direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C23.48D4, C2.D8⋊8C14, C2.8(C14×Q16), Q8⋊C4⋊7C14, (C2×C28).339D4, C22⋊C8.4C14, C14.55(C2×Q16), (C2×C14).12Q16, C23.48(C7×D4), C22⋊Q8.6C14, C22.3(C7×Q16), C28.321(C4○D4), (C2×C28).940C23, (C2×C56).263C22, (C22×C14).170D4, C22.105(D4×C14), C14.144(C8⋊C22), (Q8×C14).170C22, (C22×C28).432C22, C14.99(C22.D4), (C7×C2.D8)⋊23C2, C4.33(C7×C4○D4), (C2×C4).40(C7×D4), (C2×C4⋊C4).17C14, (C14×C4⋊C4).46C2, C4⋊C4.61(C2×C14), (C2×C8).10(C2×C14), C2.19(C7×C8⋊C22), (C7×Q8⋊C4)⋊30C2, (C2×C14).661(C2×D4), (C7×C22⋊C8).13C2, (C2×Q8).14(C2×C14), (C7×C22⋊Q8).16C2, (C7×C4⋊C4).384C22, (C22×C4).50(C2×C14), (C2×C4).115(C22×C14), C2.15(C7×C22.D4), SmallGroup(448,892)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.48D4
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=fbf-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >
Subgroups: 178 in 104 conjugacy classes, 54 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, Q8⋊C4, C2.D8, C2×C4⋊C4, C22⋊Q8, C56, C2×C28, C2×C28, C7×Q8, C22×C14, C23.48D4, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C22×C28, C22×C28, Q8×C14, C7×C22⋊C8, C7×Q8⋊C4, C7×C2.D8, C14×C4⋊C4, C7×C22⋊Q8, C7×C23.48D4
Quotients: C1, C2, C22, C7, D4, C23, C14, Q16, C2×D4, C4○D4, C2×C14, C22.D4, C2×Q16, C8⋊C22, C7×D4, C22×C14, C23.48D4, C7×Q16, D4×C14, C7×C4○D4, C7×C22.D4, C14×Q16, C7×C8⋊C22, C7×C23.48D4
►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 64 133 80 125 72)(34 120 57 134 73 126 65)(35 113 58 135 74 127 66)(36 114 59 136 75 128 67)(37 115 60 129 76 121 68)(38 116 61 130 77 122 69)(39 117 62 131 78 123 70)(40 118 63 132 79 124 71)(81 105 158 97 150 89 142)(82 106 159 98 151 90 143)(83 107 160 99 152 91 144)(84 108 153 100 145 92 137)(85 109 154 101 146 93 138)(86 110 155 102 147 94 139)(87 111 156 103 148 95 140)(88 112 157 104 149 96 141)
(1 5)(2 65)(3 7)(4 67)(6 69)(8 71)(9 13)(10 137)(11 15)(12 139)(14 141)(16 143)(17 21)(18 145)(19 23)(20 147)(22 149)(24 151)(25 29)(26 153)(27 31)(28 155)(30 157)(32 159)(33 37)(34 171)(35 39)(36 173)(38 175)(40 169)(41 45)(42 79)(43 47)(44 73)(46 75)(48 77)(49 53)(50 63)(51 55)(52 57)(54 59)(56 61)(58 62)(60 64)(66 70)(68 72)(74 78)(76 80)(81 85)(82 168)(83 87)(84 162)(86 164)(88 166)(89 93)(90 184)(91 95)(92 178)(94 180)(96 182)(97 101)(98 192)(99 103)(100 186)(102 188)(104 190)(105 109)(106 200)(107 111)(108 194)(110 196)(112 198)(113 117)(114 222)(115 119)(116 224)(118 218)(120 220)(121 125)(122 208)(123 127)(124 202)(126 204)(128 206)(129 133)(130 216)(131 135)(132 210)(134 212)(136 214)(138 142)(140 144)(146 150)(148 152)(154 158)(156 160)(161 165)(163 167)(170 174)(172 176)(177 181)(179 183)(185 189)(187 191)(193 197)(195 199)(201 205)(203 207)(209 213)(211 215)(217 221)(219 223)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 65)(7 66)(8 67)(9 140)(10 141)(11 142)(12 143)(13 144)(14 137)(15 138)(16 139)(17 148)(18 149)(19 150)(20 151)(21 152)(22 145)(23 146)(24 147)(25 156)(26 157)(27 158)(28 159)(29 160)(30 153)(31 154)(32 155)(33 174)(34 175)(35 176)(36 169)(37 170)(38 171)(39 172)(40 173)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 73)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)(81 163)(82 164)(83 165)(84 166)(85 167)(86 168)(87 161)(88 162)(89 179)(90 180)(91 181)(92 182)(93 183)(94 184)(95 177)(96 178)(97 187)(98 188)(99 189)(100 190)(101 191)(102 192)(103 185)(104 186)(105 195)(106 196)(107 197)(108 198)(109 199)(110 200)(111 193)(112 194)(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)
(1 162 5 166)(2 87 6 83)(3 168 7 164)(4 85 8 81)(9 126 13 122)(10 207 14 203)(11 124 15 128)(12 205 16 201)(17 134 21 130)(18 215 22 211)(19 132 23 136)(20 213 24 209)(25 120 29 116)(26 223 30 219)(27 118 31 114)(28 221 32 217)(33 108 37 112)(34 197 38 193)(35 106 39 110)(36 195 40 199)(41 180 45 184)(42 89 46 93)(43 178 47 182)(44 95 48 91)(49 188 53 192)(50 97 54 101)(51 186 55 190)(52 103 56 99)(57 189 61 185)(58 98 62 102)(59 187 63 191)(60 104 64 100)(65 165 69 161)(66 82 70 86)(67 163 71 167)(68 88 72 84)(73 181 77 177)(74 90 78 94)(75 179 79 183)(76 96 80 92)(105 173 109 169)(107 171 111 175)(113 159 117 155)(115 157 119 153)(121 141 125 137)(123 139 127 143)(129 149 133 145)(131 147 135 151)(138 202 142 206)(140 208 144 204)(146 210 150 214)(148 216 152 212)(154 218 158 222)(156 224 160 220)(170 194 174 198)(172 200 176 196)
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)(33,119,64,133,80,125,72)(34,120,57,134,73,126,65)(35,113,58,135,74,127,66)(36,114,59,136,75,128,67)(37,115,60,129,76,121,68)(38,116,61,130,77,122,69)(39,117,62,131,78,123,70)(40,118,63,132,79,124,71)(81,105,158,97,150,89,142)(82,106,159,98,151,90,143)(83,107,160,99,152,91,144)(84,108,153,100,145,92,137)(85,109,154,101,146,93,138)(86,110,155,102,147,94,139)(87,111,156,103,148,95,140)(88,112,157,104,149,96,141), (1,5)(2,65)(3,7)(4,67)(6,69)(8,71)(9,13)(10,137)(11,15)(12,139)(14,141)(16,143)(17,21)(18,145)(19,23)(20,147)(22,149)(24,151)(25,29)(26,153)(27,31)(28,155)(30,157)(32,159)(33,37)(34,171)(35,39)(36,173)(38,175)(40,169)(41,45)(42,79)(43,47)(44,73)(46,75)(48,77)(49,53)(50,63)(51,55)(52,57)(54,59)(56,61)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80)(81,85)(82,168)(83,87)(84,162)(86,164)(88,166)(89,93)(90,184)(91,95)(92,178)(94,180)(96,182)(97,101)(98,192)(99,103)(100,186)(102,188)(104,190)(105,109)(106,200)(107,111)(108,194)(110,196)(112,198)(113,117)(114,222)(115,119)(116,224)(118,218)(120,220)(121,125)(122,208)(123,127)(124,202)(126,204)(128,206)(129,133)(130,216)(131,135)(132,210)(134,212)(136,214)(138,142)(140,144)(146,150)(148,152)(154,158)(156,160)(161,165)(163,167)(170,174)(172,176)(177,181)(179,183)(185,189)(187,191)(193,197)(195,199)(201,205)(203,207)(209,213)(211,215)(217,221)(219,223), (1,68)(2,69)(3,70)(4,71)(5,72)(6,65)(7,66)(8,67)(9,140)(10,141)(11,142)(12,143)(13,144)(14,137)(15,138)(16,139)(17,148)(18,149)(19,150)(20,151)(21,152)(22,145)(23,146)(24,147)(25,156)(26,157)(27,158)(28,159)(29,160)(30,153)(31,154)(32,155)(33,174)(34,175)(35,176)(36,169)(37,170)(38,171)(39,172)(40,173)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,73)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57)(81,163)(82,164)(83,165)(84,166)(85,167)(86,168)(87,161)(88,162)(89,179)(90,180)(91,181)(92,182)(93,183)(94,184)(95,177)(96,178)(97,187)(98,188)(99,189)(100,190)(101,191)(102,192)(103,185)(104,186)(105,195)(106,196)(107,197)(108,198)(109,199)(110,200)(111,193)(112,194)(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), (1,162,5,166)(2,87,6,83)(3,168,7,164)(4,85,8,81)(9,126,13,122)(10,207,14,203)(11,124,15,128)(12,205,16,201)(17,134,21,130)(18,215,22,211)(19,132,23,136)(20,213,24,209)(25,120,29,116)(26,223,30,219)(27,118,31,114)(28,221,32,217)(33,108,37,112)(34,197,38,193)(35,106,39,110)(36,195,40,199)(41,180,45,184)(42,89,46,93)(43,178,47,182)(44,95,48,91)(49,188,53,192)(50,97,54,101)(51,186,55,190)(52,103,56,99)(57,189,61,185)(58,98,62,102)(59,187,63,191)(60,104,64,100)(65,165,69,161)(66,82,70,86)(67,163,71,167)(68,88,72,84)(73,181,77,177)(74,90,78,94)(75,179,79,183)(76,96,80,92)(105,173,109,169)(107,171,111,175)(113,159,117,155)(115,157,119,153)(121,141,125,137)(123,139,127,143)(129,149,133,145)(131,147,135,151)(138,202,142,206)(140,208,144,204)(146,210,150,214)(148,216,152,212)(154,218,158,222)(156,224,160,220)(170,194,174,198)(172,200,176,196)>;
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)(33,119,64,133,80,125,72)(34,120,57,134,73,126,65)(35,113,58,135,74,127,66)(36,114,59,136,75,128,67)(37,115,60,129,76,121,68)(38,116,61,130,77,122,69)(39,117,62,131,78,123,70)(40,118,63,132,79,124,71)(81,105,158,97,150,89,142)(82,106,159,98,151,90,143)(83,107,160,99,152,91,144)(84,108,153,100,145,92,137)(85,109,154,101,146,93,138)(86,110,155,102,147,94,139)(87,111,156,103,148,95,140)(88,112,157,104,149,96,141), (1,5)(2,65)(3,7)(4,67)(6,69)(8,71)(9,13)(10,137)(11,15)(12,139)(14,141)(16,143)(17,21)(18,145)(19,23)(20,147)(22,149)(24,151)(25,29)(26,153)(27,31)(28,155)(30,157)(32,159)(33,37)(34,171)(35,39)(36,173)(38,175)(40,169)(41,45)(42,79)(43,47)(44,73)(46,75)(48,77)(49,53)(50,63)(51,55)(52,57)(54,59)(56,61)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80)(81,85)(82,168)(83,87)(84,162)(86,164)(88,166)(89,93)(90,184)(91,95)(92,178)(94,180)(96,182)(97,101)(98,192)(99,103)(100,186)(102,188)(104,190)(105,109)(106,200)(107,111)(108,194)(110,196)(112,198)(113,117)(114,222)(115,119)(116,224)(118,218)(120,220)(121,125)(122,208)(123,127)(124,202)(126,204)(128,206)(129,133)(130,216)(131,135)(132,210)(134,212)(136,214)(138,142)(140,144)(146,150)(148,152)(154,158)(156,160)(161,165)(163,167)(170,174)(172,176)(177,181)(179,183)(185,189)(187,191)(193,197)(195,199)(201,205)(203,207)(209,213)(211,215)(217,221)(219,223), (1,68)(2,69)(3,70)(4,71)(5,72)(6,65)(7,66)(8,67)(9,140)(10,141)(11,142)(12,143)(13,144)(14,137)(15,138)(16,139)(17,148)(18,149)(19,150)(20,151)(21,152)(22,145)(23,146)(24,147)(25,156)(26,157)(27,158)(28,159)(29,160)(30,153)(31,154)(32,155)(33,174)(34,175)(35,176)(36,169)(37,170)(38,171)(39,172)(40,173)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,73)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57)(81,163)(82,164)(83,165)(84,166)(85,167)(86,168)(87,161)(88,162)(89,179)(90,180)(91,181)(92,182)(93,183)(94,184)(95,177)(96,178)(97,187)(98,188)(99,189)(100,190)(101,191)(102,192)(103,185)(104,186)(105,195)(106,196)(107,197)(108,198)(109,199)(110,200)(111,193)(112,194)(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), (1,162,5,166)(2,87,6,83)(3,168,7,164)(4,85,8,81)(9,126,13,122)(10,207,14,203)(11,124,15,128)(12,205,16,201)(17,134,21,130)(18,215,22,211)(19,132,23,136)(20,213,24,209)(25,120,29,116)(26,223,30,219)(27,118,31,114)(28,221,32,217)(33,108,37,112)(34,197,38,193)(35,106,39,110)(36,195,40,199)(41,180,45,184)(42,89,46,93)(43,178,47,182)(44,95,48,91)(49,188,53,192)(50,97,54,101)(51,186,55,190)(52,103,56,99)(57,189,61,185)(58,98,62,102)(59,187,63,191)(60,104,64,100)(65,165,69,161)(66,82,70,86)(67,163,71,167)(68,88,72,84)(73,181,77,177)(74,90,78,94)(75,179,79,183)(76,96,80,92)(105,173,109,169)(107,171,111,175)(113,159,117,155)(115,157,119,153)(121,141,125,137)(123,139,127,143)(129,149,133,145)(131,147,135,151)(138,202,142,206)(140,208,144,204)(146,210,150,214)(148,216,152,212)(154,218,158,222)(156,224,160,220)(170,194,174,198)(172,200,176,196) );
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),(33,119,64,133,80,125,72),(34,120,57,134,73,126,65),(35,113,58,135,74,127,66),(36,114,59,136,75,128,67),(37,115,60,129,76,121,68),(38,116,61,130,77,122,69),(39,117,62,131,78,123,70),(40,118,63,132,79,124,71),(81,105,158,97,150,89,142),(82,106,159,98,151,90,143),(83,107,160,99,152,91,144),(84,108,153,100,145,92,137),(85,109,154,101,146,93,138),(86,110,155,102,147,94,139),(87,111,156,103,148,95,140),(88,112,157,104,149,96,141)], [(1,5),(2,65),(3,7),(4,67),(6,69),(8,71),(9,13),(10,137),(11,15),(12,139),(14,141),(16,143),(17,21),(18,145),(19,23),(20,147),(22,149),(24,151),(25,29),(26,153),(27,31),(28,155),(30,157),(32,159),(33,37),(34,171),(35,39),(36,173),(38,175),(40,169),(41,45),(42,79),(43,47),(44,73),(46,75),(48,77),(49,53),(50,63),(51,55),(52,57),(54,59),(56,61),(58,62),(60,64),(66,70),(68,72),(74,78),(76,80),(81,85),(82,168),(83,87),(84,162),(86,164),(88,166),(89,93),(90,184),(91,95),(92,178),(94,180),(96,182),(97,101),(98,192),(99,103),(100,186),(102,188),(104,190),(105,109),(106,200),(107,111),(108,194),(110,196),(112,198),(113,117),(114,222),(115,119),(116,224),(118,218),(120,220),(121,125),(122,208),(123,127),(124,202),(126,204),(128,206),(129,133),(130,216),(131,135),(132,210),(134,212),(136,214),(138,142),(140,144),(146,150),(148,152),(154,158),(156,160),(161,165),(163,167),(170,174),(172,176),(177,181),(179,183),(185,189),(187,191),(193,197),(195,199),(201,205),(203,207),(209,213),(211,215),(217,221),(219,223)], [(1,68),(2,69),(3,70),(4,71),(5,72),(6,65),(7,66),(8,67),(9,140),(10,141),(11,142),(12,143),(13,144),(14,137),(15,138),(16,139),(17,148),(18,149),(19,150),(20,151),(21,152),(22,145),(23,146),(24,147),(25,156),(26,157),(27,158),(28,159),(29,160),(30,153),(31,154),(32,155),(33,174),(34,175),(35,176),(36,169),(37,170),(38,171),(39,172),(40,173),(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,73),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(56,57),(81,163),(82,164),(83,165),(84,166),(85,167),(86,168),(87,161),(88,162),(89,179),(90,180),(91,181),(92,182),(93,183),(94,184),(95,177),(96,178),(97,187),(98,188),(99,189),(100,190),(101,191),(102,192),(103,185),(104,186),(105,195),(106,196),(107,197),(108,198),(109,199),(110,200),(111,193),(112,194),(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)], [(1,162,5,166),(2,87,6,83),(3,168,7,164),(4,85,8,81),(9,126,13,122),(10,207,14,203),(11,124,15,128),(12,205,16,201),(17,134,21,130),(18,215,22,211),(19,132,23,136),(20,213,24,209),(25,120,29,116),(26,223,30,219),(27,118,31,114),(28,221,32,217),(33,108,37,112),(34,197,38,193),(35,106,39,110),(36,195,40,199),(41,180,45,184),(42,89,46,93),(43,178,47,182),(44,95,48,91),(49,188,53,192),(50,97,54,101),(51,186,55,190),(52,103,56,99),(57,189,61,185),(58,98,62,102),(59,187,63,191),(60,104,64,100),(65,165,69,161),(66,82,70,86),(67,163,71,167),(68,88,72,84),(73,181,77,177),(74,90,78,94),(75,179,79,183),(76,96,80,92),(105,173,109,169),(107,171,111,175),(113,159,117,155),(115,157,119,153),(121,141,125,137),(123,139,127,143),(129,149,133,145),(131,147,135,151),(138,202,142,206),(140,208,144,204),(146,210,150,214),(148,216,152,212),(154,218,158,222),(156,224,160,220),(170,194,174,198),(172,200,176,196)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28L | 28M | ··· | 28AP | 28AQ | ··· | 28BB | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
133 irreducible representations
| dim | 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 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | C4○D4 | Q16 | C7×D4 | C7×D4 | C7×C4○D4 | C7×Q16 | C8⋊C22 | C7×C8⋊C22 |
| kernel | C7×C23.48D4 | C7×C22⋊C8 | C7×Q8⋊C4 | C7×C2.D8 | C14×C4⋊C4 | C7×C22⋊Q8 | C23.48D4 | C22⋊C8 | Q8⋊C4 | C2.D8 | C2×C4⋊C4 | C22⋊Q8 | C2×C28 | C22×C14 | C28 | C2×C14 | C2×C4 | C23 | C4 | C22 | C14 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 6 | 6 | 12 | 12 | 6 | 6 | 1 | 1 | 4 | 4 | 6 | 6 | 24 | 24 | 1 | 6 |
Matrix representation of C7×C23.48D4 ►in GL4(𝔽113) generated by
| 109 | 0 | 0 | 0 |
| 0 | 109 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 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 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 15 | 0 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 0 | 82 | 31 |
| 0 | 0 | 82 | 82 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 98 |
| 0 | 0 | 98 | 0 |
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,30,0,0,0,0,30],[1,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[0,98,0,0,15,0,0,0,0,0,82,82,0,0,31,82],[0,1,0,0,1,0,0,0,0,0,0,98,0,0,98,0] >;
C7×C23.48D4 in GAP, Magma, Sage, TeX
C_7\times C_2^3._{48}D_4 % in TeX
G:=Group("C7xC2^3.48D4"); // GroupNames label
G:=SmallGroup(448,892);
// by ID
G=gap.SmallGroup(448,892);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,2438,1486,14117,3547,124]);
// 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=f*b*f^-1=b*c=c*b,b*d=d*b,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*d*e^3>;
// generators/relations