direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C23.20D4, C2.D8⋊9C14, C4.Q8⋊12C14, Q8⋊C4⋊8C14, (C2×C28).462D4, C22⋊C8.5C14, C23.19(C7×D4), C22⋊Q8.7C14, (C22×C14).37D4, C28.322(C4○D4), C14.129(C4○D8), (C2×C28).941C23, (C2×C56).264C22, C42⋊C2.9C14, C22.106(D4×C14), (Q8×C14).171C22, C14.144(C8.C22), (C22×C28).433C22, C14.100(C22.D4), (C7×C4.Q8)⋊27C2, (C7×C2.D8)⋊24C2, C2.16(C7×C4○D8), C4.34(C7×C4○D4), C4⋊C4.62(C2×C14), (C2×C8).11(C2×C14), (C2×C4).108(C7×D4), (C7×Q8⋊C4)⋊36C2, (C2×C14).662(C2×D4), (C7×C22⋊C8).14C2, (C2×Q8).15(C2×C14), C2.19(C7×C8.C22), (C7×C22⋊Q8).17C2, (C7×C4⋊C4).385C22, (C22×C4).51(C2×C14), (C7×C42⋊C2).23C2, (C2×C4).116(C22×C14), C2.16(C7×C22.D4), SmallGroup(448,893)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.20D4
G = < a,b,c,d,e,f | a7=b2=c2=d2=1, e4=f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce3 >
Subgroups: 162 in 96 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, C28, C28, C2×C14, C2×C14, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C56, C2×C28, C2×C28, C7×Q8, C22×C14, C23.20D4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C22×C28, Q8×C14, C7×C22⋊C8, C7×Q8⋊C4, C7×C4.Q8, C7×C2.D8, C7×C42⋊C2, C7×C22⋊Q8, C7×C23.20D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C22.D4, C4○D8, C8.C22, C7×D4, C22×C14, C23.20D4, D4×C14, C7×C4○D4, C7×C22.D4, C7×C4○D8, C7×C8.C22, C7×C23.20D4
►On 224 points
Generators in S224
(1 175 223 55 215 47 207)(2 176 224 56 216 48 208)(3 169 217 49 209 41 201)(4 170 218 50 210 42 202)(5 171 219 51 211 43 203)(6 172 220 52 212 44 204)(7 173 221 53 213 45 205)(8 174 222 54 214 46 206)(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)(33 129 77 121 69 119 63)(34 130 78 122 70 120 64)(35 131 79 123 71 113 57)(36 132 80 124 72 114 58)(37 133 73 125 65 115 59)(38 134 74 126 66 116 60)(39 135 75 127 67 117 61)(40 136 76 128 68 118 62)(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 68)(4 70)(6 72)(8 66)(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 50)(36 52)(38 54)(40 56)(42 78)(44 80)(46 74)(48 76)(58 220)(60 222)(62 224)(64 218)(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)(114 172)(116 174)(118 176)(120 170)(122 202)(124 204)(126 206)(128 208)(130 210)(132 212)(134 214)(136 216)(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 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 65)(8 66)(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 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(41 77)(42 78)(43 79)(44 80)(45 73)(46 74)(47 75)(48 76)(57 219)(58 220)(59 221)(60 222)(61 223)(62 224)(63 217)(64 218)(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)(113 171)(114 172)(115 173)(116 174)(117 175)(118 176)(119 169)(120 170)(121 201)(122 202)(123 203)(124 204)(125 205)(126 206)(127 207)(128 208)(129 209)(130 210)(131 211)(132 212)(133 213)(134 214)(135 215)(136 216)
(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 5 162)(2 84 6 88)(3 164 7 168)(4 82 8 86)(9 124 13 128)(10 207 14 203)(11 122 15 126)(12 205 16 201)(17 132 21 136)(18 215 22 211)(19 130 23 134)(20 213 24 209)(25 58 29 62)(26 223 30 219)(27 64 31 60)(28 221 32 217)(33 103 37 99)(34 191 38 187)(35 101 39 97)(36 189 40 185)(41 180 45 184)(42 90 46 94)(43 178 47 182)(44 96 48 92)(49 188 53 192)(50 98 54 102)(51 186 55 190)(52 104 56 100)(57 153 61 157)(59 159 63 155)(65 83 69 87)(66 163 70 167)(67 81 71 85)(68 161 72 165)(73 91 77 95)(74 179 78 183)(75 89 79 93)(76 177 80 181)(105 113 109 117)(106 174 110 170)(107 119 111 115)(108 172 112 176)(114 197 118 193)(116 195 120 199)(121 139 125 143)(123 137 127 141)(129 147 133 151)(131 145 135 149)(138 202 142 206)(140 208 144 204)(146 210 150 214)(148 216 152 212)(154 218 158 222)(156 224 160 220)(169 196 173 200)(171 194 175 198)
G:=sub<Sym(224)| (1,175,223,55,215,47,207)(2,176,224,56,216,48,208)(3,169,217,49,209,41,201)(4,170,218,50,210,42,202)(5,171,219,51,211,43,203)(6,172,220,52,212,44,204)(7,173,221,53,213,45,205)(8,174,222,54,214,46,206)(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)(33,129,77,121,69,119,63)(34,130,78,122,70,120,64)(35,131,79,123,71,113,57)(36,132,80,124,72,114,58)(37,133,73,125,65,115,59)(38,134,74,126,66,116,60)(39,135,75,127,67,117,61)(40,136,76,128,68,118,62)(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), (2,68)(4,70)(6,72)(8,66)(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)(34,50)(36,52)(38,54)(40,56)(42,78)(44,80)(46,74)(48,76)(58,220)(60,222)(62,224)(64,218)(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)(114,172)(116,174)(118,176)(120,170)(122,202)(124,204)(126,206)(128,208)(130,210)(132,212)(134,214)(136,216)(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), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,65)(8,66)(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)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76)(57,219)(58,220)(59,221)(60,222)(61,223)(62,224)(63,217)(64,218)(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)(113,171)(114,172)(115,173)(116,174)(117,175)(118,176)(119,169)(120,170)(121,201)(122,202)(123,203)(124,204)(125,205)(126,206)(127,207)(128,208)(129,209)(130,210)(131,211)(132,212)(133,213)(134,214)(135,215)(136,216), (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), (1,166,5,162)(2,84,6,88)(3,164,7,168)(4,82,8,86)(9,124,13,128)(10,207,14,203)(11,122,15,126)(12,205,16,201)(17,132,21,136)(18,215,22,211)(19,130,23,134)(20,213,24,209)(25,58,29,62)(26,223,30,219)(27,64,31,60)(28,221,32,217)(33,103,37,99)(34,191,38,187)(35,101,39,97)(36,189,40,185)(41,180,45,184)(42,90,46,94)(43,178,47,182)(44,96,48,92)(49,188,53,192)(50,98,54,102)(51,186,55,190)(52,104,56,100)(57,153,61,157)(59,159,63,155)(65,83,69,87)(66,163,70,167)(67,81,71,85)(68,161,72,165)(73,91,77,95)(74,179,78,183)(75,89,79,93)(76,177,80,181)(105,113,109,117)(106,174,110,170)(107,119,111,115)(108,172,112,176)(114,197,118,193)(116,195,120,199)(121,139,125,143)(123,137,127,141)(129,147,133,151)(131,145,135,149)(138,202,142,206)(140,208,144,204)(146,210,150,214)(148,216,152,212)(154,218,158,222)(156,224,160,220)(169,196,173,200)(171,194,175,198)>;
G:=Group( (1,175,223,55,215,47,207)(2,176,224,56,216,48,208)(3,169,217,49,209,41,201)(4,170,218,50,210,42,202)(5,171,219,51,211,43,203)(6,172,220,52,212,44,204)(7,173,221,53,213,45,205)(8,174,222,54,214,46,206)(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)(33,129,77,121,69,119,63)(34,130,78,122,70,120,64)(35,131,79,123,71,113,57)(36,132,80,124,72,114,58)(37,133,73,125,65,115,59)(38,134,74,126,66,116,60)(39,135,75,127,67,117,61)(40,136,76,128,68,118,62)(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), (2,68)(4,70)(6,72)(8,66)(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)(34,50)(36,52)(38,54)(40,56)(42,78)(44,80)(46,74)(48,76)(58,220)(60,222)(62,224)(64,218)(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)(114,172)(116,174)(118,176)(120,170)(122,202)(124,204)(126,206)(128,208)(130,210)(132,212)(134,214)(136,216)(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), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,65)(8,66)(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)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76)(57,219)(58,220)(59,221)(60,222)(61,223)(62,224)(63,217)(64,218)(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)(113,171)(114,172)(115,173)(116,174)(117,175)(118,176)(119,169)(120,170)(121,201)(122,202)(123,203)(124,204)(125,205)(126,206)(127,207)(128,208)(129,209)(130,210)(131,211)(132,212)(133,213)(134,214)(135,215)(136,216), (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), (1,166,5,162)(2,84,6,88)(3,164,7,168)(4,82,8,86)(9,124,13,128)(10,207,14,203)(11,122,15,126)(12,205,16,201)(17,132,21,136)(18,215,22,211)(19,130,23,134)(20,213,24,209)(25,58,29,62)(26,223,30,219)(27,64,31,60)(28,221,32,217)(33,103,37,99)(34,191,38,187)(35,101,39,97)(36,189,40,185)(41,180,45,184)(42,90,46,94)(43,178,47,182)(44,96,48,92)(49,188,53,192)(50,98,54,102)(51,186,55,190)(52,104,56,100)(57,153,61,157)(59,159,63,155)(65,83,69,87)(66,163,70,167)(67,81,71,85)(68,161,72,165)(73,91,77,95)(74,179,78,183)(75,89,79,93)(76,177,80,181)(105,113,109,117)(106,174,110,170)(107,119,111,115)(108,172,112,176)(114,197,118,193)(116,195,120,199)(121,139,125,143)(123,137,127,141)(129,147,133,151)(131,145,135,149)(138,202,142,206)(140,208,144,204)(146,210,150,214)(148,216,152,212)(154,218,158,222)(156,224,160,220)(169,196,173,200)(171,194,175,198) );
G=PermutationGroup([[(1,175,223,55,215,47,207),(2,176,224,56,216,48,208),(3,169,217,49,209,41,201),(4,170,218,50,210,42,202),(5,171,219,51,211,43,203),(6,172,220,52,212,44,204),(7,173,221,53,213,45,205),(8,174,222,54,214,46,206),(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),(33,129,77,121,69,119,63),(34,130,78,122,70,120,64),(35,131,79,123,71,113,57),(36,132,80,124,72,114,58),(37,133,73,125,65,115,59),(38,134,74,126,66,116,60),(39,135,75,127,67,117,61),(40,136,76,128,68,118,62),(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)], [(2,68),(4,70),(6,72),(8,66),(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),(34,50),(36,52),(38,54),(40,56),(42,78),(44,80),(46,74),(48,76),(58,220),(60,222),(62,224),(64,218),(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),(114,172),(116,174),(118,176),(120,170),(122,202),(124,204),(126,206),(128,208),(130,210),(132,212),(134,214),(136,216),(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)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,65),(8,66),(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),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56),(41,77),(42,78),(43,79),(44,80),(45,73),(46,74),(47,75),(48,76),(57,219),(58,220),(59,221),(60,222),(61,223),(62,224),(63,217),(64,218),(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),(113,171),(114,172),(115,173),(116,174),(117,175),(118,176),(119,169),(120,170),(121,201),(122,202),(123,203),(124,204),(125,205),(126,206),(127,207),(128,208),(129,209),(130,210),(131,211),(132,212),(133,213),(134,214),(135,215),(136,216)], [(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)], [(1,166,5,162),(2,84,6,88),(3,164,7,168),(4,82,8,86),(9,124,13,128),(10,207,14,203),(11,122,15,126),(12,205,16,201),(17,132,21,136),(18,215,22,211),(19,130,23,134),(20,213,24,209),(25,58,29,62),(26,223,30,219),(27,64,31,60),(28,221,32,217),(33,103,37,99),(34,191,38,187),(35,101,39,97),(36,189,40,185),(41,180,45,184),(42,90,46,94),(43,178,47,182),(44,96,48,92),(49,188,53,192),(50,98,54,102),(51,186,55,190),(52,104,56,100),(57,153,61,157),(59,159,63,155),(65,83,69,87),(66,163,70,167),(67,81,71,85),(68,161,72,165),(73,91,77,95),(74,179,78,183),(75,89,79,93),(76,177,80,181),(105,113,109,117),(106,174,110,170),(107,119,111,115),(108,172,112,176),(114,197,118,193),(116,195,120,199),(121,139,125,143),(123,137,127,141),(129,147,133,151),(131,145,135,149),(138,202,142,206),(140,208,144,204),(146,210,150,214),(148,216,152,212),(154,218,158,222),(156,224,160,220),(169,196,173,200),(171,194,175,198)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14X | 28A | ··· | 28X | 28Y | ··· | 28AV | 28AW | ··· | 28BH | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 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×C23.20D4 | C7×C22⋊C8 | C7×Q8⋊C4 | C7×C4.Q8 | C7×C2.D8 | C7×C42⋊C2 | C7×C22⋊Q8 | C23.20D4 | C22⋊C8 | Q8⋊C4 | C4.Q8 | C2.D8 | C42⋊C2 | C22⋊Q8 | 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×C23.20D4 ►in GL4(𝔽113) generated by
| 109 | 0 | 0 | 0 |
| 0 | 109 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 1 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 61 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 64 | 17 | 0 | 0 |
| 98 | 49 | 0 | 0 |
| 0 | 0 | 44 | 0 |
| 0 | 0 | 83 | 95 |
| 57 | 84 | 0 | 0 |
| 112 | 56 | 0 | 0 |
| 0 | 0 | 32 | 36 |
| 0 | 0 | 50 | 81 |
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,1,112,0,0,0,0,1,61,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[64,98,0,0,17,49,0,0,0,0,44,83,0,0,0,95],[57,112,0,0,84,56,0,0,0,0,32,50,0,0,36,81] >;
C7×C23.20D4 in GAP, Magma, Sage, TeX
C_7\times C_2^3._{20}D_4 % in TeX
G:=Group("C7xC2^3.20D4"); // GroupNames label
G:=SmallGroup(448,893);
// by ID
G=gap.SmallGroup(448,893);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,1576,2438,310,14117,3547,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=1,e^4=f^2=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^-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^-1=c*e^3>;
// generators/relations