direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2xC4:C4:7D7, C4:C4:51D14, (C2xC14).46C24, C28.88(C22xC4), C14.12(C23xC4), C4:Dic7:70C22, C14:3(C42:C2), (C2xC28).578C23, (C4xDic7):75C22, D14.18(C22xC4), (C22xC4).359D14, C22.22(C23xD7), D14:C4.115C22, Dic7.23(C22xC4), (C23xD7).97C22, C23.325(C22xD7), C22.73(D4:2D7), (C22xC14).395C23, (C22xC28).358C22, C22.33(Q8:2D7), (C2xDic7).185C23, (C22xD7).153C23, (C22xDic7).210C22, (C2xC4xD7):6C4, (C14xC4:C4):8C2, C4.91(C2xC4xD7), (C2xC4:C4):25D7, (C4xD7):12(C2xC4), C7:3(C2xC42:C2), (C2xC4xDic7):32C2, (D7xC22xC4).4C2, C2.14(D7xC22xC4), (C7xC4:C4):43C22, C22.72(C2xC4xD7), (C2xC4:Dic7):38C2, C14.71(C2xC4oD4), (C2xC4).161(C4xD7), C2.4(C2xD4:2D7), C2.1(C2xQ8:2D7), (C2xC28).128(C2xC4), (C2xD14:C4).23C2, (C2xC4xD7).243C22, (C22xD7).65(C2xC4), (C2xC4).265(C22xD7), (C2xC14).171(C4oD4), (C2xC14).151(C22xC4), (C2xDic7).111(C2xC4), SmallGroup(448,955)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xC4:C4:7D7
G = < a,b,c,d,e | a2=b4=c4=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 1284 in 330 conjugacy classes, 167 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2xC4, C2xC4, C23, C23, D7, C14, C14, C42, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C22xC4, C24, Dic7, Dic7, C28, C28, D14, D14, C2xC14, C2xC14, C2xC42, C2xC22:C4, C2xC4:C4, C2xC4:C4, C42:C2, C23xC4, C4xD7, C2xDic7, C2xDic7, C2xC28, C2xC28, C22xD7, C22xD7, C22xC14, C2xC42:C2, C4xDic7, C4:Dic7, D14:C4, C7xC4:C4, C2xC4xD7, C22xDic7, C22xDic7, C22xC28, C22xC28, C23xD7, C4:C4:7D7, C2xC4xDic7, C2xC4:Dic7, C2xD14:C4, C14xC4:C4, D7xC22xC4, C2xC4:C4:7D7
Quotients: C1, C2, C4, C22, C2xC4, C23, D7, C22xC4, C4oD4, C24, D14, C42:C2, C23xC4, C2xC4oD4, C4xD7, C22xD7, C2xC42:C2, C2xC4xD7, D4:2D7, Q8:2D7, C23xD7, C4:C4:7D7, D7xC22xC4, C2xD4:2D7, C2xQ8:2D7, C2xC4:C4:7D7
►On 224 points
Generators in S224
(1 134)(2 135)(3 136)(4 137)(5 138)(6 139)(7 140)(8 127)(9 128)(10 129)(11 130)(12 131)(13 132)(14 133)(15 120)(16 121)(17 122)(18 123)(19 124)(20 125)(21 126)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 162)(30 163)(31 164)(32 165)(33 166)(34 167)(35 168)(36 155)(37 156)(38 157)(39 158)(40 159)(41 160)(42 161)(43 148)(44 149)(45 150)(46 151)(47 152)(48 153)(49 154)(50 141)(51 142)(52 143)(53 144)(54 145)(55 146)(56 147)(57 190)(58 191)(59 192)(60 193)(61 194)(62 195)(63 196)(64 183)(65 184)(66 185)(67 186)(68 187)(69 188)(70 189)(71 176)(72 177)(73 178)(74 179)(75 180)(76 181)(77 182)(78 169)(79 170)(80 171)(81 172)(82 173)(83 174)(84 175)(85 218)(86 219)(87 220)(88 221)(89 222)(90 223)(91 224)(92 211)(93 212)(94 213)(95 214)(96 215)(97 216)(98 217)(99 204)(100 205)(101 206)(102 207)(103 208)(104 209)(105 210)(106 197)(107 198)(108 199)(109 200)(110 201)(111 202)(112 203)
(1 183 8 190)(2 184 9 191)(3 185 10 192)(4 186 11 193)(5 187 12 194)(6 188 13 195)(7 189 14 196)(15 169 22 176)(16 170 23 177)(17 171 24 178)(18 172 25 179)(19 173 26 180)(20 174 27 181)(21 175 28 182)(29 211 36 218)(30 212 37 219)(31 213 38 220)(32 214 39 221)(33 215 40 222)(34 216 41 223)(35 217 42 224)(43 197 50 204)(44 198 51 205)(45 199 52 206)(46 200 53 207)(47 201 54 208)(48 202 55 209)(49 203 56 210)(57 134 64 127)(58 135 65 128)(59 136 66 129)(60 137 67 130)(61 138 68 131)(62 139 69 132)(63 140 70 133)(71 120 78 113)(72 121 79 114)(73 122 80 115)(74 123 81 116)(75 124 82 117)(76 125 83 118)(77 126 84 119)(85 162 92 155)(86 163 93 156)(87 164 94 157)(88 165 95 158)(89 166 96 159)(90 167 97 160)(91 168 98 161)(99 148 106 141)(100 149 107 142)(101 150 108 143)(102 151 109 144)(103 152 110 145)(104 153 111 146)(105 154 112 147)
(1 99 15 85)(2 100 16 86)(3 101 17 87)(4 102 18 88)(5 103 19 89)(6 104 20 90)(7 105 21 91)(8 106 22 92)(9 107 23 93)(10 108 24 94)(11 109 25 95)(12 110 26 96)(13 111 27 97)(14 112 28 98)(29 57 43 71)(30 58 44 72)(31 59 45 73)(32 60 46 74)(33 61 47 75)(34 62 48 76)(35 63 49 77)(36 64 50 78)(37 65 51 79)(38 66 52 80)(39 67 53 81)(40 68 54 82)(41 69 55 83)(42 70 56 84)(113 211 127 197)(114 212 128 198)(115 213 129 199)(116 214 130 200)(117 215 131 201)(118 216 132 202)(119 217 133 203)(120 218 134 204)(121 219 135 205)(122 220 136 206)(123 221 137 207)(124 222 138 208)(125 223 139 209)(126 224 140 210)(141 169 155 183)(142 170 156 184)(143 171 157 185)(144 172 158 186)(145 173 159 187)(146 174 160 188)(147 175 161 189)(148 176 162 190)(149 177 163 191)(150 178 164 192)(151 179 165 193)(152 180 166 194)(153 181 167 195)(154 182 168 196)
(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)
(1 119)(2 118)(3 117)(4 116)(5 115)(6 114)(7 113)(8 126)(9 125)(10 124)(11 123)(12 122)(13 121)(14 120)(15 133)(16 132)(17 131)(18 130)(19 129)(20 128)(21 127)(22 140)(23 139)(24 138)(25 137)(26 136)(27 135)(28 134)(29 154)(30 153)(31 152)(32 151)(33 150)(34 149)(35 148)(36 147)(37 146)(38 145)(39 144)(40 143)(41 142)(42 141)(43 168)(44 167)(45 166)(46 165)(47 164)(48 163)(49 162)(50 161)(51 160)(52 159)(53 158)(54 157)(55 156)(56 155)(57 175)(58 174)(59 173)(60 172)(61 171)(62 170)(63 169)(64 182)(65 181)(66 180)(67 179)(68 178)(69 177)(70 176)(71 189)(72 188)(73 187)(74 186)(75 185)(76 184)(77 183)(78 196)(79 195)(80 194)(81 193)(82 192)(83 191)(84 190)(85 210)(86 209)(87 208)(88 207)(89 206)(90 205)(91 204)(92 203)(93 202)(94 201)(95 200)(96 199)(97 198)(98 197)(99 224)(100 223)(101 222)(102 221)(103 220)(104 219)(105 218)(106 217)(107 216)(108 215)(109 214)(110 213)(111 212)(112 211)
G:=sub<Sym(224)| (1,134)(2,135)(3,136)(4,137)(5,138)(6,139)(7,140)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,120)(16,121)(17,122)(18,123)(19,124)(20,125)(21,126)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,162)(30,163)(31,164)(32,165)(33,166)(34,167)(35,168)(36,155)(37,156)(38,157)(39,158)(40,159)(41,160)(42,161)(43,148)(44,149)(45,150)(46,151)(47,152)(48,153)(49,154)(50,141)(51,142)(52,143)(53,144)(54,145)(55,146)(56,147)(57,190)(58,191)(59,192)(60,193)(61,194)(62,195)(63,196)(64,183)(65,184)(66,185)(67,186)(68,187)(69,188)(70,189)(71,176)(72,177)(73,178)(74,179)(75,180)(76,181)(77,182)(78,169)(79,170)(80,171)(81,172)(82,173)(83,174)(84,175)(85,218)(86,219)(87,220)(88,221)(89,222)(90,223)(91,224)(92,211)(93,212)(94,213)(95,214)(96,215)(97,216)(98,217)(99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)(106,197)(107,198)(108,199)(109,200)(110,201)(111,202)(112,203), (1,183,8,190)(2,184,9,191)(3,185,10,192)(4,186,11,193)(5,187,12,194)(6,188,13,195)(7,189,14,196)(15,169,22,176)(16,170,23,177)(17,171,24,178)(18,172,25,179)(19,173,26,180)(20,174,27,181)(21,175,28,182)(29,211,36,218)(30,212,37,219)(31,213,38,220)(32,214,39,221)(33,215,40,222)(34,216,41,223)(35,217,42,224)(43,197,50,204)(44,198,51,205)(45,199,52,206)(46,200,53,207)(47,201,54,208)(48,202,55,209)(49,203,56,210)(57,134,64,127)(58,135,65,128)(59,136,66,129)(60,137,67,130)(61,138,68,131)(62,139,69,132)(63,140,70,133)(71,120,78,113)(72,121,79,114)(73,122,80,115)(74,123,81,116)(75,124,82,117)(76,125,83,118)(77,126,84,119)(85,162,92,155)(86,163,93,156)(87,164,94,157)(88,165,95,158)(89,166,96,159)(90,167,97,160)(91,168,98,161)(99,148,106,141)(100,149,107,142)(101,150,108,143)(102,151,109,144)(103,152,110,145)(104,153,111,146)(105,154,112,147), (1,99,15,85)(2,100,16,86)(3,101,17,87)(4,102,18,88)(5,103,19,89)(6,104,20,90)(7,105,21,91)(8,106,22,92)(9,107,23,93)(10,108,24,94)(11,109,25,95)(12,110,26,96)(13,111,27,97)(14,112,28,98)(29,57,43,71)(30,58,44,72)(31,59,45,73)(32,60,46,74)(33,61,47,75)(34,62,48,76)(35,63,49,77)(36,64,50,78)(37,65,51,79)(38,66,52,80)(39,67,53,81)(40,68,54,82)(41,69,55,83)(42,70,56,84)(113,211,127,197)(114,212,128,198)(115,213,129,199)(116,214,130,200)(117,215,131,201)(118,216,132,202)(119,217,133,203)(120,218,134,204)(121,219,135,205)(122,220,136,206)(123,221,137,207)(124,222,138,208)(125,223,139,209)(126,224,140,210)(141,169,155,183)(142,170,156,184)(143,171,157,185)(144,172,158,186)(145,173,159,187)(146,174,160,188)(147,175,161,189)(148,176,162,190)(149,177,163,191)(150,178,164,192)(151,179,165,193)(152,180,166,194)(153,181,167,195)(154,182,168,196), (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), (1,119)(2,118)(3,117)(4,116)(5,115)(6,114)(7,113)(8,126)(9,125)(10,124)(11,123)(12,122)(13,121)(14,120)(15,133)(16,132)(17,131)(18,130)(19,129)(20,128)(21,127)(22,140)(23,139)(24,138)(25,137)(26,136)(27,135)(28,134)(29,154)(30,153)(31,152)(32,151)(33,150)(34,149)(35,148)(36,147)(37,146)(38,145)(39,144)(40,143)(41,142)(42,141)(43,168)(44,167)(45,166)(46,165)(47,164)(48,163)(49,162)(50,161)(51,160)(52,159)(53,158)(54,157)(55,156)(56,155)(57,175)(58,174)(59,173)(60,172)(61,171)(62,170)(63,169)(64,182)(65,181)(66,180)(67,179)(68,178)(69,177)(70,176)(71,189)(72,188)(73,187)(74,186)(75,185)(76,184)(77,183)(78,196)(79,195)(80,194)(81,193)(82,192)(83,191)(84,190)(85,210)(86,209)(87,208)(88,207)(89,206)(90,205)(91,204)(92,203)(93,202)(94,201)(95,200)(96,199)(97,198)(98,197)(99,224)(100,223)(101,222)(102,221)(103,220)(104,219)(105,218)(106,217)(107,216)(108,215)(109,214)(110,213)(111,212)(112,211)>;
G:=Group( (1,134)(2,135)(3,136)(4,137)(5,138)(6,139)(7,140)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,120)(16,121)(17,122)(18,123)(19,124)(20,125)(21,126)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,162)(30,163)(31,164)(32,165)(33,166)(34,167)(35,168)(36,155)(37,156)(38,157)(39,158)(40,159)(41,160)(42,161)(43,148)(44,149)(45,150)(46,151)(47,152)(48,153)(49,154)(50,141)(51,142)(52,143)(53,144)(54,145)(55,146)(56,147)(57,190)(58,191)(59,192)(60,193)(61,194)(62,195)(63,196)(64,183)(65,184)(66,185)(67,186)(68,187)(69,188)(70,189)(71,176)(72,177)(73,178)(74,179)(75,180)(76,181)(77,182)(78,169)(79,170)(80,171)(81,172)(82,173)(83,174)(84,175)(85,218)(86,219)(87,220)(88,221)(89,222)(90,223)(91,224)(92,211)(93,212)(94,213)(95,214)(96,215)(97,216)(98,217)(99,204)(100,205)(101,206)(102,207)(103,208)(104,209)(105,210)(106,197)(107,198)(108,199)(109,200)(110,201)(111,202)(112,203), (1,183,8,190)(2,184,9,191)(3,185,10,192)(4,186,11,193)(5,187,12,194)(6,188,13,195)(7,189,14,196)(15,169,22,176)(16,170,23,177)(17,171,24,178)(18,172,25,179)(19,173,26,180)(20,174,27,181)(21,175,28,182)(29,211,36,218)(30,212,37,219)(31,213,38,220)(32,214,39,221)(33,215,40,222)(34,216,41,223)(35,217,42,224)(43,197,50,204)(44,198,51,205)(45,199,52,206)(46,200,53,207)(47,201,54,208)(48,202,55,209)(49,203,56,210)(57,134,64,127)(58,135,65,128)(59,136,66,129)(60,137,67,130)(61,138,68,131)(62,139,69,132)(63,140,70,133)(71,120,78,113)(72,121,79,114)(73,122,80,115)(74,123,81,116)(75,124,82,117)(76,125,83,118)(77,126,84,119)(85,162,92,155)(86,163,93,156)(87,164,94,157)(88,165,95,158)(89,166,96,159)(90,167,97,160)(91,168,98,161)(99,148,106,141)(100,149,107,142)(101,150,108,143)(102,151,109,144)(103,152,110,145)(104,153,111,146)(105,154,112,147), (1,99,15,85)(2,100,16,86)(3,101,17,87)(4,102,18,88)(5,103,19,89)(6,104,20,90)(7,105,21,91)(8,106,22,92)(9,107,23,93)(10,108,24,94)(11,109,25,95)(12,110,26,96)(13,111,27,97)(14,112,28,98)(29,57,43,71)(30,58,44,72)(31,59,45,73)(32,60,46,74)(33,61,47,75)(34,62,48,76)(35,63,49,77)(36,64,50,78)(37,65,51,79)(38,66,52,80)(39,67,53,81)(40,68,54,82)(41,69,55,83)(42,70,56,84)(113,211,127,197)(114,212,128,198)(115,213,129,199)(116,214,130,200)(117,215,131,201)(118,216,132,202)(119,217,133,203)(120,218,134,204)(121,219,135,205)(122,220,136,206)(123,221,137,207)(124,222,138,208)(125,223,139,209)(126,224,140,210)(141,169,155,183)(142,170,156,184)(143,171,157,185)(144,172,158,186)(145,173,159,187)(146,174,160,188)(147,175,161,189)(148,176,162,190)(149,177,163,191)(150,178,164,192)(151,179,165,193)(152,180,166,194)(153,181,167,195)(154,182,168,196), (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), (1,119)(2,118)(3,117)(4,116)(5,115)(6,114)(7,113)(8,126)(9,125)(10,124)(11,123)(12,122)(13,121)(14,120)(15,133)(16,132)(17,131)(18,130)(19,129)(20,128)(21,127)(22,140)(23,139)(24,138)(25,137)(26,136)(27,135)(28,134)(29,154)(30,153)(31,152)(32,151)(33,150)(34,149)(35,148)(36,147)(37,146)(38,145)(39,144)(40,143)(41,142)(42,141)(43,168)(44,167)(45,166)(46,165)(47,164)(48,163)(49,162)(50,161)(51,160)(52,159)(53,158)(54,157)(55,156)(56,155)(57,175)(58,174)(59,173)(60,172)(61,171)(62,170)(63,169)(64,182)(65,181)(66,180)(67,179)(68,178)(69,177)(70,176)(71,189)(72,188)(73,187)(74,186)(75,185)(76,184)(77,183)(78,196)(79,195)(80,194)(81,193)(82,192)(83,191)(84,190)(85,210)(86,209)(87,208)(88,207)(89,206)(90,205)(91,204)(92,203)(93,202)(94,201)(95,200)(96,199)(97,198)(98,197)(99,224)(100,223)(101,222)(102,221)(103,220)(104,219)(105,218)(106,217)(107,216)(108,215)(109,214)(110,213)(111,212)(112,211) );
G=PermutationGroup([[(1,134),(2,135),(3,136),(4,137),(5,138),(6,139),(7,140),(8,127),(9,128),(10,129),(11,130),(12,131),(13,132),(14,133),(15,120),(16,121),(17,122),(18,123),(19,124),(20,125),(21,126),(22,113),(23,114),(24,115),(25,116),(26,117),(27,118),(28,119),(29,162),(30,163),(31,164),(32,165),(33,166),(34,167),(35,168),(36,155),(37,156),(38,157),(39,158),(40,159),(41,160),(42,161),(43,148),(44,149),(45,150),(46,151),(47,152),(48,153),(49,154),(50,141),(51,142),(52,143),(53,144),(54,145),(55,146),(56,147),(57,190),(58,191),(59,192),(60,193),(61,194),(62,195),(63,196),(64,183),(65,184),(66,185),(67,186),(68,187),(69,188),(70,189),(71,176),(72,177),(73,178),(74,179),(75,180),(76,181),(77,182),(78,169),(79,170),(80,171),(81,172),(82,173),(83,174),(84,175),(85,218),(86,219),(87,220),(88,221),(89,222),(90,223),(91,224),(92,211),(93,212),(94,213),(95,214),(96,215),(97,216),(98,217),(99,204),(100,205),(101,206),(102,207),(103,208),(104,209),(105,210),(106,197),(107,198),(108,199),(109,200),(110,201),(111,202),(112,203)], [(1,183,8,190),(2,184,9,191),(3,185,10,192),(4,186,11,193),(5,187,12,194),(6,188,13,195),(7,189,14,196),(15,169,22,176),(16,170,23,177),(17,171,24,178),(18,172,25,179),(19,173,26,180),(20,174,27,181),(21,175,28,182),(29,211,36,218),(30,212,37,219),(31,213,38,220),(32,214,39,221),(33,215,40,222),(34,216,41,223),(35,217,42,224),(43,197,50,204),(44,198,51,205),(45,199,52,206),(46,200,53,207),(47,201,54,208),(48,202,55,209),(49,203,56,210),(57,134,64,127),(58,135,65,128),(59,136,66,129),(60,137,67,130),(61,138,68,131),(62,139,69,132),(63,140,70,133),(71,120,78,113),(72,121,79,114),(73,122,80,115),(74,123,81,116),(75,124,82,117),(76,125,83,118),(77,126,84,119),(85,162,92,155),(86,163,93,156),(87,164,94,157),(88,165,95,158),(89,166,96,159),(90,167,97,160),(91,168,98,161),(99,148,106,141),(100,149,107,142),(101,150,108,143),(102,151,109,144),(103,152,110,145),(104,153,111,146),(105,154,112,147)], [(1,99,15,85),(2,100,16,86),(3,101,17,87),(4,102,18,88),(5,103,19,89),(6,104,20,90),(7,105,21,91),(8,106,22,92),(9,107,23,93),(10,108,24,94),(11,109,25,95),(12,110,26,96),(13,111,27,97),(14,112,28,98),(29,57,43,71),(30,58,44,72),(31,59,45,73),(32,60,46,74),(33,61,47,75),(34,62,48,76),(35,63,49,77),(36,64,50,78),(37,65,51,79),(38,66,52,80),(39,67,53,81),(40,68,54,82),(41,69,55,83),(42,70,56,84),(113,211,127,197),(114,212,128,198),(115,213,129,199),(116,214,130,200),(117,215,131,201),(118,216,132,202),(119,217,133,203),(120,218,134,204),(121,219,135,205),(122,220,136,206),(123,221,137,207),(124,222,138,208),(125,223,139,209),(126,224,140,210),(141,169,155,183),(142,170,156,184),(143,171,157,185),(144,172,158,186),(145,173,159,187),(146,174,160,188),(147,175,161,189),(148,176,162,190),(149,177,163,191),(150,178,164,192),(151,179,165,193),(152,180,166,194),(153,181,167,195),(154,182,168,196)], [(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)], [(1,119),(2,118),(3,117),(4,116),(5,115),(6,114),(7,113),(8,126),(9,125),(10,124),(11,123),(12,122),(13,121),(14,120),(15,133),(16,132),(17,131),(18,130),(19,129),(20,128),(21,127),(22,140),(23,139),(24,138),(25,137),(26,136),(27,135),(28,134),(29,154),(30,153),(31,152),(32,151),(33,150),(34,149),(35,148),(36,147),(37,146),(38,145),(39,144),(40,143),(41,142),(42,141),(43,168),(44,167),(45,166),(46,165),(47,164),(48,163),(49,162),(50,161),(51,160),(52,159),(53,158),(54,157),(55,156),(56,155),(57,175),(58,174),(59,173),(60,172),(61,171),(62,170),(63,169),(64,182),(65,181),(66,180),(67,179),(68,178),(69,177),(70,176),(71,189),(72,188),(73,187),(74,186),(75,185),(76,184),(77,183),(78,196),(79,195),(80,194),(81,193),(82,192),(83,191),(84,190),(85,210),(86,209),(87,208),(88,207),(89,206),(90,205),(91,204),(92,203),(93,202),(94,201),(95,200),(96,199),(97,198),(98,197),(99,224),(100,223),(101,222),(102,221),(103,220),(104,219),(105,218),(106,217),(107,216),(108,215),(109,214),(110,213),(111,212),(112,211)]])
100 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4L | 4M | ··· | 4T | 4U | ··· | 4AB | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D7 | C4oD4 | D14 | D14 | C4xD7 | D4:2D7 | Q8:2D7 |
| kernel | C2xC4:C4:7D7 | C4:C4:7D7 | C2xC4xDic7 | C2xC4:Dic7 | C2xD14:C4 | C14xC4:C4 | D7xC22xC4 | C2xC4xD7 | C2xC4:C4 | C2xC14 | C4:C4 | C22xC4 | C2xC4 | C22 | C22 |
| # reps | 1 | 8 | 2 | 1 | 2 | 1 | 1 | 16 | 3 | 8 | 12 | 9 | 24 | 6 | 6 |
Matrix representation of C2xC4:C4:7D7 ►in GL6(F29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 17 | 12 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 24 |
| 0 | 0 | 0 | 0 | 17 | 12 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 18 | 4 | 0 | 0 | 0 | 0 |
| 28 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 10 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,17,0,0,0,0,0,12],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,17,0,0,0,0,24,12],[0,1,0,0,0,0,28,18,0,0,0,0,0,0,0,1,0,0,0,0,28,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,28,0,0,0,0,4,11,0,0,0,0,0,0,7,10,0,0,0,0,1,22,0,0,0,0,0,0,1,1,0,0,0,0,0,28] >;
C2xC4:C4:7D7 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes C_4\rtimes_7D_7
% in TeX
G:=Group("C2xC4:C4:7D7"); // GroupNames label
G:=SmallGroup(448,955);
// by ID
G=gap.SmallGroup(448,955);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,1123,297,80,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations