direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C23.19D4, C2.D8⋊7C14, C22⋊C8⋊7C14, C4.Q8⋊10C14, C4⋊D4.7C14, (C2×C28).461D4, D4⋊C4⋊13C14, C23.18(C7×D4), C42⋊C2⋊5C14, (C22×C14).36D4, C28.319(C4○D4), C14.128(C4○D8), (C2×C28).938C23, (C2×C56).306C22, C22.103(D4×C14), C14.143(C8⋊C22), (D4×C14).197C22, (C22×C28).430C22, C14.97(C22.D4), (C7×C4.Q8)⋊25C2, (C7×C2.D8)⋊22C2, C2.15(C7×C4○D8), C4.31(C7×C4○D4), C4⋊C4.59(C2×C14), (C7×C22⋊C8)⋊24C2, (C2×C8).43(C2×C14), (C2×C4).107(C7×D4), C2.18(C7×C8⋊C22), (C7×D4⋊C4)⋊31C2, (C2×D4).20(C2×C14), (C2×C14).659(C2×D4), (C7×C4⋊D4).17C2, (C7×C42⋊C2)⋊26C2, (C7×C4⋊C4).382C22, (C22×C4).48(C2×C14), (C2×C4).113(C22×C14), C2.13(C7×C22.D4), SmallGroup(448,890)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.19D4
G = < a,b,c,d,e,f | a7=b2=c2=d2=f2=1, e4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=cde3 >
Subgroups: 210 in 106 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×D4, C28, C28, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C56, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C23.19D4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C22×C28, D4×C14, D4×C14, C7×C22⋊C8, C7×D4⋊C4, C7×C4.Q8, C7×C2.D8, C7×C42⋊C2, C7×C4⋊D4, C7×C23.19D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C22.D4, C4○D8, C8⋊C22, C7×D4, C22×C14, C23.19D4, D4×C14, C7×C4○D4, C7×C22.D4, C7×C4○D8, C7×C8⋊C22, C7×C23.19D4
►On 224 points
Generators in S224
(1 170 219 51 211 43 203)(2 171 220 52 212 44 204)(3 172 221 53 213 45 205)(4 173 222 54 214 46 206)(5 174 223 55 215 47 207)(6 175 224 56 216 48 208)(7 176 217 49 209 41 201)(8 169 218 50 210 42 202)(9 161 193 25 185 17 177)(10 162 194 26 186 18 178)(11 163 195 27 187 19 179)(12 164 196 28 188 20 180)(13 165 197 29 189 21 181)(14 166 198 30 190 22 182)(15 167 199 31 191 23 183)(16 168 200 32 192 24 184)(33 119 57 133 73 125 65)(34 120 58 134 74 126 66)(35 113 59 135 75 127 67)(36 114 60 136 76 128 68)(37 115 61 129 77 121 69)(38 116 62 130 78 122 70)(39 117 63 131 79 123 71)(40 118 64 132 80 124 72)(81 105 157 97 149 89 141)(82 106 158 98 150 90 142)(83 107 159 99 151 91 143)(84 108 160 100 152 92 144)(85 109 153 101 145 93 137)(86 110 154 102 146 94 138)(87 111 155 103 147 95 139)(88 112 156 104 148 96 140)
(2 70)(4 72)(6 66)(8 68)(9 13)(10 141)(11 15)(12 143)(14 137)(16 139)(17 21)(18 149)(19 23)(20 151)(22 145)(24 147)(25 29)(26 157)(27 31)(28 159)(30 153)(32 155)(34 175)(36 169)(38 171)(40 173)(42 76)(44 78)(46 80)(48 74)(50 60)(52 62)(54 64)(56 58)(81 162)(82 86)(83 164)(84 88)(85 166)(87 168)(89 178)(90 94)(91 180)(92 96)(93 182)(95 184)(97 186)(98 102)(99 188)(100 104)(101 190)(103 192)(105 194)(106 110)(107 196)(108 112)(109 198)(111 200)(114 218)(116 220)(118 222)(120 224)(122 204)(124 206)(126 208)(128 202)(130 212)(132 214)(134 216)(136 210)(138 142)(140 144)(146 150)(148 152)(154 158)(156 160)(161 165)(163 167)(177 181)(179 183)(185 189)(187 191)(193 197)(195 199)
(1 69)(2 70)(3 71)(4 72)(5 65)(6 66)(7 67)(8 68)(9 144)(10 137)(11 138)(12 139)(13 140)(14 141)(15 142)(16 143)(17 152)(18 145)(19 146)(20 147)(21 148)(22 149)(23 150)(24 151)(25 160)(26 153)(27 154)(28 155)(29 156)(30 157)(31 158)(32 159)(33 174)(34 175)(35 176)(36 169)(37 170)(38 171)(39 172)(40 173)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 73)(48 74)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 57)(56 58)(81 166)(82 167)(83 168)(84 161)(85 162)(86 163)(87 164)(88 165)(89 182)(90 183)(91 184)(92 177)(93 178)(94 179)(95 180)(96 181)(97 190)(98 191)(99 192)(100 185)(101 186)(102 187)(103 188)(104 189)(105 198)(106 199)(107 200)(108 193)(109 194)(110 195)(111 196)(112 197)(113 217)(114 218)(115 219)(116 220)(117 221)(118 222)(119 223)(120 224)(121 203)(122 204)(123 205)(124 206)(125 207)(126 208)(127 201)(128 202)(129 211)(130 212)(131 213)(132 214)(133 215)(134 216)(135 209)(136 210)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)(161 165)(162 166)(163 167)(164 168)(169 173)(170 174)(171 175)(172 176)(177 181)(178 182)(179 183)(180 184)(185 189)(186 190)(187 191)(188 192)(193 197)(194 198)(195 199)(196 200)(201 205)(202 206)(203 207)(204 208)(209 213)(210 214)(211 215)(212 216)(217 221)(218 222)(219 223)(220 224)
(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)
(1 166)(2 88)(3 164)(4 86)(5 162)(6 84)(7 168)(8 82)(9 126)(10 207)(11 124)(12 205)(13 122)(14 203)(15 128)(16 201)(17 134)(18 215)(19 132)(20 213)(21 130)(22 211)(23 136)(24 209)(25 120)(26 223)(27 118)(28 221)(29 116)(30 219)(31 114)(32 217)(33 109)(34 193)(35 107)(36 199)(37 105)(38 197)(39 111)(40 195)(41 184)(42 90)(43 182)(44 96)(45 180)(46 94)(47 178)(48 92)(49 192)(50 98)(51 190)(52 104)(53 188)(54 102)(55 186)(56 100)(57 101)(58 185)(59 99)(60 191)(61 97)(62 189)(63 103)(64 187)(65 85)(66 161)(67 83)(68 167)(69 81)(70 165)(71 87)(72 163)(73 93)(74 177)(75 91)(76 183)(77 89)(78 181)(79 95)(80 179)(106 169)(108 175)(110 173)(112 171)(113 159)(115 157)(117 155)(119 153)(121 141)(123 139)(125 137)(127 143)(129 149)(131 147)(133 145)(135 151)(138 206)(140 204)(142 202)(144 208)(146 214)(148 212)(150 210)(152 216)(154 222)(156 220)(158 218)(160 224)(170 198)(172 196)(174 194)(176 200)
G:=sub<Sym(224)| (1,170,219,51,211,43,203)(2,171,220,52,212,44,204)(3,172,221,53,213,45,205)(4,173,222,54,214,46,206)(5,174,223,55,215,47,207)(6,175,224,56,216,48,208)(7,176,217,49,209,41,201)(8,169,218,50,210,42,202)(9,161,193,25,185,17,177)(10,162,194,26,186,18,178)(11,163,195,27,187,19,179)(12,164,196,28,188,20,180)(13,165,197,29,189,21,181)(14,166,198,30,190,22,182)(15,167,199,31,191,23,183)(16,168,200,32,192,24,184)(33,119,57,133,73,125,65)(34,120,58,134,74,126,66)(35,113,59,135,75,127,67)(36,114,60,136,76,128,68)(37,115,61,129,77,121,69)(38,116,62,130,78,122,70)(39,117,63,131,79,123,71)(40,118,64,132,80,124,72)(81,105,157,97,149,89,141)(82,106,158,98,150,90,142)(83,107,159,99,151,91,143)(84,108,160,100,152,92,144)(85,109,153,101,145,93,137)(86,110,154,102,146,94,138)(87,111,155,103,147,95,139)(88,112,156,104,148,96,140), (2,70)(4,72)(6,66)(8,68)(9,13)(10,141)(11,15)(12,143)(14,137)(16,139)(17,21)(18,149)(19,23)(20,151)(22,145)(24,147)(25,29)(26,157)(27,31)(28,159)(30,153)(32,155)(34,175)(36,169)(38,171)(40,173)(42,76)(44,78)(46,80)(48,74)(50,60)(52,62)(54,64)(56,58)(81,162)(82,86)(83,164)(84,88)(85,166)(87,168)(89,178)(90,94)(91,180)(92,96)(93,182)(95,184)(97,186)(98,102)(99,188)(100,104)(101,190)(103,192)(105,194)(106,110)(107,196)(108,112)(109,198)(111,200)(114,218)(116,220)(118,222)(120,224)(122,204)(124,206)(126,208)(128,202)(130,212)(132,214)(134,216)(136,210)(138,142)(140,144)(146,150)(148,152)(154,158)(156,160)(161,165)(163,167)(177,181)(179,183)(185,189)(187,191)(193,197)(195,199), (1,69)(2,70)(3,71)(4,72)(5,65)(6,66)(7,67)(8,68)(9,144)(10,137)(11,138)(12,139)(13,140)(14,141)(15,142)(16,143)(17,152)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,151)(25,160)(26,153)(27,154)(28,155)(29,156)(30,157)(31,158)(32,159)(33,174)(34,175)(35,176)(36,169)(37,170)(38,171)(39,172)(40,173)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,73)(48,74)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58)(81,166)(82,167)(83,168)(84,161)(85,162)(86,163)(87,164)(88,165)(89,182)(90,183)(91,184)(92,177)(93,178)(94,179)(95,180)(96,181)(97,190)(98,191)(99,192)(100,185)(101,186)(102,187)(103,188)(104,189)(105,198)(106,199)(107,200)(108,193)(109,194)(110,195)(111,196)(112,197)(113,217)(114,218)(115,219)(116,220)(117,221)(118,222)(119,223)(120,224)(121,203)(122,204)(123,205)(124,206)(125,207)(126,208)(127,201)(128,202)(129,211)(130,212)(131,213)(132,214)(133,215)(134,216)(135,209)(136,210), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)(172,176)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192)(193,197)(194,198)(195,199)(196,200)(201,205)(202,206)(203,207)(204,208)(209,213)(210,214)(211,215)(212,216)(217,221)(218,222)(219,223)(220,224), (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), (1,166)(2,88)(3,164)(4,86)(5,162)(6,84)(7,168)(8,82)(9,126)(10,207)(11,124)(12,205)(13,122)(14,203)(15,128)(16,201)(17,134)(18,215)(19,132)(20,213)(21,130)(22,211)(23,136)(24,209)(25,120)(26,223)(27,118)(28,221)(29,116)(30,219)(31,114)(32,217)(33,109)(34,193)(35,107)(36,199)(37,105)(38,197)(39,111)(40,195)(41,184)(42,90)(43,182)(44,96)(45,180)(46,94)(47,178)(48,92)(49,192)(50,98)(51,190)(52,104)(53,188)(54,102)(55,186)(56,100)(57,101)(58,185)(59,99)(60,191)(61,97)(62,189)(63,103)(64,187)(65,85)(66,161)(67,83)(68,167)(69,81)(70,165)(71,87)(72,163)(73,93)(74,177)(75,91)(76,183)(77,89)(78,181)(79,95)(80,179)(106,169)(108,175)(110,173)(112,171)(113,159)(115,157)(117,155)(119,153)(121,141)(123,139)(125,137)(127,143)(129,149)(131,147)(133,145)(135,151)(138,206)(140,204)(142,202)(144,208)(146,214)(148,212)(150,210)(152,216)(154,222)(156,220)(158,218)(160,224)(170,198)(172,196)(174,194)(176,200)>;
G:=Group( (1,170,219,51,211,43,203)(2,171,220,52,212,44,204)(3,172,221,53,213,45,205)(4,173,222,54,214,46,206)(5,174,223,55,215,47,207)(6,175,224,56,216,48,208)(7,176,217,49,209,41,201)(8,169,218,50,210,42,202)(9,161,193,25,185,17,177)(10,162,194,26,186,18,178)(11,163,195,27,187,19,179)(12,164,196,28,188,20,180)(13,165,197,29,189,21,181)(14,166,198,30,190,22,182)(15,167,199,31,191,23,183)(16,168,200,32,192,24,184)(33,119,57,133,73,125,65)(34,120,58,134,74,126,66)(35,113,59,135,75,127,67)(36,114,60,136,76,128,68)(37,115,61,129,77,121,69)(38,116,62,130,78,122,70)(39,117,63,131,79,123,71)(40,118,64,132,80,124,72)(81,105,157,97,149,89,141)(82,106,158,98,150,90,142)(83,107,159,99,151,91,143)(84,108,160,100,152,92,144)(85,109,153,101,145,93,137)(86,110,154,102,146,94,138)(87,111,155,103,147,95,139)(88,112,156,104,148,96,140), (2,70)(4,72)(6,66)(8,68)(9,13)(10,141)(11,15)(12,143)(14,137)(16,139)(17,21)(18,149)(19,23)(20,151)(22,145)(24,147)(25,29)(26,157)(27,31)(28,159)(30,153)(32,155)(34,175)(36,169)(38,171)(40,173)(42,76)(44,78)(46,80)(48,74)(50,60)(52,62)(54,64)(56,58)(81,162)(82,86)(83,164)(84,88)(85,166)(87,168)(89,178)(90,94)(91,180)(92,96)(93,182)(95,184)(97,186)(98,102)(99,188)(100,104)(101,190)(103,192)(105,194)(106,110)(107,196)(108,112)(109,198)(111,200)(114,218)(116,220)(118,222)(120,224)(122,204)(124,206)(126,208)(128,202)(130,212)(132,214)(134,216)(136,210)(138,142)(140,144)(146,150)(148,152)(154,158)(156,160)(161,165)(163,167)(177,181)(179,183)(185,189)(187,191)(193,197)(195,199), (1,69)(2,70)(3,71)(4,72)(5,65)(6,66)(7,67)(8,68)(9,144)(10,137)(11,138)(12,139)(13,140)(14,141)(15,142)(16,143)(17,152)(18,145)(19,146)(20,147)(21,148)(22,149)(23,150)(24,151)(25,160)(26,153)(27,154)(28,155)(29,156)(30,157)(31,158)(32,159)(33,174)(34,175)(35,176)(36,169)(37,170)(38,171)(39,172)(40,173)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,73)(48,74)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58)(81,166)(82,167)(83,168)(84,161)(85,162)(86,163)(87,164)(88,165)(89,182)(90,183)(91,184)(92,177)(93,178)(94,179)(95,180)(96,181)(97,190)(98,191)(99,192)(100,185)(101,186)(102,187)(103,188)(104,189)(105,198)(106,199)(107,200)(108,193)(109,194)(110,195)(111,196)(112,197)(113,217)(114,218)(115,219)(116,220)(117,221)(118,222)(119,223)(120,224)(121,203)(122,204)(123,205)(124,206)(125,207)(126,208)(127,201)(128,202)(129,211)(130,212)(131,213)(132,214)(133,215)(134,216)(135,209)(136,210), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)(172,176)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192)(193,197)(194,198)(195,199)(196,200)(201,205)(202,206)(203,207)(204,208)(209,213)(210,214)(211,215)(212,216)(217,221)(218,222)(219,223)(220,224), (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), (1,166)(2,88)(3,164)(4,86)(5,162)(6,84)(7,168)(8,82)(9,126)(10,207)(11,124)(12,205)(13,122)(14,203)(15,128)(16,201)(17,134)(18,215)(19,132)(20,213)(21,130)(22,211)(23,136)(24,209)(25,120)(26,223)(27,118)(28,221)(29,116)(30,219)(31,114)(32,217)(33,109)(34,193)(35,107)(36,199)(37,105)(38,197)(39,111)(40,195)(41,184)(42,90)(43,182)(44,96)(45,180)(46,94)(47,178)(48,92)(49,192)(50,98)(51,190)(52,104)(53,188)(54,102)(55,186)(56,100)(57,101)(58,185)(59,99)(60,191)(61,97)(62,189)(63,103)(64,187)(65,85)(66,161)(67,83)(68,167)(69,81)(70,165)(71,87)(72,163)(73,93)(74,177)(75,91)(76,183)(77,89)(78,181)(79,95)(80,179)(106,169)(108,175)(110,173)(112,171)(113,159)(115,157)(117,155)(119,153)(121,141)(123,139)(125,137)(127,143)(129,149)(131,147)(133,145)(135,151)(138,206)(140,204)(142,202)(144,208)(146,214)(148,212)(150,210)(152,216)(154,222)(156,220)(158,218)(160,224)(170,198)(172,196)(174,194)(176,200) );
G=PermutationGroup([[(1,170,219,51,211,43,203),(2,171,220,52,212,44,204),(3,172,221,53,213,45,205),(4,173,222,54,214,46,206),(5,174,223,55,215,47,207),(6,175,224,56,216,48,208),(7,176,217,49,209,41,201),(8,169,218,50,210,42,202),(9,161,193,25,185,17,177),(10,162,194,26,186,18,178),(11,163,195,27,187,19,179),(12,164,196,28,188,20,180),(13,165,197,29,189,21,181),(14,166,198,30,190,22,182),(15,167,199,31,191,23,183),(16,168,200,32,192,24,184),(33,119,57,133,73,125,65),(34,120,58,134,74,126,66),(35,113,59,135,75,127,67),(36,114,60,136,76,128,68),(37,115,61,129,77,121,69),(38,116,62,130,78,122,70),(39,117,63,131,79,123,71),(40,118,64,132,80,124,72),(81,105,157,97,149,89,141),(82,106,158,98,150,90,142),(83,107,159,99,151,91,143),(84,108,160,100,152,92,144),(85,109,153,101,145,93,137),(86,110,154,102,146,94,138),(87,111,155,103,147,95,139),(88,112,156,104,148,96,140)], [(2,70),(4,72),(6,66),(8,68),(9,13),(10,141),(11,15),(12,143),(14,137),(16,139),(17,21),(18,149),(19,23),(20,151),(22,145),(24,147),(25,29),(26,157),(27,31),(28,159),(30,153),(32,155),(34,175),(36,169),(38,171),(40,173),(42,76),(44,78),(46,80),(48,74),(50,60),(52,62),(54,64),(56,58),(81,162),(82,86),(83,164),(84,88),(85,166),(87,168),(89,178),(90,94),(91,180),(92,96),(93,182),(95,184),(97,186),(98,102),(99,188),(100,104),(101,190),(103,192),(105,194),(106,110),(107,196),(108,112),(109,198),(111,200),(114,218),(116,220),(118,222),(120,224),(122,204),(124,206),(126,208),(128,202),(130,212),(132,214),(134,216),(136,210),(138,142),(140,144),(146,150),(148,152),(154,158),(156,160),(161,165),(163,167),(177,181),(179,183),(185,189),(187,191),(193,197),(195,199)], [(1,69),(2,70),(3,71),(4,72),(5,65),(6,66),(7,67),(8,68),(9,144),(10,137),(11,138),(12,139),(13,140),(14,141),(15,142),(16,143),(17,152),(18,145),(19,146),(20,147),(21,148),(22,149),(23,150),(24,151),(25,160),(26,153),(27,154),(28,155),(29,156),(30,157),(31,158),(32,159),(33,174),(34,175),(35,176),(36,169),(37,170),(38,171),(39,172),(40,173),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,73),(48,74),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,57),(56,58),(81,166),(82,167),(83,168),(84,161),(85,162),(86,163),(87,164),(88,165),(89,182),(90,183),(91,184),(92,177),(93,178),(94,179),(95,180),(96,181),(97,190),(98,191),(99,192),(100,185),(101,186),(102,187),(103,188),(104,189),(105,198),(106,199),(107,200),(108,193),(109,194),(110,195),(111,196),(112,197),(113,217),(114,218),(115,219),(116,220),(117,221),(118,222),(119,223),(120,224),(121,203),(122,204),(123,205),(124,206),(125,207),(126,208),(127,201),(128,202),(129,211),(130,212),(131,213),(132,214),(133,215),(134,216),(135,209),(136,210)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144),(145,149),(146,150),(147,151),(148,152),(153,157),(154,158),(155,159),(156,160),(161,165),(162,166),(163,167),(164,168),(169,173),(170,174),(171,175),(172,176),(177,181),(178,182),(179,183),(180,184),(185,189),(186,190),(187,191),(188,192),(193,197),(194,198),(195,199),(196,200),(201,205),(202,206),(203,207),(204,208),(209,213),(210,214),(211,215),(212,216),(217,221),(218,222),(219,223),(220,224)], [(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)], [(1,166),(2,88),(3,164),(4,86),(5,162),(6,84),(7,168),(8,82),(9,126),(10,207),(11,124),(12,205),(13,122),(14,203),(15,128),(16,201),(17,134),(18,215),(19,132),(20,213),(21,130),(22,211),(23,136),(24,209),(25,120),(26,223),(27,118),(28,221),(29,116),(30,219),(31,114),(32,217),(33,109),(34,193),(35,107),(36,199),(37,105),(38,197),(39,111),(40,195),(41,184),(42,90),(43,182),(44,96),(45,180),(46,94),(47,178),(48,92),(49,192),(50,98),(51,190),(52,104),(53,188),(54,102),(55,186),(56,100),(57,101),(58,185),(59,99),(60,191),(61,97),(62,189),(63,103),(64,187),(65,85),(66,161),(67,83),(68,167),(69,81),(70,165),(71,87),(72,163),(73,93),(74,177),(75,91),(76,183),(77,89),(78,181),(79,95),(80,179),(106,169),(108,175),(110,173),(112,171),(113,159),(115,157),(117,155),(119,153),(121,141),(123,139),(125,137),(127,143),(129,149),(131,147),(133,145),(135,151),(138,206),(140,204),(142,202),(144,208),(146,214),(148,212),(150,210),(152,216),(154,222),(156,220),(158,218),(160,224),(170,198),(172,196),(174,194),(176,200)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14X | 14Y | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28AV | 28AW | ··· | 28BB | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | C4○D4 | C4○D8 | C7×D4 | C7×D4 | C7×C4○D4 | C7×C4○D8 | C8⋊C22 | C7×C8⋊C22 |
| kernel | C7×C23.19D4 | C7×C22⋊C8 | C7×D4⋊C4 | C7×C4.Q8 | C7×C2.D8 | C7×C42⋊C2 | C7×C4⋊D4 | C23.19D4 | C22⋊C8 | D4⋊C4 | C4.Q8 | C2.D8 | C42⋊C2 | C4⋊D4 | C2×C28 | C22×C14 | C28 | C14 | C2×C4 | C23 | C4 | C2 | C14 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 6 | 6 | 6 | 6 | 1 | 1 | 4 | 4 | 6 | 6 | 24 | 24 | 1 | 6 |
Matrix representation of C7×C23.19D4 ►in GL4(𝔽113) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 49 | 0 |
| 0 | 0 | 0 | 49 |
| 1 | 0 | 0 | 0 |
| 48 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 95 | 0 | 0 | 0 |
| 59 | 69 | 0 | 0 |
| 0 | 0 | 15 | 83 |
| 0 | 0 | 0 | 98 |
| 39 | 69 | 0 | 0 |
| 14 | 74 | 0 | 0 |
| 0 | 0 | 1 | 111 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [30,0,0,0,0,30,0,0,0,0,49,0,0,0,0,49],[1,48,0,0,0,112,0,0,0,0,1,1,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[95,59,0,0,0,69,0,0,0,0,15,0,0,0,83,98],[39,14,0,0,69,74,0,0,0,0,1,0,0,0,111,112] >;
C7×C23.19D4 in GAP, Magma, Sage, TeX
C_7\times C_2^3._{19}D_4 % in TeX
G:=Group("C7xC2^3.19D4"); // GroupNames label
G:=SmallGroup(448,890);
// by ID
G=gap.SmallGroup(448,890);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,1576,2438,310,14117,3547,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=f^2=1,e^4=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=b*c=c*b,b*d=d*b,f*b*f=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=c*d*e^3>;
// generators/relations