metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14⋊C8⋊2C2, (C22×C8)⋊3D7, (C22×C56)⋊2C2, (C2×C4).95D28, C4.86(C2×D28), (C2×D28).10C4, (C2×C28).482D4, C28.436(C2×D4), (C2×C8).293D14, C23.34(C4×D7), C4.26(D14⋊C4), C14.18(C8○D4), C28.51(C22⋊C4), (C2×C56).354C22, (C2×C28).860C23, C22.5(D14⋊C4), (C2×Dic14).10C4, (C22×C4).427D14, C2.18(D28.2C4), (C22×C28).541C22, (C2×C7⋊D4).8C4, C2.23(C2×D14⋊C4), (C2×C4○D28).4C2, (C2×C4).115(C4×D7), C4.126(C2×C7⋊D4), (C2×C4.Dic7)⋊6C2, C22.141(C2×C4×D7), (C2×C28).228(C2×C4), C7⋊2((C22×C8)⋊C2), (C2×C7⋊C8).204C22, C14.51(C2×C22⋊C4), (C2×C4×D7).183C22, (C2×C4).253(C7⋊D4), (C22×C14).95(C2×C4), (C2×Dic7).31(C2×C4), (C22×D7).23(C2×C4), (C2×C4).802(C22×D7), (C2×C14).62(C22⋊C4), (C2×C14).130(C22×C4), SmallGroup(448,644)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C22×C8)⋊D7
G = < a,b,c,d,e | a2=b2=c8=d7=e2=1, ab=ba, ece=ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bc4, cd=dc, 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, C4.Dic7, C2×C56, C2×C56, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, D14⋊C8, C2×C4.Dic7, C22×C56, C2×C4○D28, (C22×C8)⋊D7
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, D28.2C4, C2×D14⋊C4, (C22×C8)⋊D7
►On 224 points
Generators in S224
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 165)(10 166)(11 167)(12 168)(13 161)(14 162)(15 163)(16 164)(17 110)(18 111)(19 112)(20 105)(21 106)(22 107)(23 108)(24 109)(25 150)(26 151)(27 152)(28 145)(29 146)(30 147)(31 148)(32 149)(33 185)(34 186)(35 187)(36 188)(37 189)(38 190)(39 191)(40 192)(49 179)(50 180)(51 181)(52 182)(53 183)(54 184)(55 177)(56 178)(57 196)(58 197)(59 198)(60 199)(61 200)(62 193)(63 194)(64 195)(65 173)(66 174)(67 175)(68 176)(69 169)(70 170)(71 171)(72 172)(73 144)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 113)(89 122)(90 123)(91 124)(92 125)(93 126)(94 127)(95 128)(96 121)(97 134)(98 135)(99 136)(100 129)(101 130)(102 131)(103 132)(104 133)(153 206)(154 207)(155 208)(156 201)(157 202)(158 203)(159 204)(160 205)(209 218)(210 219)(211 220)(212 221)(213 222)(214 223)(215 224)(216 217)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 165)(10 166)(11 167)(12 168)(13 161)(14 162)(15 163)(16 164)(17 110)(18 111)(19 112)(20 105)(21 106)(22 107)(23 108)(24 109)(25 146)(26 147)(27 148)(28 149)(29 150)(30 151)(31 152)(32 145)(33 189)(34 190)(35 191)(36 192)(37 185)(38 186)(39 187)(40 188)(49 179)(50 180)(51 181)(52 182)(53 183)(54 184)(55 177)(56 178)(57 200)(58 193)(59 194)(60 195)(61 196)(62 197)(63 198)(64 199)(65 173)(66 174)(67 175)(68 176)(69 169)(70 170)(71 171)(72 172)(73 144)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(81 118)(82 119)(83 120)(84 113)(85 114)(86 115)(87 116)(88 117)(89 126)(90 127)(91 128)(92 121)(93 122)(94 123)(95 124)(96 125)(97 134)(98 135)(99 136)(100 129)(101 130)(102 131)(103 132)(104 133)(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 9 78 179 176 99 20)(2 10 79 180 169 100 21)(3 11 80 181 170 101 22)(4 12 73 182 171 102 23)(5 13 74 183 172 103 24)(6 14 75 184 173 104 17)(7 15 76 177 174 97 18)(8 16 77 178 175 98 19)(25 36 199 125 160 209 116)(26 37 200 126 153 210 117)(27 38 193 127 154 211 118)(28 39 194 128 155 212 119)(29 40 195 121 156 213 120)(30 33 196 122 157 214 113)(31 34 197 123 158 215 114)(32 35 198 124 159 216 115)(41 163 139 55 66 134 111)(42 164 140 56 67 135 112)(43 165 141 49 68 136 105)(44 166 142 50 69 129 106)(45 167 143 51 70 130 107)(46 168 144 52 71 131 108)(47 161 137 53 72 132 109)(48 162 138 54 65 133 110)(57 89 202 223 88 147 185)(58 90 203 224 81 148 186)(59 91 204 217 82 149 187)(60 92 205 218 83 150 188)(61 93 206 219 84 151 189)(62 94 207 220 85 152 190)(63 95 208 221 86 145 191)(64 96 201 222 87 146 192)
(1 91)(2 125)(3 93)(4 127)(5 95)(6 121)(7 89)(8 123)(9 59)(10 199)(11 61)(12 193)(13 63)(14 195)(15 57)(16 197)(17 156)(18 202)(19 158)(20 204)(21 160)(22 206)(23 154)(24 208)(25 180)(26 51)(27 182)(28 53)(29 184)(30 55)(31 178)(32 49)(33 139)(34 77)(35 141)(36 79)(37 143)(38 73)(39 137)(40 75)(41 122)(42 90)(43 124)(44 92)(45 126)(46 94)(47 128)(48 96)(50 150)(52 152)(54 146)(56 148)(58 164)(60 166)(62 168)(64 162)(65 87)(66 113)(67 81)(68 115)(69 83)(70 117)(71 85)(72 119)(74 191)(76 185)(78 187)(80 189)(82 176)(84 170)(86 172)(88 174)(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)(114 175)(116 169)(118 171)(120 173)(129 218)(130 210)(131 220)(132 212)(133 222)(134 214)(135 224)(136 216)(138 192)(140 186)(142 188)(144 190)(145 183)(147 177)(149 179)(151 181)(161 194)(163 196)(165 198)(167 200)
G:=sub<Sym(224)| (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,165)(10,166)(11,167)(12,168)(13,161)(14,162)(15,163)(16,164)(17,110)(18,111)(19,112)(20,105)(21,106)(22,107)(23,108)(24,109)(25,150)(26,151)(27,152)(28,145)(29,146)(30,147)(31,148)(32,149)(33,185)(34,186)(35,187)(36,188)(37,189)(38,190)(39,191)(40,192)(49,179)(50,180)(51,181)(52,182)(53,183)(54,184)(55,177)(56,178)(57,196)(58,197)(59,198)(60,199)(61,200)(62,193)(63,194)(64,195)(65,173)(66,174)(67,175)(68,176)(69,169)(70,170)(71,171)(72,172)(73,144)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,113)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,121)(97,134)(98,135)(99,136)(100,129)(101,130)(102,131)(103,132)(104,133)(153,206)(154,207)(155,208)(156,201)(157,202)(158,203)(159,204)(160,205)(209,218)(210,219)(211,220)(212,221)(213,222)(214,223)(215,224)(216,217), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,165)(10,166)(11,167)(12,168)(13,161)(14,162)(15,163)(16,164)(17,110)(18,111)(19,112)(20,105)(21,106)(22,107)(23,108)(24,109)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,145)(33,189)(34,190)(35,191)(36,192)(37,185)(38,186)(39,187)(40,188)(49,179)(50,180)(51,181)(52,182)(53,183)(54,184)(55,177)(56,178)(57,200)(58,193)(59,194)(60,195)(61,196)(62,197)(63,198)(64,199)(65,173)(66,174)(67,175)(68,176)(69,169)(70,170)(71,171)(72,172)(73,144)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,118)(82,119)(83,120)(84,113)(85,114)(86,115)(87,116)(88,117)(89,126)(90,127)(91,128)(92,121)(93,122)(94,123)(95,124)(96,125)(97,134)(98,135)(99,136)(100,129)(101,130)(102,131)(103,132)(104,133)(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), (1,9,78,179,176,99,20)(2,10,79,180,169,100,21)(3,11,80,181,170,101,22)(4,12,73,182,171,102,23)(5,13,74,183,172,103,24)(6,14,75,184,173,104,17)(7,15,76,177,174,97,18)(8,16,77,178,175,98,19)(25,36,199,125,160,209,116)(26,37,200,126,153,210,117)(27,38,193,127,154,211,118)(28,39,194,128,155,212,119)(29,40,195,121,156,213,120)(30,33,196,122,157,214,113)(31,34,197,123,158,215,114)(32,35,198,124,159,216,115)(41,163,139,55,66,134,111)(42,164,140,56,67,135,112)(43,165,141,49,68,136,105)(44,166,142,50,69,129,106)(45,167,143,51,70,130,107)(46,168,144,52,71,131,108)(47,161,137,53,72,132,109)(48,162,138,54,65,133,110)(57,89,202,223,88,147,185)(58,90,203,224,81,148,186)(59,91,204,217,82,149,187)(60,92,205,218,83,150,188)(61,93,206,219,84,151,189)(62,94,207,220,85,152,190)(63,95,208,221,86,145,191)(64,96,201,222,87,146,192), (1,91)(2,125)(3,93)(4,127)(5,95)(6,121)(7,89)(8,123)(9,59)(10,199)(11,61)(12,193)(13,63)(14,195)(15,57)(16,197)(17,156)(18,202)(19,158)(20,204)(21,160)(22,206)(23,154)(24,208)(25,180)(26,51)(27,182)(28,53)(29,184)(30,55)(31,178)(32,49)(33,139)(34,77)(35,141)(36,79)(37,143)(38,73)(39,137)(40,75)(41,122)(42,90)(43,124)(44,92)(45,126)(46,94)(47,128)(48,96)(50,150)(52,152)(54,146)(56,148)(58,164)(60,166)(62,168)(64,162)(65,87)(66,113)(67,81)(68,115)(69,83)(70,117)(71,85)(72,119)(74,191)(76,185)(78,187)(80,189)(82,176)(84,170)(86,172)(88,174)(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)(114,175)(116,169)(118,171)(120,173)(129,218)(130,210)(131,220)(132,212)(133,222)(134,214)(135,224)(136,216)(138,192)(140,186)(142,188)(144,190)(145,183)(147,177)(149,179)(151,181)(161,194)(163,196)(165,198)(167,200)>;
G:=Group( (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,165)(10,166)(11,167)(12,168)(13,161)(14,162)(15,163)(16,164)(17,110)(18,111)(19,112)(20,105)(21,106)(22,107)(23,108)(24,109)(25,150)(26,151)(27,152)(28,145)(29,146)(30,147)(31,148)(32,149)(33,185)(34,186)(35,187)(36,188)(37,189)(38,190)(39,191)(40,192)(49,179)(50,180)(51,181)(52,182)(53,183)(54,184)(55,177)(56,178)(57,196)(58,197)(59,198)(60,199)(61,200)(62,193)(63,194)(64,195)(65,173)(66,174)(67,175)(68,176)(69,169)(70,170)(71,171)(72,172)(73,144)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,114)(82,115)(83,116)(84,117)(85,118)(86,119)(87,120)(88,113)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,121)(97,134)(98,135)(99,136)(100,129)(101,130)(102,131)(103,132)(104,133)(153,206)(154,207)(155,208)(156,201)(157,202)(158,203)(159,204)(160,205)(209,218)(210,219)(211,220)(212,221)(213,222)(214,223)(215,224)(216,217), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,41)(8,42)(9,165)(10,166)(11,167)(12,168)(13,161)(14,162)(15,163)(16,164)(17,110)(18,111)(19,112)(20,105)(21,106)(22,107)(23,108)(24,109)(25,146)(26,147)(27,148)(28,149)(29,150)(30,151)(31,152)(32,145)(33,189)(34,190)(35,191)(36,192)(37,185)(38,186)(39,187)(40,188)(49,179)(50,180)(51,181)(52,182)(53,183)(54,184)(55,177)(56,178)(57,200)(58,193)(59,194)(60,195)(61,196)(62,197)(63,198)(64,199)(65,173)(66,174)(67,175)(68,176)(69,169)(70,170)(71,171)(72,172)(73,144)(74,137)(75,138)(76,139)(77,140)(78,141)(79,142)(80,143)(81,118)(82,119)(83,120)(84,113)(85,114)(86,115)(87,116)(88,117)(89,126)(90,127)(91,128)(92,121)(93,122)(94,123)(95,124)(96,125)(97,134)(98,135)(99,136)(100,129)(101,130)(102,131)(103,132)(104,133)(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), (1,9,78,179,176,99,20)(2,10,79,180,169,100,21)(3,11,80,181,170,101,22)(4,12,73,182,171,102,23)(5,13,74,183,172,103,24)(6,14,75,184,173,104,17)(7,15,76,177,174,97,18)(8,16,77,178,175,98,19)(25,36,199,125,160,209,116)(26,37,200,126,153,210,117)(27,38,193,127,154,211,118)(28,39,194,128,155,212,119)(29,40,195,121,156,213,120)(30,33,196,122,157,214,113)(31,34,197,123,158,215,114)(32,35,198,124,159,216,115)(41,163,139,55,66,134,111)(42,164,140,56,67,135,112)(43,165,141,49,68,136,105)(44,166,142,50,69,129,106)(45,167,143,51,70,130,107)(46,168,144,52,71,131,108)(47,161,137,53,72,132,109)(48,162,138,54,65,133,110)(57,89,202,223,88,147,185)(58,90,203,224,81,148,186)(59,91,204,217,82,149,187)(60,92,205,218,83,150,188)(61,93,206,219,84,151,189)(62,94,207,220,85,152,190)(63,95,208,221,86,145,191)(64,96,201,222,87,146,192), (1,91)(2,125)(3,93)(4,127)(5,95)(6,121)(7,89)(8,123)(9,59)(10,199)(11,61)(12,193)(13,63)(14,195)(15,57)(16,197)(17,156)(18,202)(19,158)(20,204)(21,160)(22,206)(23,154)(24,208)(25,180)(26,51)(27,182)(28,53)(29,184)(30,55)(31,178)(32,49)(33,139)(34,77)(35,141)(36,79)(37,143)(38,73)(39,137)(40,75)(41,122)(42,90)(43,124)(44,92)(45,126)(46,94)(47,128)(48,96)(50,150)(52,152)(54,146)(56,148)(58,164)(60,166)(62,168)(64,162)(65,87)(66,113)(67,81)(68,115)(69,83)(70,117)(71,85)(72,119)(74,191)(76,185)(78,187)(80,189)(82,176)(84,170)(86,172)(88,174)(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)(114,175)(116,169)(118,171)(120,173)(129,218)(130,210)(131,220)(132,212)(133,222)(134,214)(135,224)(136,216)(138,192)(140,186)(142,188)(144,190)(145,183)(147,177)(149,179)(151,181)(161,194)(163,196)(165,198)(167,200) );
G=PermutationGroup([[(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,41),(8,42),(9,165),(10,166),(11,167),(12,168),(13,161),(14,162),(15,163),(16,164),(17,110),(18,111),(19,112),(20,105),(21,106),(22,107),(23,108),(24,109),(25,150),(26,151),(27,152),(28,145),(29,146),(30,147),(31,148),(32,149),(33,185),(34,186),(35,187),(36,188),(37,189),(38,190),(39,191),(40,192),(49,179),(50,180),(51,181),(52,182),(53,183),(54,184),(55,177),(56,178),(57,196),(58,197),(59,198),(60,199),(61,200),(62,193),(63,194),(64,195),(65,173),(66,174),(67,175),(68,176),(69,169),(70,170),(71,171),(72,172),(73,144),(74,137),(75,138),(76,139),(77,140),(78,141),(79,142),(80,143),(81,114),(82,115),(83,116),(84,117),(85,118),(86,119),(87,120),(88,113),(89,122),(90,123),(91,124),(92,125),(93,126),(94,127),(95,128),(96,121),(97,134),(98,135),(99,136),(100,129),(101,130),(102,131),(103,132),(104,133),(153,206),(154,207),(155,208),(156,201),(157,202),(158,203),(159,204),(160,205),(209,218),(210,219),(211,220),(212,221),(213,222),(214,223),(215,224),(216,217)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,41),(8,42),(9,165),(10,166),(11,167),(12,168),(13,161),(14,162),(15,163),(16,164),(17,110),(18,111),(19,112),(20,105),(21,106),(22,107),(23,108),(24,109),(25,146),(26,147),(27,148),(28,149),(29,150),(30,151),(31,152),(32,145),(33,189),(34,190),(35,191),(36,192),(37,185),(38,186),(39,187),(40,188),(49,179),(50,180),(51,181),(52,182),(53,183),(54,184),(55,177),(56,178),(57,200),(58,193),(59,194),(60,195),(61,196),(62,197),(63,198),(64,199),(65,173),(66,174),(67,175),(68,176),(69,169),(70,170),(71,171),(72,172),(73,144),(74,137),(75,138),(76,139),(77,140),(78,141),(79,142),(80,143),(81,118),(82,119),(83,120),(84,113),(85,114),(86,115),(87,116),(88,117),(89,126),(90,127),(91,128),(92,121),(93,122),(94,123),(95,124),(96,125),(97,134),(98,135),(99,136),(100,129),(101,130),(102,131),(103,132),(104,133),(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)], [(1,9,78,179,176,99,20),(2,10,79,180,169,100,21),(3,11,80,181,170,101,22),(4,12,73,182,171,102,23),(5,13,74,183,172,103,24),(6,14,75,184,173,104,17),(7,15,76,177,174,97,18),(8,16,77,178,175,98,19),(25,36,199,125,160,209,116),(26,37,200,126,153,210,117),(27,38,193,127,154,211,118),(28,39,194,128,155,212,119),(29,40,195,121,156,213,120),(30,33,196,122,157,214,113),(31,34,197,123,158,215,114),(32,35,198,124,159,216,115),(41,163,139,55,66,134,111),(42,164,140,56,67,135,112),(43,165,141,49,68,136,105),(44,166,142,50,69,129,106),(45,167,143,51,70,130,107),(46,168,144,52,71,131,108),(47,161,137,53,72,132,109),(48,162,138,54,65,133,110),(57,89,202,223,88,147,185),(58,90,203,224,81,148,186),(59,91,204,217,82,149,187),(60,92,205,218,83,150,188),(61,93,206,219,84,151,189),(62,94,207,220,85,152,190),(63,95,208,221,86,145,191),(64,96,201,222,87,146,192)], [(1,91),(2,125),(3,93),(4,127),(5,95),(6,121),(7,89),(8,123),(9,59),(10,199),(11,61),(12,193),(13,63),(14,195),(15,57),(16,197),(17,156),(18,202),(19,158),(20,204),(21,160),(22,206),(23,154),(24,208),(25,180),(26,51),(27,182),(28,53),(29,184),(30,55),(31,178),(32,49),(33,139),(34,77),(35,141),(36,79),(37,143),(38,73),(39,137),(40,75),(41,122),(42,90),(43,124),(44,92),(45,126),(46,94),(47,128),(48,96),(50,150),(52,152),(54,146),(56,148),(58,164),(60,166),(62,168),(64,162),(65,87),(66,113),(67,81),(68,115),(69,83),(70,117),(71,85),(72,119),(74,191),(76,185),(78,187),(80,189),(82,176),(84,170),(86,172),(88,174),(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),(114,175),(116,169),(118,171),(120,173),(129,218),(130,210),(131,220),(132,212),(133,222),(134,214),(135,224),(136,216),(138,192),(140,186),(142,188),(144,190),(145,183),(147,177),(149,179),(151,181),(161,194),(163,196),(165,198),(167,200)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| 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 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D7 | D14 | D14 | C8○D4 | C4×D7 | D28 | C7⋊D4 | C4×D7 | D28.2C4 |
| kernel | (C22×C8)⋊D7 | D14⋊C8 | C2×C4.Dic7 | C22×C56 | C2×C4○D28 | C2×Dic14 | C2×D28 | C2×C7⋊D4 | C2×C28 | C22×C8 | 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 | 48 |
Matrix representation of (C22×C8)⋊D7 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 54 | 112 |
| 95 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 95 | 0 |
| 0 | 0 | 0 | 95 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 73 | 109 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 91 | 5 |
| 0 | 0 | 39 | 22 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,1,0,0,0,0,1,54,0,0,0,112],[95,0,0,0,0,18,0,0,0,0,95,0,0,0,0,95],[1,0,0,0,0,1,0,0,0,0,28,73,0,0,0,109],[0,1,0,0,1,0,0,0,0,0,91,39,0,0,5,22] >;
(C22×C8)⋊D7 in GAP, Magma, Sage, TeX
(C_2^2\times C_8)\rtimes D_7
% in TeX
G:=Group("(C2^2xC8):D7"); // GroupNames label
G:=SmallGroup(448,644);
// by ID
G=gap.SmallGroup(448,644);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,422,58,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^8=d^7=e^2=1,a*b=b*a,e*c*e=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b*c^4,c*d=d*c,e*d*e=d^-1>;
// generators/relations