126,171,184,130)(45,101,53,127,172,177,131)(46,102,54,128,173,178,132)(47,103,55,121,174,179,133)(48,104,56,122,175,180,134)(73,223,155,163,146,193,141)(74,224,156,164,147,194,142)(75,217,157,165,148,195,143)(76,218,158,166,149,196,144)(77,219,159,167,150,197,137)(78,220,160,168,151,198,138)(79,221,153,161,152,199,139)(80,222,154,162,145,200,140), (1,111)(2,104)(3,105)(4,98)(5,107)(6,100)(7,109)(8,102)(9,194)(10,14)(11,196)(12,16)(13,198)(15,200)(17,62)(18,138)(19,64)(20,140)(21,58)(22,142)(23,60)(24,144)(25,118)(26,48)(27,120)(28,42)(29,114)(30,44)(31,116)(32,46)(33,134)(34,85)(35,136)(36,87)(37,130)(38,81)(39,132)(40,83)(41,123)(43,125)(45,127)(47,121)(49,97)(50,106)(51,99)(52,108)(53,101)(54,110)(55,103)(56,112)(57,149)(59,151)(61,145)(63,147)(65,80)(66,192)(67,74)(68,186)(69,76)(70,188)(71,78)(72,190)(73,167)(75,161)(77,163)(79,165)(82,173)(84,175)(86,169)(88,171)(89,178)(91,180)(93,182)(95,184)(113,124)(115,126)(117,128)(119,122)(129,170)(131,172)(133,174)(135,176)(137,146)(139,148)(141,150)(143,152)(153,217)(154,201)(155,219)(156,203)(157,221)(158,205)(159,223)(160,207)(162,187)(164,189)(166,191)(168,185)(193,197)(195,199)(202,215)(204,209)(206,211)(208,213)(210,222)(212,224)(214,218)(216,220)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,49)(8,50)(9,194)(10,195)(11,196)(12,197)(13,198)(14,199)(15,200)(16,193)(17,150)(18,151)(19,152)(20,145)(21,146)(22,147)(23,148)(24,149)(25,125)(26,126)(27,127)(28,128)(29,121)(30,122)(31,123)(32,124)(33,171)(34,172)(35,173)(36,174)(37,175)(38,176)(39,169)(40,170)(41,116)(42,117)(43,118)(44,119)(45,120)(46,113)(47,114)(48,115)(57,144)(58,137)(59,138)(60,139)(61,140)(62,141)(63,142)(64,143)(65,162)(66,163)(67,164)(68,165)(69,166)(70,167)(71,168)(72,161)(73,188)(74,189)(75,190)(76,191)(77,192)(78,185)(79,186)(80,187)(81,135)(82,136)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,182)(90,183)(91,184)(92,177)(93,178)(94,179)(95,180)(96,181)(97,109)(98,110)(99,111)(100,112)(101,105)(102,106)(103,107)(104,108)(153,209)(154,210)(155,211)(156,212)(157,213)(158,214)(159,215)(160,216)(201,222)(202,223)(203,224)(204,217)(205,218)(206,219)(207,220)(208,221), (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), (1,212,51,156)(2,211,52,155)(3,210,53,154)(4,209,54,153)(5,216,55,160)(6,215,56,159)(7,214,49,158)(8,213,50,157)(9,183,194,90)(10,182,195,89)(11,181,196,96)(12,180,197,95)(13,179,198,94)(14,178,199,93)(15,177,200,92)(16,184,193,91)(17,175,150,37)(18,174,151,36)(19,173,152,35)(20,172,145,34)(21,171,146,33)(22,170,147,40)(23,169,148,39)(24,176,149,38)(25,67,125,164)(26,66,126,163)(27,65,127,162)(28,72,128,161)(29,71,121,168)(30,70,122,167)(31,69,123,166)(32,68,124,165)(41,76,116,191)(42,75,117,190)(43,74,118,189)(44,73,119,188)(45,80,120,187)(46,79,113,186)(47,78,114,185)(48,77,115,192)(57,135,144,81)(58,134,137,88)(59,133,138,87)(60,132,139,86)(61,131,140,85)(62,130,141,84)(63,129,142,83)(64,136,143,82)(97,218,109,205)(98,217,110,204)(99,224,111,203)(100,223,112,202)(101,222,105,201)(102,221,106,208)(103,220,107,207)(104,219,108,206), (1,25,40,90,83,118,111)(2,26,33,91,84,119,112)(3,27,34,92,85,120,105)(4,28,35,93,86,113,106)(5,29,36,94,87,114,107)(6,30,37,95,88,115,108)(7,31,38,96,81,116,109)(8,32,39,89,82,117,110)(9,63,189,203,212,67,22)(10,64,190,204,213,68,23)(11,57,191,205,214,69,24)(12,58,192,206,215,70,17)(13,59,185,207,216,71,18)(14,60,186,208,209,72,19)(15,61,187,201,210,65,20)(16,62,188,202,211,66,21)(41,97,49,123,176,181,135)(42,98,50,124,169,182,136)(43,99,51,125,170,183,129)(44,100,52,126,171,184,130)(45,101,53,127,172,177,131)(46,102,54,128,173,178,132)(47,103,55,121,174,179,133)(48,104,56,122,175,180,134)(73,223,155,163,146,193,141)(74,224,156,164,147,194,142)(75,217,157,165,148,195,143)(76,218,158,166,149,196,144)(77,219,159,167,150,197,137)(78,220,160,168,151,198,138)(79,221,153,161,152,199,139)(80,222,154,162,145,200,140), (1,111)(2,104)(3,105)(4,98)(5,107)(6,100)(7,109)(8,102)(9,194)(10,14)(11,196)(12,16)(13,198)(15,200)(17,62)(18,138)(19,64)(20,140)(21,58)(22,142)(23,60)(24,144)(25,118)(26,48)(27,120)(28,42)(29,114)(30,44)(31,116)(32,46)(33,134)(34,85)(35,136)(36,87)(37,130)(38,81)(39,132)(40,83)(41,123)(43,125)(45,127)(47,121)(49,97)(50,106)(51,99)(52,108)(53,101)(54,110)(55,103)(56,112)(57,149)(59,151)(61,145)(63,147)(65,80)(66,192)(67,74)(68,186)(69,76)(70,188)(71,78)(72,190)(73,167)(75,161)(77,163)(79,165)(82,173)(84,175)(86,169)(88,171)(89,178)(91,180)(93,182)(95,184)(113,124)(115,126)(117,128)(119,122)(129,170)(131,172)(133,174)(135,176)(137,146)(139,148)(141,150)(143,152)(153,217)(154,201)(155,219)(156,203)(157,221)(158,205)(159,223)(160,207)(162,187)(164,189)(166,191)(168,185)(193,197)(195,199)(202,215)(204,209)(206,211)(208,213)(210,222)(212,224)(214,218)(216,220) );
G=PermutationGroup([[(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,49),(8,50),(9,194),(10,195),(11,196),(12,197),(13,198),(14,199),(15,200),(16,193),(17,150),(18,151),(19,152),(20,145),(21,146),(22,147),(23,148),(24,149),(25,125),(26,126),(27,127),(28,128),(29,121),(30,122),(31,123),(32,124),(33,171),(34,172),(35,173),(36,174),(37,175),(38,176),(39,169),(40,170),(41,116),(42,117),(43,118),(44,119),(45,120),(46,113),(47,114),(48,115),(57,144),(58,137),(59,138),(60,139),(61,140),(62,141),(63,142),(64,143),(65,162),(66,163),(67,164),(68,165),(69,166),(70,167),(71,168),(72,161),(73,188),(74,189),(75,190),(76,191),(77,192),(78,185),(79,186),(80,187),(81,135),(82,136),(83,129),(84,130),(85,131),(86,132),(87,133),(88,134),(89,182),(90,183),(91,184),(92,177),(93,178),(94,179),(95,180),(96,181),(97,109),(98,110),(99,111),(100,112),(101,105),(102,106),(103,107),(104,108),(153,209),(154,210),(155,211),(156,212),(157,213),(158,214),(159,215),(160,216),(201,222),(202,223),(203,224),(204,217),(205,218),(206,219),(207,220),(208,221)], [(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)], [(1,212,51,156),(2,211,52,155),(3,210,53,154),(4,209,54,153),(5,216,55,160),(6,215,56,159),(7,214,49,158),(8,213,50,157),(9,183,194,90),(10,182,195,89),(11,181,196,96),(12,180,197,95),(13,179,198,94),(14,178,199,93),(15,177,200,92),(16,184,193,91),(17,175,150,37),(18,174,151,36),(19,173,152,35),(20,172,145,34),(21,171,146,33),(22,170,147,40),(23,169,148,39),(24,176,149,38),(25,67,125,164),(26,66,126,163),(27,65,127,162),(28,72,128,161),(29,71,121,168),(30,70,122,167),(31,69,123,166),(32,68,124,165),(41,76,116,191),(42,75,117,190),(43,74,118,189),(44,73,119,188),(45,80,120,187),(46,79,113,186),(47,78,114,185),(48,77,115,192),(57,135,144,81),(58,134,137,88),(59,133,138,87),(60,132,139,86),(61,131,140,85),(62,130,141,84),(63,129,142,83),(64,136,143,82),(97,218,109,205),(98,217,110,204),(99,224,111,203),(100,223,112,202),(101,222,105,201),(102,221,106,208),(103,220,107,207),(104,219,108,206)], [(1,25,40,90,83,118,111),(2,26,33,91,84,119,112),(3,27,34,92,85,120,105),(4,28,35,93,86,113,106),(5,29,36,94,87,114,107),(6,30,37,95,88,115,108),(7,31,38,96,81,116,109),(8,32,39,89,82,117,110),(9,63,189,203,212,67,22),(10,64,190,204,213,68,23),(11,57,191,205,214,69,24),(12,58,192,206,215,70,17),(13,59,185,207,216,71,18),(14,60,186,208,209,72,19),(15,61,187,201,210,65,20),(16,62,188,202,211,66,21),(41,97,49,123,176,181,135),(42,98,50,124,169,182,136),(43,99,51,125,170,183,129),(44,100,52,126,171,184,130),(45,101,53,127,172,177,131),(46,102,54,128,173,178,132),(47,103,55,121,174,179,133),(48,104,56,122,175,180,134),(73,223,155,163,146,193,141),(74,224,156,164,147,194,142),(75,217,157,165,148,195,143),(76,218,158,166,149,196,144),(77,219,159,167,150,197,137),(78,220,160,168,151,198,138),(79,221,153,161,152,199,139),(80,222,154,162,145,200,140)], [(1,111),(2,104),(3,105),(4,98),(5,107),(6,100),(7,109),(8,102),(9,194),(10,14),(11,196),(12,16),(13,198),(15,200),(17,62),(18,138),(19,64),(20,140),(21,58),(22,142),(23,60),(24,144),(25,118),(26,48),(27,120),(28,42),(29,114),(30,44),(31,116),(32,46),(33,134),(34,85),(35,136),(36,87),(37,130),(38,81),(39,132),(40,83),(41,123),(43,125),(45,127),(47,121),(49,97),(50,106),(51,99),(52,108),(53,101),(54,110),(55,103),(56,112),(57,149),(59,151),(61,145),(63,147),(65,80),(66,192),(67,74),(68,186),(69,76),(70,188),(71,78),(72,190),(73,167),(75,161),(77,163),(79,165),(82,173),(84,175),(86,169),(88,171),(89,178),(91,180),(93,182),(95,184),(113,124),(115,126),(117,128),(119,122),(129,170),(131,172),(133,174),(135,176),(137,146),(139,148),(141,150),(143,152),(153,217),(154,201),(155,219),(156,203),(157,221),(158,205),(159,223),(160,207),(162,187),(164,189),(166,191),(168,185),(193,197),(195,199),(202,215),(204,209),(206,211),(208,213),(210,222),(212,224),(214,218),(216,220)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 28 | 56 | 2 | 2 | 4 | 4 | 8 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C4○D8 | C4○D28 | C8⋊C22 | Q8⋊2D7 | D4×D7 | D8⋊D7 | Q8.D14 |
| kernel | C2.D8⋊D7 | C4.Dic14 | C14.D8 | D14⋊C8 | C2.D56 | C7×C2.D8 | C4⋊C4⋊7D7 | C4⋊D28 | C2×Dic7 | C22×D7 | C2.D8 | C28 | C4⋊C4 | C2×C8 | C14 | C4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 6 | 3 | 4 | 12 | 1 | 3 | 3 | 6 | 6 |
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 |
| 44 | 82 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 84 | 12 |
| 0 | 0 | 43 | 29 |
| 18 | 13 | 0 | 0 |
| 88 | 95 | 0 | 0 |
| 0 | 0 | 96 | 46 |
| 0 | 0 | 33 | 17 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 103 | 1 |
| 0 | 0 | 13 | 89 |
| 1 | 15 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 89 | 79 |
| 0 | 0 | 100 | 24 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[44,0,0,0,82,18,0,0,0,0,84,43,0,0,12,29],[18,88,0,0,13,95,0,0,0,0,96,33,0,0,46,17],[1,0,0,0,0,1,0,0,0,0,103,13,0,0,1,89],[1,0,0,0,15,112,0,0,0,0,89,100,0,0,79,24] >;
C2.D8⋊D7 in GAP, Magma, Sage, TeX
C_2.D_8\rtimes D_7
% in TeX
G:=Group("C2.D8:D7"); // GroupNames label
G:=SmallGroup(448,419);
// by ID
G=gap.SmallGroup(448,419);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,64,926,219,268,851,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=d^7=e^2=1,c^2=a,a*b=b*a,e*c*e=a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e=a*b^5,c*d=d*c,e*d*e=d^-1>;
// generators/relations