direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C23⋊2D4, (C2×C28)⋊24D4, C23⋊3(C7×D4), (C22×C14)⋊5D4, (C22×D4)⋊1C14, C24.7(C2×C14), C14.90C22≀C2, C22.68(D4×C14), C14.40(C4⋊1D4), C2.C42⋊9C14, (C23×C14).4C22, C14.137(C4⋊D4), C23.75(C22×C14), (C22×C14).456C23, (C22×C28).400C22, (C2×C4)⋊2(C7×D4), (D4×C2×C14)⋊13C2, C2.6(C7×C4⋊D4), C2.3(C7×C4⋊1D4), (C2×C22⋊C4)⋊6C14, C2.4(C7×C22≀C2), (C14×C22⋊C4)⋊27C2, (C2×C14).608(C2×D4), (C22×C4).4(C2×C14), C22.35(C7×C4○D4), (C2×C14).216(C4○D4), (C7×C2.C42)⋊25C2, SmallGroup(448,800)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23⋊2D4
G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf=bc=cb, bd=db, ebe-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
Subgroups: 634 in 322 conjugacy classes, 86 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C22×D4, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C23⋊2D4, C7×C22⋊C4, C22×C28, C22×C28, D4×C14, C23×C14, C7×C2.C42, C14×C22⋊C4, D4×C2×C14, C7×C23⋊2D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C22≀C2, C4⋊D4, C4⋊1D4, C7×D4, C22×C14, C23⋊2D4, D4×C14, C7×C4○D4, C7×C22≀C2, C7×C4⋊D4, C7×C4⋊1D4, C7×C23⋊2D4
►On 224 points
Generators in S224
(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)
(1 132)(2 133)(3 127)(4 128)(5 129)(6 130)(7 131)(8 185)(9 186)(10 187)(11 188)(12 189)(13 183)(14 184)(15 164)(16 165)(17 166)(18 167)(19 168)(20 162)(21 163)(22 171)(23 172)(24 173)(25 174)(26 175)(27 169)(28 170)(29 205)(30 206)(31 207)(32 208)(33 209)(34 210)(35 204)(36 156)(37 157)(38 158)(39 159)(40 160)(41 161)(42 155)(43 192)(44 193)(45 194)(46 195)(47 196)(48 190)(49 191)(50 151)(51 152)(52 153)(53 154)(54 148)(55 149)(56 150)(57 102)(58 103)(59 104)(60 105)(61 99)(62 100)(63 101)(64 138)(65 139)(66 140)(67 134)(68 135)(69 136)(70 137)(71 145)(72 146)(73 147)(74 141)(75 142)(76 143)(77 144)(78 123)(79 124)(80 125)(81 126)(82 120)(83 121)(84 122)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)(91 119)(92 110)(93 111)(94 112)(95 106)(96 107)(97 108)(98 109)(176 222)(177 223)(178 224)(179 218)(180 219)(181 220)(182 221)(197 214)(198 215)(199 216)(200 217)(201 211)(202 212)(203 213)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 57)(7 58)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 36)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 216)(23 217)(24 211)(25 212)(26 213)(27 214)(28 215)(29 223)(30 224)(31 218)(32 219)(33 220)(34 221)(35 222)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 89)(72 90)(73 91)(74 85)(75 86)(76 87)(77 88)(99 127)(100 128)(101 129)(102 130)(103 131)(104 132)(105 133)(106 134)(107 135)(108 136)(109 137)(110 138)(111 139)(112 140)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(155 183)(156 184)(157 185)(158 186)(159 187)(160 188)(161 189)(162 190)(163 191)(164 192)(165 193)(166 194)(167 195)(168 196)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)(181 209)(182 210)
(1 78)(2 79)(3 80)(4 81)(5 82)(6 83)(7 84)(8 222)(9 223)(10 224)(11 218)(12 219)(13 220)(14 221)(15 24)(16 25)(17 26)(18 27)(19 28)(20 22)(21 23)(29 38)(30 39)(31 40)(32 41)(33 42)(34 36)(35 37)(43 211)(44 212)(45 213)(46 214)(47 215)(48 216)(49 217)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(64 73)(65 74)(66 75)(67 76)(68 77)(69 71)(70 72)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(91 92)(99 153)(100 154)(101 148)(102 149)(103 150)(104 151)(105 152)(106 115)(107 116)(108 117)(109 118)(110 119)(111 113)(112 114)(120 129)(121 130)(122 131)(123 132)(124 133)(125 127)(126 128)(134 143)(135 144)(136 145)(137 146)(138 147)(139 141)(140 142)(155 209)(156 210)(157 204)(158 205)(159 206)(160 207)(161 208)(162 171)(163 172)(164 173)(165 174)(166 175)(167 169)(168 170)(176 185)(177 186)(178 187)(179 188)(180 189)(181 183)(182 184)(190 199)(191 200)(192 201)(193 202)(194 203)(195 197)(196 198)
(1 188 76 199)(2 189 77 200)(3 183 71 201)(4 184 72 202)(5 185 73 203)(6 186 74 197)(7 187 75 198)(8 110 213 148)(9 111 214 149)(10 112 215 150)(11 106 216 151)(12 107 217 152)(13 108 211 153)(14 109 212 154)(15 127 33 145)(16 128 34 146)(17 129 35 147)(18 130 29 141)(19 131 30 142)(20 132 31 143)(21 133 32 144)(22 123 40 134)(23 124 41 135)(24 125 42 136)(25 126 36 137)(26 120 37 138)(27 121 38 139)(28 122 39 140)(43 99 220 117)(44 100 221 118)(45 101 222 119)(46 102 223 113)(47 103 224 114)(48 104 218 115)(49 105 219 116)(50 207 95 162)(51 208 96 163)(52 209 97 164)(53 210 98 165)(54 204 92 166)(55 205 93 167)(56 206 94 168)(57 158 85 169)(58 159 86 170)(59 160 87 171)(60 161 88 172)(61 155 89 173)(62 156 90 174)(63 157 91 175)(64 194 82 176)(65 195 83 177)(66 196 84 178)(67 190 78 179)(68 191 79 180)(69 192 80 181)(70 193 81 182)
(1 199)(2 200)(3 201)(4 202)(5 203)(6 197)(7 198)(8 119)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 125)(16 126)(17 120)(18 121)(19 122)(20 123)(21 124)(22 132)(23 133)(24 127)(25 128)(26 129)(27 130)(28 131)(29 139)(30 140)(31 134)(32 135)(33 136)(34 137)(35 138)(36 146)(37 147)(38 141)(39 142)(40 143)(41 144)(42 145)(43 153)(44 154)(45 148)(46 149)(47 150)(48 151)(49 152)(50 162)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 158)(86 159)(87 160)(88 161)(89 155)(90 156)(91 157)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)
G:=sub<Sym(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), (1,132)(2,133)(3,127)(4,128)(5,129)(6,130)(7,131)(8,185)(9,186)(10,187)(11,188)(12,189)(13,183)(14,184)(15,164)(16,165)(17,166)(18,167)(19,168)(20,162)(21,163)(22,171)(23,172)(24,173)(25,174)(26,175)(27,169)(28,170)(29,205)(30,206)(31,207)(32,208)(33,209)(34,210)(35,204)(36,156)(37,157)(38,158)(39,159)(40,160)(41,161)(42,155)(43,192)(44,193)(45,194)(46,195)(47,196)(48,190)(49,191)(50,151)(51,152)(52,153)(53,154)(54,148)(55,149)(56,150)(57,102)(58,103)(59,104)(60,105)(61,99)(62,100)(63,101)(64,138)(65,139)(66,140)(67,134)(68,135)(69,136)(70,137)(71,145)(72,146)(73,147)(74,141)(75,142)(76,143)(77,144)(78,123)(79,124)(80,125)(81,126)(82,120)(83,121)(84,122)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,119)(92,110)(93,111)(94,112)(95,106)(96,107)(97,108)(98,109)(176,222)(177,223)(178,224)(179,218)(180,219)(181,220)(182,221)(197,214)(198,215)(199,216)(200,217)(201,211)(202,212)(203,213), (1,59)(2,60)(3,61)(4,62)(5,63)(6,57)(7,58)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,36)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,216)(23,217)(24,211)(25,212)(26,213)(27,214)(28,215)(29,223)(30,224)(31,218)(32,219)(33,220)(34,221)(35,222)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,89)(72,90)(73,91)(74,85)(75,86)(76,87)(77,88)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,133)(106,134)(107,135)(108,136)(109,137)(110,138)(111,139)(112,140)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(155,183)(156,184)(157,185)(158,186)(159,187)(160,188)(161,189)(162,190)(163,191)(164,192)(165,193)(166,194)(167,195)(168,196)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210), (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,222)(9,223)(10,224)(11,218)(12,219)(13,220)(14,221)(15,24)(16,25)(17,26)(18,27)(19,28)(20,22)(21,23)(29,38)(30,39)(31,40)(32,41)(33,42)(34,36)(35,37)(43,211)(44,212)(45,213)(46,214)(47,215)(48,216)(49,217)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,73)(65,74)(66,75)(67,76)(68,77)(69,71)(70,72)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92)(99,153)(100,154)(101,148)(102,149)(103,150)(104,151)(105,152)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(120,129)(121,130)(122,131)(123,132)(124,133)(125,127)(126,128)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(155,209)(156,210)(157,204)(158,205)(159,206)(160,207)(161,208)(162,171)(163,172)(164,173)(165,174)(166,175)(167,169)(168,170)(176,185)(177,186)(178,187)(179,188)(180,189)(181,183)(182,184)(190,199)(191,200)(192,201)(193,202)(194,203)(195,197)(196,198), (1,188,76,199)(2,189,77,200)(3,183,71,201)(4,184,72,202)(5,185,73,203)(6,186,74,197)(7,187,75,198)(8,110,213,148)(9,111,214,149)(10,112,215,150)(11,106,216,151)(12,107,217,152)(13,108,211,153)(14,109,212,154)(15,127,33,145)(16,128,34,146)(17,129,35,147)(18,130,29,141)(19,131,30,142)(20,132,31,143)(21,133,32,144)(22,123,40,134)(23,124,41,135)(24,125,42,136)(25,126,36,137)(26,120,37,138)(27,121,38,139)(28,122,39,140)(43,99,220,117)(44,100,221,118)(45,101,222,119)(46,102,223,113)(47,103,224,114)(48,104,218,115)(49,105,219,116)(50,207,95,162)(51,208,96,163)(52,209,97,164)(53,210,98,165)(54,204,92,166)(55,205,93,167)(56,206,94,168)(57,158,85,169)(58,159,86,170)(59,160,87,171)(60,161,88,172)(61,155,89,173)(62,156,90,174)(63,157,91,175)(64,194,82,176)(65,195,83,177)(66,196,84,178)(67,190,78,179)(68,191,79,180)(69,192,80,181)(70,193,81,182), (1,199)(2,200)(3,201)(4,202)(5,203)(6,197)(7,198)(8,119)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,125)(16,126)(17,120)(18,121)(19,122)(20,123)(21,124)(22,132)(23,133)(24,127)(25,128)(26,129)(27,130)(28,131)(29,139)(30,140)(31,134)(32,135)(33,136)(34,137)(35,138)(36,146)(37,147)(38,141)(39,142)(40,143)(41,144)(42,145)(43,153)(44,154)(45,148)(46,149)(47,150)(48,151)(49,152)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,158)(86,159)(87,160)(88,161)(89,155)(90,156)(91,157)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224)>;
G:=Group( (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), (1,132)(2,133)(3,127)(4,128)(5,129)(6,130)(7,131)(8,185)(9,186)(10,187)(11,188)(12,189)(13,183)(14,184)(15,164)(16,165)(17,166)(18,167)(19,168)(20,162)(21,163)(22,171)(23,172)(24,173)(25,174)(26,175)(27,169)(28,170)(29,205)(30,206)(31,207)(32,208)(33,209)(34,210)(35,204)(36,156)(37,157)(38,158)(39,159)(40,160)(41,161)(42,155)(43,192)(44,193)(45,194)(46,195)(47,196)(48,190)(49,191)(50,151)(51,152)(52,153)(53,154)(54,148)(55,149)(56,150)(57,102)(58,103)(59,104)(60,105)(61,99)(62,100)(63,101)(64,138)(65,139)(66,140)(67,134)(68,135)(69,136)(70,137)(71,145)(72,146)(73,147)(74,141)(75,142)(76,143)(77,144)(78,123)(79,124)(80,125)(81,126)(82,120)(83,121)(84,122)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,119)(92,110)(93,111)(94,112)(95,106)(96,107)(97,108)(98,109)(176,222)(177,223)(178,224)(179,218)(180,219)(181,220)(182,221)(197,214)(198,215)(199,216)(200,217)(201,211)(202,212)(203,213), (1,59)(2,60)(3,61)(4,62)(5,63)(6,57)(7,58)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,36)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,216)(23,217)(24,211)(25,212)(26,213)(27,214)(28,215)(29,223)(30,224)(31,218)(32,219)(33,220)(34,221)(35,222)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,89)(72,90)(73,91)(74,85)(75,86)(76,87)(77,88)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,133)(106,134)(107,135)(108,136)(109,137)(110,138)(111,139)(112,140)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(155,183)(156,184)(157,185)(158,186)(159,187)(160,188)(161,189)(162,190)(163,191)(164,192)(165,193)(166,194)(167,195)(168,196)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210), (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,222)(9,223)(10,224)(11,218)(12,219)(13,220)(14,221)(15,24)(16,25)(17,26)(18,27)(19,28)(20,22)(21,23)(29,38)(30,39)(31,40)(32,41)(33,42)(34,36)(35,37)(43,211)(44,212)(45,213)(46,214)(47,215)(48,216)(49,217)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,73)(65,74)(66,75)(67,76)(68,77)(69,71)(70,72)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92)(99,153)(100,154)(101,148)(102,149)(103,150)(104,151)(105,152)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(120,129)(121,130)(122,131)(123,132)(124,133)(125,127)(126,128)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(155,209)(156,210)(157,204)(158,205)(159,206)(160,207)(161,208)(162,171)(163,172)(164,173)(165,174)(166,175)(167,169)(168,170)(176,185)(177,186)(178,187)(179,188)(180,189)(181,183)(182,184)(190,199)(191,200)(192,201)(193,202)(194,203)(195,197)(196,198), (1,188,76,199)(2,189,77,200)(3,183,71,201)(4,184,72,202)(5,185,73,203)(6,186,74,197)(7,187,75,198)(8,110,213,148)(9,111,214,149)(10,112,215,150)(11,106,216,151)(12,107,217,152)(13,108,211,153)(14,109,212,154)(15,127,33,145)(16,128,34,146)(17,129,35,147)(18,130,29,141)(19,131,30,142)(20,132,31,143)(21,133,32,144)(22,123,40,134)(23,124,41,135)(24,125,42,136)(25,126,36,137)(26,120,37,138)(27,121,38,139)(28,122,39,140)(43,99,220,117)(44,100,221,118)(45,101,222,119)(46,102,223,113)(47,103,224,114)(48,104,218,115)(49,105,219,116)(50,207,95,162)(51,208,96,163)(52,209,97,164)(53,210,98,165)(54,204,92,166)(55,205,93,167)(56,206,94,168)(57,158,85,169)(58,159,86,170)(59,160,87,171)(60,161,88,172)(61,155,89,173)(62,156,90,174)(63,157,91,175)(64,194,82,176)(65,195,83,177)(66,196,84,178)(67,190,78,179)(68,191,79,180)(69,192,80,181)(70,193,81,182), (1,199)(2,200)(3,201)(4,202)(5,203)(6,197)(7,198)(8,119)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,125)(16,126)(17,120)(18,121)(19,122)(20,123)(21,124)(22,132)(23,133)(24,127)(25,128)(26,129)(27,130)(28,131)(29,139)(30,140)(31,134)(32,135)(33,136)(34,137)(35,138)(36,146)(37,147)(38,141)(39,142)(40,143)(41,144)(42,145)(43,153)(44,154)(45,148)(46,149)(47,150)(48,151)(49,152)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,158)(86,159)(87,160)(88,161)(89,155)(90,156)(91,157)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224) );
G=PermutationGroup([[(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)], [(1,132),(2,133),(3,127),(4,128),(5,129),(6,130),(7,131),(8,185),(9,186),(10,187),(11,188),(12,189),(13,183),(14,184),(15,164),(16,165),(17,166),(18,167),(19,168),(20,162),(21,163),(22,171),(23,172),(24,173),(25,174),(26,175),(27,169),(28,170),(29,205),(30,206),(31,207),(32,208),(33,209),(34,210),(35,204),(36,156),(37,157),(38,158),(39,159),(40,160),(41,161),(42,155),(43,192),(44,193),(45,194),(46,195),(47,196),(48,190),(49,191),(50,151),(51,152),(52,153),(53,154),(54,148),(55,149),(56,150),(57,102),(58,103),(59,104),(60,105),(61,99),(62,100),(63,101),(64,138),(65,139),(66,140),(67,134),(68,135),(69,136),(70,137),(71,145),(72,146),(73,147),(74,141),(75,142),(76,143),(77,144),(78,123),(79,124),(80,125),(81,126),(82,120),(83,121),(84,122),(85,113),(86,114),(87,115),(88,116),(89,117),(90,118),(91,119),(92,110),(93,111),(94,112),(95,106),(96,107),(97,108),(98,109),(176,222),(177,223),(178,224),(179,218),(180,219),(181,220),(182,221),(197,214),(198,215),(199,216),(200,217),(201,211),(202,212),(203,213)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,57),(7,58),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,36),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,216),(23,217),(24,211),(25,212),(26,213),(27,214),(28,215),(29,223),(30,224),(31,218),(32,219),(33,220),(34,221),(35,222),(50,78),(51,79),(52,80),(53,81),(54,82),(55,83),(56,84),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,89),(72,90),(73,91),(74,85),(75,86),(76,87),(77,88),(99,127),(100,128),(101,129),(102,130),(103,131),(104,132),(105,133),(106,134),(107,135),(108,136),(109,137),(110,138),(111,139),(112,140),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(155,183),(156,184),(157,185),(158,186),(159,187),(160,188),(161,189),(162,190),(163,191),(164,192),(165,193),(166,194),(167,195),(168,196),(169,197),(170,198),(171,199),(172,200),(173,201),(174,202),(175,203),(176,204),(177,205),(178,206),(179,207),(180,208),(181,209),(182,210)], [(1,78),(2,79),(3,80),(4,81),(5,82),(6,83),(7,84),(8,222),(9,223),(10,224),(11,218),(12,219),(13,220),(14,221),(15,24),(16,25),(17,26),(18,27),(19,28),(20,22),(21,23),(29,38),(30,39),(31,40),(32,41),(33,42),(34,36),(35,37),(43,211),(44,212),(45,213),(46,214),(47,215),(48,216),(49,217),(50,59),(51,60),(52,61),(53,62),(54,63),(55,57),(56,58),(64,73),(65,74),(66,75),(67,76),(68,77),(69,71),(70,72),(85,93),(86,94),(87,95),(88,96),(89,97),(90,98),(91,92),(99,153),(100,154),(101,148),(102,149),(103,150),(104,151),(105,152),(106,115),(107,116),(108,117),(109,118),(110,119),(111,113),(112,114),(120,129),(121,130),(122,131),(123,132),(124,133),(125,127),(126,128),(134,143),(135,144),(136,145),(137,146),(138,147),(139,141),(140,142),(155,209),(156,210),(157,204),(158,205),(159,206),(160,207),(161,208),(162,171),(163,172),(164,173),(165,174),(166,175),(167,169),(168,170),(176,185),(177,186),(178,187),(179,188),(180,189),(181,183),(182,184),(190,199),(191,200),(192,201),(193,202),(194,203),(195,197),(196,198)], [(1,188,76,199),(2,189,77,200),(3,183,71,201),(4,184,72,202),(5,185,73,203),(6,186,74,197),(7,187,75,198),(8,110,213,148),(9,111,214,149),(10,112,215,150),(11,106,216,151),(12,107,217,152),(13,108,211,153),(14,109,212,154),(15,127,33,145),(16,128,34,146),(17,129,35,147),(18,130,29,141),(19,131,30,142),(20,132,31,143),(21,133,32,144),(22,123,40,134),(23,124,41,135),(24,125,42,136),(25,126,36,137),(26,120,37,138),(27,121,38,139),(28,122,39,140),(43,99,220,117),(44,100,221,118),(45,101,222,119),(46,102,223,113),(47,103,224,114),(48,104,218,115),(49,105,219,116),(50,207,95,162),(51,208,96,163),(52,209,97,164),(53,210,98,165),(54,204,92,166),(55,205,93,167),(56,206,94,168),(57,158,85,169),(58,159,86,170),(59,160,87,171),(60,161,88,172),(61,155,89,173),(62,156,90,174),(63,157,91,175),(64,194,82,176),(65,195,83,177),(66,196,84,178),(67,190,78,179),(68,191,79,180),(69,192,80,181),(70,193,81,182)], [(1,199),(2,200),(3,201),(4,202),(5,203),(6,197),(7,198),(8,119),(9,113),(10,114),(11,115),(12,116),(13,117),(14,118),(15,125),(16,126),(17,120),(18,121),(19,122),(20,123),(21,124),(22,132),(23,133),(24,127),(25,128),(26,129),(27,130),(28,131),(29,139),(30,140),(31,134),(32,135),(33,136),(34,137),(35,138),(36,146),(37,147),(38,141),(39,142),(40,143),(41,144),(42,145),(43,153),(44,154),(45,148),(46,149),(47,150),(48,151),(49,152),(50,162),(51,163),(52,164),(53,165),(54,166),(55,167),(56,168),(57,169),(58,170),(59,171),(60,172),(61,173),(62,174),(63,175),(64,176),(65,177),(66,178),(67,179),(68,180),(69,181),(70,182),(71,183),(72,184),(73,185),(74,186),(75,187),(76,188),(77,189),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,158),(86,159),(87,160),(88,161),(89,155),(90,156),(91,157),(92,204),(93,205),(94,206),(95,207),(96,208),(97,209),(98,210),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,218),(107,219),(108,220),(109,221),(110,222),(111,223),(112,224)]])
154 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2M | 4A | ··· | 4H | 7A | ··· | 7F | 14A | ··· | 14AP | 14AQ | ··· | 14BZ | 28A | ··· | 28AV |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | D4 | C4○D4 | C7×D4 | C7×D4 | C7×C4○D4 |
| kernel | C7×C23⋊2D4 | C7×C2.C42 | C14×C22⋊C4 | D4×C2×C14 | C23⋊2D4 | C2.C42 | C2×C22⋊C4 | C22×D4 | C2×C28 | C22×C14 | C2×C14 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 3 | 3 | 6 | 6 | 18 | 18 | 6 | 6 | 2 | 36 | 36 | 12 |
Matrix representation of C7×C23⋊2D4 ►in GL6(𝔽29)
| 24 | 0 | 0 | 0 | 0 | 0 |
| 0 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 0 | 0 | 0 |
| 0 | 0 | 0 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 28 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 13 |
| 0 | 0 | 0 | 0 | 13 | 8 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 13 |
| 0 | 0 | 0 | 0 | 13 | 8 |
G:=sub<GL(6,GF(29))| [24,0,0,0,0,0,0,24,0,0,0,0,0,0,24,0,0,0,0,0,0,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,28,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,21,13,0,0,0,0,13,8],[0,17,0,0,0,0,12,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,21,13,0,0,0,0,13,8] >;
C7×C23⋊2D4 in GAP, Magma, Sage, TeX
C_7\times C_2^3\rtimes_2D_4
% in TeX
G:=Group("C7xC2^3:2D4"); // GroupNames label
G:=SmallGroup(448,800);
// by ID
G=gap.SmallGroup(448,800);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,2438,2403]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f=b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations