metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14⋊C8⋊40C2, C4.89(C2×D28), (C2×D28).14C4, C28.444(C2×D4), (C2×C28).171D4, (C2×C4).152D28, (C2×C8).189D14, (C2×M4(2))⋊9D7, C23.29(C4×D7), C4.12(D14⋊C4), C14.34(C8○D4), C28.26(C22⋊C4), (C14×M4(2))⋊17C2, (C2×C28).869C23, C2.19(D28.C4), (C2×C56).319C22, C22.2(D14⋊C4), (C2×Dic14).14C4, (C22×C4).350D14, (C22×C28).186C22, (C22×C7⋊C8)⋊6C2, (C2×C4).84(C4×D7), C2.28(C2×D14⋊C4), (C2×C7⋊D4).12C4, C4.135(C2×C7⋊D4), C22.148(C2×C4×D7), (C2×C28).106(C2×C4), C7⋊3((C22×C8)⋊C2), (C2×C4○D28).11C2, (C2×C7⋊C8).325C22, C14.56(C2×C22⋊C4), (C2×C4×D7).186C22, (C2×C4).141(C7⋊D4), (C22×C14).69(C2×C4), (C2×Dic7).35(C2×C4), (C22×D7).26(C2×C4), (C2×C4).811(C22×D7), (C2×C14).20(C22⋊C4), (C2×C14).139(C22×C4), SmallGroup(448,663)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×D28).14C4
G = < a,b,c,d,e | a2=b8=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b5, bd=db, ebe=ab5, cd=dc, ece=b4c, ede=d-1 >
Subgroups: 740 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C22⋊C8, C22×C8, C2×M4(2), C2×C4○D4, C7⋊C8, C56, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, (C22×C8)⋊C2, C2×C7⋊C8, C2×C7⋊C8, C2×C56, C7×M4(2), C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, D14⋊C8, C22×C7⋊C8, C14×M4(2), C2×C4○D28, (C2×D28).14C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, D14, C2×C22⋊C4, C8○D4, C4×D7, D28, C7⋊D4, C22×D7, (C22×C8)⋊C2, D14⋊C4, C2×C4×D7, C2×D28, C2×C7⋊D4, D28.C4, C2×D14⋊C4, (C2×D28).14C4
►On 224 points
Generators in S224
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 41)(9 138)(10 139)(11 140)(12 141)(13 142)(14 143)(15 144)(16 137)(17 108)(18 109)(19 110)(20 111)(21 112)(22 105)(23 106)(24 107)(25 125)(26 126)(27 127)(28 128)(29 121)(30 122)(31 123)(32 124)(33 189)(34 190)(35 191)(36 192)(37 185)(38 186)(39 187)(40 188)(49 183)(50 184)(51 177)(52 178)(53 179)(54 180)(55 181)(56 182)(57 115)(58 116)(59 117)(60 118)(61 119)(62 120)(63 113)(64 114)(65 167)(66 168)(67 161)(68 162)(69 163)(70 164)(71 165)(72 166)(73 169)(74 170)(75 171)(76 172)(77 173)(78 174)(79 175)(80 176)(81 195)(82 196)(83 197)(84 198)(85 199)(86 200)(87 193)(88 194)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 145)(96 146)(97 130)(98 131)(99 132)(100 133)(101 134)(102 135)(103 136)(104 129)(153 202)(154 203)(155 204)(156 205)(157 206)(158 207)(159 208)(160 201)(209 222)(210 223)(211 224)(212 217)(213 218)(214 219)(215 220)(216 221)
(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 46)(2 43)(3 48)(4 45)(5 42)(6 47)(7 44)(8 41)(9 138)(10 143)(11 140)(12 137)(13 142)(14 139)(15 144)(16 141)(17 108)(18 105)(19 110)(20 107)(21 112)(22 109)(23 106)(24 111)(25 121)(26 126)(27 123)(28 128)(29 125)(30 122)(31 127)(32 124)(33 189)(34 186)(35 191)(36 188)(37 185)(38 190)(39 187)(40 192)(49 179)(50 184)(51 181)(52 178)(53 183)(54 180)(55 177)(56 182)(57 115)(58 120)(59 117)(60 114)(61 119)(62 116)(63 113)(64 118)(65 163)(66 168)(67 165)(68 162)(69 167)(70 164)(71 161)(72 166)(73 169)(74 174)(75 171)(76 176)(77 173)(78 170)(79 175)(80 172)(81 199)(82 196)(83 193)(84 198)(85 195)(86 200)(87 197)(88 194)(89 147)(90 152)(91 149)(92 146)(93 151)(94 148)(95 145)(96 150)(97 134)(98 131)(99 136)(100 133)(101 130)(102 135)(103 132)(104 129)(153 202)(154 207)(155 204)(156 201)(157 206)(158 203)(159 208)(160 205)(209 218)(210 223)(211 220)(212 217)(213 222)(214 219)(215 224)(216 221)
(1 67 53 172 141 99 18)(2 68 54 173 142 100 19)(3 69 55 174 143 101 20)(4 70 56 175 144 102 21)(5 71 49 176 137 103 22)(6 72 50 169 138 104 23)(7 65 51 170 139 97 24)(8 66 52 171 140 98 17)(9 129 106 47 166 184 73)(10 130 107 48 167 177 74)(11 131 108 41 168 178 75)(12 132 109 42 161 179 76)(13 133 110 43 162 180 77)(14 134 111 44 163 181 78)(15 135 112 45 164 182 79)(16 136 105 46 165 183 80)(25 36 199 146 160 209 116)(26 37 200 147 153 210 117)(27 38 193 148 154 211 118)(28 39 194 149 155 212 119)(29 40 195 150 156 213 120)(30 33 196 151 157 214 113)(31 34 197 152 158 215 114)(32 35 198 145 159 216 115)(57 124 191 84 95 208 221)(58 125 192 85 96 201 222)(59 126 185 86 89 202 223)(60 127 186 87 90 203 224)(61 128 187 88 91 204 217)(62 121 188 81 92 205 218)(63 122 189 82 93 206 219)(64 123 190 83 94 207 220)
(1 91)(2 146)(3 93)(4 148)(5 95)(6 150)(7 89)(8 152)(9 62)(10 117)(11 64)(12 119)(13 58)(14 113)(15 60)(16 115)(17 158)(18 204)(19 160)(20 206)(21 154)(22 208)(23 156)(24 202)(25 173)(26 74)(27 175)(28 76)(29 169)(30 78)(31 171)(32 80)(33 181)(34 52)(35 183)(36 54)(37 177)(38 56)(39 179)(40 50)(41 94)(42 149)(43 96)(44 151)(45 90)(46 145)(47 92)(48 147)(49 191)(51 185)(53 187)(55 189)(57 137)(59 139)(61 141)(63 143)(65 86)(66 197)(67 88)(68 199)(69 82)(70 193)(71 84)(72 195)(73 121)(75 123)(77 125)(79 127)(81 166)(83 168)(85 162)(87 164)(97 223)(98 215)(99 217)(100 209)(101 219)(102 211)(103 221)(104 213)(105 159)(106 205)(107 153)(108 207)(109 155)(110 201)(111 157)(112 203)(114 140)(116 142)(118 144)(120 138)(122 174)(124 176)(126 170)(128 172)(129 218)(130 210)(131 220)(132 212)(133 222)(134 214)(135 224)(136 216)(161 194)(163 196)(165 198)(167 200)(178 190)(180 192)(182 186)(184 188)
G:=sub<Sym(224)| (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,41)(9,138)(10,139)(11,140)(12,141)(13,142)(14,143)(15,144)(16,137)(17,108)(18,109)(19,110)(20,111)(21,112)(22,105)(23,106)(24,107)(25,125)(26,126)(27,127)(28,128)(29,121)(30,122)(31,123)(32,124)(33,189)(34,190)(35,191)(36,192)(37,185)(38,186)(39,187)(40,188)(49,183)(50,184)(51,177)(52,178)(53,179)(54,180)(55,181)(56,182)(57,115)(58,116)(59,117)(60,118)(61,119)(62,120)(63,113)(64,114)(65,167)(66,168)(67,161)(68,162)(69,163)(70,164)(71,165)(72,166)(73,169)(74,170)(75,171)(76,172)(77,173)(78,174)(79,175)(80,176)(81,195)(82,196)(83,197)(84,198)(85,199)(86,200)(87,193)(88,194)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,145)(96,146)(97,130)(98,131)(99,132)(100,133)(101,134)(102,135)(103,136)(104,129)(153,202)(154,203)(155,204)(156,205)(157,206)(158,207)(159,208)(160,201)(209,222)(210,223)(211,224)(212,217)(213,218)(214,219)(215,220)(216,221), (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), (1,46)(2,43)(3,48)(4,45)(5,42)(6,47)(7,44)(8,41)(9,138)(10,143)(11,140)(12,137)(13,142)(14,139)(15,144)(16,141)(17,108)(18,105)(19,110)(20,107)(21,112)(22,109)(23,106)(24,111)(25,121)(26,126)(27,123)(28,128)(29,125)(30,122)(31,127)(32,124)(33,189)(34,186)(35,191)(36,188)(37,185)(38,190)(39,187)(40,192)(49,179)(50,184)(51,181)(52,178)(53,183)(54,180)(55,177)(56,182)(57,115)(58,120)(59,117)(60,114)(61,119)(62,116)(63,113)(64,118)(65,163)(66,168)(67,165)(68,162)(69,167)(70,164)(71,161)(72,166)(73,169)(74,174)(75,171)(76,176)(77,173)(78,170)(79,175)(80,172)(81,199)(82,196)(83,193)(84,198)(85,195)(86,200)(87,197)(88,194)(89,147)(90,152)(91,149)(92,146)(93,151)(94,148)(95,145)(96,150)(97,134)(98,131)(99,136)(100,133)(101,130)(102,135)(103,132)(104,129)(153,202)(154,207)(155,204)(156,201)(157,206)(158,203)(159,208)(160,205)(209,218)(210,223)(211,220)(212,217)(213,222)(214,219)(215,224)(216,221), (1,67,53,172,141,99,18)(2,68,54,173,142,100,19)(3,69,55,174,143,101,20)(4,70,56,175,144,102,21)(5,71,49,176,137,103,22)(6,72,50,169,138,104,23)(7,65,51,170,139,97,24)(8,66,52,171,140,98,17)(9,129,106,47,166,184,73)(10,130,107,48,167,177,74)(11,131,108,41,168,178,75)(12,132,109,42,161,179,76)(13,133,110,43,162,180,77)(14,134,111,44,163,181,78)(15,135,112,45,164,182,79)(16,136,105,46,165,183,80)(25,36,199,146,160,209,116)(26,37,200,147,153,210,117)(27,38,193,148,154,211,118)(28,39,194,149,155,212,119)(29,40,195,150,156,213,120)(30,33,196,151,157,214,113)(31,34,197,152,158,215,114)(32,35,198,145,159,216,115)(57,124,191,84,95,208,221)(58,125,192,85,96,201,222)(59,126,185,86,89,202,223)(60,127,186,87,90,203,224)(61,128,187,88,91,204,217)(62,121,188,81,92,205,218)(63,122,189,82,93,206,219)(64,123,190,83,94,207,220), (1,91)(2,146)(3,93)(4,148)(5,95)(6,150)(7,89)(8,152)(9,62)(10,117)(11,64)(12,119)(13,58)(14,113)(15,60)(16,115)(17,158)(18,204)(19,160)(20,206)(21,154)(22,208)(23,156)(24,202)(25,173)(26,74)(27,175)(28,76)(29,169)(30,78)(31,171)(32,80)(33,181)(34,52)(35,183)(36,54)(37,177)(38,56)(39,179)(40,50)(41,94)(42,149)(43,96)(44,151)(45,90)(46,145)(47,92)(48,147)(49,191)(51,185)(53,187)(55,189)(57,137)(59,139)(61,141)(63,143)(65,86)(66,197)(67,88)(68,199)(69,82)(70,193)(71,84)(72,195)(73,121)(75,123)(77,125)(79,127)(81,166)(83,168)(85,162)(87,164)(97,223)(98,215)(99,217)(100,209)(101,219)(102,211)(103,221)(104,213)(105,159)(106,205)(107,153)(108,207)(109,155)(110,201)(111,157)(112,203)(114,140)(116,142)(118,144)(120,138)(122,174)(124,176)(126,170)(128,172)(129,218)(130,210)(131,220)(132,212)(133,222)(134,214)(135,224)(136,216)(161,194)(163,196)(165,198)(167,200)(178,190)(180,192)(182,186)(184,188)>;
G:=Group( (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,41)(9,138)(10,139)(11,140)(12,141)(13,142)(14,143)(15,144)(16,137)(17,108)(18,109)(19,110)(20,111)(21,112)(22,105)(23,106)(24,107)(25,125)(26,126)(27,127)(28,128)(29,121)(30,122)(31,123)(32,124)(33,189)(34,190)(35,191)(36,192)(37,185)(38,186)(39,187)(40,188)(49,183)(50,184)(51,177)(52,178)(53,179)(54,180)(55,181)(56,182)(57,115)(58,116)(59,117)(60,118)(61,119)(62,120)(63,113)(64,114)(65,167)(66,168)(67,161)(68,162)(69,163)(70,164)(71,165)(72,166)(73,169)(74,170)(75,171)(76,172)(77,173)(78,174)(79,175)(80,176)(81,195)(82,196)(83,197)(84,198)(85,199)(86,200)(87,193)(88,194)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,145)(96,146)(97,130)(98,131)(99,132)(100,133)(101,134)(102,135)(103,136)(104,129)(153,202)(154,203)(155,204)(156,205)(157,206)(158,207)(159,208)(160,201)(209,222)(210,223)(211,224)(212,217)(213,218)(214,219)(215,220)(216,221), (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), (1,46)(2,43)(3,48)(4,45)(5,42)(6,47)(7,44)(8,41)(9,138)(10,143)(11,140)(12,137)(13,142)(14,139)(15,144)(16,141)(17,108)(18,105)(19,110)(20,107)(21,112)(22,109)(23,106)(24,111)(25,121)(26,126)(27,123)(28,128)(29,125)(30,122)(31,127)(32,124)(33,189)(34,186)(35,191)(36,188)(37,185)(38,190)(39,187)(40,192)(49,179)(50,184)(51,181)(52,178)(53,183)(54,180)(55,177)(56,182)(57,115)(58,120)(59,117)(60,114)(61,119)(62,116)(63,113)(64,118)(65,163)(66,168)(67,165)(68,162)(69,167)(70,164)(71,161)(72,166)(73,169)(74,174)(75,171)(76,176)(77,173)(78,170)(79,175)(80,172)(81,199)(82,196)(83,193)(84,198)(85,195)(86,200)(87,197)(88,194)(89,147)(90,152)(91,149)(92,146)(93,151)(94,148)(95,145)(96,150)(97,134)(98,131)(99,136)(100,133)(101,130)(102,135)(103,132)(104,129)(153,202)(154,207)(155,204)(156,201)(157,206)(158,203)(159,208)(160,205)(209,218)(210,223)(211,220)(212,217)(213,222)(214,219)(215,224)(216,221), (1,67,53,172,141,99,18)(2,68,54,173,142,100,19)(3,69,55,174,143,101,20)(4,70,56,175,144,102,21)(5,71,49,176,137,103,22)(6,72,50,169,138,104,23)(7,65,51,170,139,97,24)(8,66,52,171,140,98,17)(9,129,106,47,166,184,73)(10,130,107,48,167,177,74)(11,131,108,41,168,178,75)(12,132,109,42,161,179,76)(13,133,110,43,162,180,77)(14,134,111,44,163,181,78)(15,135,112,45,164,182,79)(16,136,105,46,165,183,80)(25,36,199,146,160,209,116)(26,37,200,147,153,210,117)(27,38,193,148,154,211,118)(28,39,194,149,155,212,119)(29,40,195,150,156,213,120)(30,33,196,151,157,214,113)(31,34,197,152,158,215,114)(32,35,198,145,159,216,115)(57,124,191,84,95,208,221)(58,125,192,85,96,201,222)(59,126,185,86,89,202,223)(60,127,186,87,90,203,224)(61,128,187,88,91,204,217)(62,121,188,81,92,205,218)(63,122,189,82,93,206,219)(64,123,190,83,94,207,220), (1,91)(2,146)(3,93)(4,148)(5,95)(6,150)(7,89)(8,152)(9,62)(10,117)(11,64)(12,119)(13,58)(14,113)(15,60)(16,115)(17,158)(18,204)(19,160)(20,206)(21,154)(22,208)(23,156)(24,202)(25,173)(26,74)(27,175)(28,76)(29,169)(30,78)(31,171)(32,80)(33,181)(34,52)(35,183)(36,54)(37,177)(38,56)(39,179)(40,50)(41,94)(42,149)(43,96)(44,151)(45,90)(46,145)(47,92)(48,147)(49,191)(51,185)(53,187)(55,189)(57,137)(59,139)(61,141)(63,143)(65,86)(66,197)(67,88)(68,199)(69,82)(70,193)(71,84)(72,195)(73,121)(75,123)(77,125)(79,127)(81,166)(83,168)(85,162)(87,164)(97,223)(98,215)(99,217)(100,209)(101,219)(102,211)(103,221)(104,213)(105,159)(106,205)(107,153)(108,207)(109,155)(110,201)(111,157)(112,203)(114,140)(116,142)(118,144)(120,138)(122,174)(124,176)(126,170)(128,172)(129,218)(130,210)(131,220)(132,212)(133,222)(134,214)(135,224)(136,216)(161,194)(163,196)(165,198)(167,200)(178,190)(180,192)(182,186)(184,188) );
G=PermutationGroup([[(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,41),(9,138),(10,139),(11,140),(12,141),(13,142),(14,143),(15,144),(16,137),(17,108),(18,109),(19,110),(20,111),(21,112),(22,105),(23,106),(24,107),(25,125),(26,126),(27,127),(28,128),(29,121),(30,122),(31,123),(32,124),(33,189),(34,190),(35,191),(36,192),(37,185),(38,186),(39,187),(40,188),(49,183),(50,184),(51,177),(52,178),(53,179),(54,180),(55,181),(56,182),(57,115),(58,116),(59,117),(60,118),(61,119),(62,120),(63,113),(64,114),(65,167),(66,168),(67,161),(68,162),(69,163),(70,164),(71,165),(72,166),(73,169),(74,170),(75,171),(76,172),(77,173),(78,174),(79,175),(80,176),(81,195),(82,196),(83,197),(84,198),(85,199),(86,200),(87,193),(88,194),(89,147),(90,148),(91,149),(92,150),(93,151),(94,152),(95,145),(96,146),(97,130),(98,131),(99,132),(100,133),(101,134),(102,135),(103,136),(104,129),(153,202),(154,203),(155,204),(156,205),(157,206),(158,207),(159,208),(160,201),(209,222),(210,223),(211,224),(212,217),(213,218),(214,219),(215,220),(216,221)], [(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)], [(1,46),(2,43),(3,48),(4,45),(5,42),(6,47),(7,44),(8,41),(9,138),(10,143),(11,140),(12,137),(13,142),(14,139),(15,144),(16,141),(17,108),(18,105),(19,110),(20,107),(21,112),(22,109),(23,106),(24,111),(25,121),(26,126),(27,123),(28,128),(29,125),(30,122),(31,127),(32,124),(33,189),(34,186),(35,191),(36,188),(37,185),(38,190),(39,187),(40,192),(49,179),(50,184),(51,181),(52,178),(53,183),(54,180),(55,177),(56,182),(57,115),(58,120),(59,117),(60,114),(61,119),(62,116),(63,113),(64,118),(65,163),(66,168),(67,165),(68,162),(69,167),(70,164),(71,161),(72,166),(73,169),(74,174),(75,171),(76,176),(77,173),(78,170),(79,175),(80,172),(81,199),(82,196),(83,193),(84,198),(85,195),(86,200),(87,197),(88,194),(89,147),(90,152),(91,149),(92,146),(93,151),(94,148),(95,145),(96,150),(97,134),(98,131),(99,136),(100,133),(101,130),(102,135),(103,132),(104,129),(153,202),(154,207),(155,204),(156,201),(157,206),(158,203),(159,208),(160,205),(209,218),(210,223),(211,220),(212,217),(213,222),(214,219),(215,224),(216,221)], [(1,67,53,172,141,99,18),(2,68,54,173,142,100,19),(3,69,55,174,143,101,20),(4,70,56,175,144,102,21),(5,71,49,176,137,103,22),(6,72,50,169,138,104,23),(7,65,51,170,139,97,24),(8,66,52,171,140,98,17),(9,129,106,47,166,184,73),(10,130,107,48,167,177,74),(11,131,108,41,168,178,75),(12,132,109,42,161,179,76),(13,133,110,43,162,180,77),(14,134,111,44,163,181,78),(15,135,112,45,164,182,79),(16,136,105,46,165,183,80),(25,36,199,146,160,209,116),(26,37,200,147,153,210,117),(27,38,193,148,154,211,118),(28,39,194,149,155,212,119),(29,40,195,150,156,213,120),(30,33,196,151,157,214,113),(31,34,197,152,158,215,114),(32,35,198,145,159,216,115),(57,124,191,84,95,208,221),(58,125,192,85,96,201,222),(59,126,185,86,89,202,223),(60,127,186,87,90,203,224),(61,128,187,88,91,204,217),(62,121,188,81,92,205,218),(63,122,189,82,93,206,219),(64,123,190,83,94,207,220)], [(1,91),(2,146),(3,93),(4,148),(5,95),(6,150),(7,89),(8,152),(9,62),(10,117),(11,64),(12,119),(13,58),(14,113),(15,60),(16,115),(17,158),(18,204),(19,160),(20,206),(21,154),(22,208),(23,156),(24,202),(25,173),(26,74),(27,175),(28,76),(29,169),(30,78),(31,171),(32,80),(33,181),(34,52),(35,183),(36,54),(37,177),(38,56),(39,179),(40,50),(41,94),(42,149),(43,96),(44,151),(45,90),(46,145),(47,92),(48,147),(49,191),(51,185),(53,187),(55,189),(57,137),(59,139),(61,141),(63,143),(65,86),(66,197),(67,88),(68,199),(69,82),(70,193),(71,84),(72,195),(73,121),(75,123),(77,125),(79,127),(81,166),(83,168),(85,162),(87,164),(97,223),(98,215),(99,217),(100,209),(101,219),(102,211),(103,221),(104,213),(105,159),(106,205),(107,153),(108,207),(109,155),(110,201),(111,157),(112,203),(114,140),(116,142),(118,144),(120,138),(122,174),(124,176),(126,170),(128,172),(129,218),(130,210),(131,220),(132,212),(133,222),(134,214),(135,224),(136,216),(161,194),(163,196),(165,198),(167,200),(178,190),(180,192),(182,186),(184,188)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | D14 | D14 | C8○D4 | C4×D7 | D28 | C7⋊D4 | C4×D7 | D28.C4 |
| kernel | (C2×D28).14C4 | D14⋊C8 | C22×C7⋊C8 | C14×M4(2) | C2×C4○D28 | C2×Dic14 | C2×D28 | C2×C7⋊D4 | C2×C28 | C2×M4(2) | C2×C8 | C22×C4 | C14 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 3 | 6 | 3 | 8 | 6 | 12 | 12 | 6 | 12 |
Matrix representation of (C2×D28).14C4 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 58 | 75 | 0 | 0 |
| 38 | 55 | 0 | 0 |
| 0 | 0 | 98 | 2 |
| 0 | 0 | 106 | 15 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 15 | 112 |
| 0 | 1 | 0 | 0 |
| 112 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 104 | 1 | 0 | 0 |
| 33 | 9 | 0 | 0 |
| 0 | 0 | 69 | 36 |
| 0 | 0 | 31 | 44 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[58,38,0,0,75,55,0,0,0,0,98,106,0,0,2,15],[112,0,0,0,0,112,0,0,0,0,1,15,0,0,0,112],[0,112,0,0,1,9,0,0,0,0,1,0,0,0,0,1],[104,33,0,0,1,9,0,0,0,0,69,31,0,0,36,44] >;
(C2×D28).14C4 in GAP, Magma, Sage, TeX
(C_2\times D_{28})._{14}C_4 % in TeX
G:=Group("(C2xD28).14C4"); // GroupNames label
G:=SmallGroup(448,663);
// by ID
G=gap.SmallGroup(448,663);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,422,387,58,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^5,b*d=d*b,e*b*e=a*b^5,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations