direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C23.23D4, (C2×D4)⋊3C28, (C2×C28)⋊36D4, C2.5(D4×C28), (D4×C14)⋊15C4, (C23×C4)⋊3C14, (C23×C28)⋊2C2, C23⋊3(C2×C28), C14.106(C4×D4), C23.22(C7×D4), C14.89C22≀C2, C24.30(C2×C14), (C22×D4).1C14, C22.35(D4×C14), C2.C42⋊8C14, (C22×C14).156D4, C14.133(C4⋊D4), C23.59(C22×C14), (C23×C14).87C22, C22.35(C22×C28), (C22×C28).494C22, (C22×C14).450C23, C14.87(C22.D4), (C2×C4)⋊9(C7×D4), (C2×C4)⋊3(C2×C28), (C2×C28)⋊24(C2×C4), (D4×C2×C14).13C2, C2.2(C7×C4⋊D4), (C2×C22⋊C4)⋊2C14, (C14×C22⋊C4)⋊6C2, (C22×C14)⋊4(C2×C4), C2.3(C7×C22≀C2), C2.7(C14×C22⋊C4), C22⋊1(C7×C22⋊C4), (C2×C14)⋊4(C22⋊C4), (C2×C14).602(C2×D4), C14.94(C2×C22⋊C4), C22.20(C7×C4○D4), (C22×C4).87(C2×C14), (C2×C14).210(C4○D4), C2.3(C7×C22.D4), (C2×C14).222(C22×C4), (C7×C2.C42)⋊24C2, SmallGroup(448,794)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.23D4
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, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de-1 >
Subgroups: 498 in 286 conjugacy classes, 106 normal (30 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, C22×C4, C2×D4, C2×D4, C24, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C22×D4, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C23.23D4, C7×C22⋊C4, C22×C28, C22×C28, C22×C28, D4×C14, D4×C14, C23×C14, C7×C2.C42, C14×C22⋊C4, C14×C22⋊C4, C23×C28, D4×C2×C14, C7×C23.23D4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C4○D4, C28, C2×C14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C2×C28, C7×D4, C22×C14, C23.23D4, C7×C22⋊C4, C22×C28, D4×C14, C7×C4○D4, C14×C22⋊C4, D4×C28, C7×C22≀C2, C7×C4⋊D4, C7×C22.D4, C7×C23.23D4
►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)
(8 213)(9 214)(10 215)(11 216)(12 217)(13 211)(14 212)(15 33)(16 34)(17 35)(18 29)(19 30)(20 31)(21 32)(22 40)(23 41)(24 42)(25 36)(26 37)(27 38)(28 39)(43 220)(44 221)(45 222)(46 223)(47 224)(48 218)(49 219)(155 173)(156 174)(157 175)(158 169)(159 170)(160 171)(161 172)(162 207)(163 208)(164 209)(165 210)(166 204)(167 205)(168 206)(176 194)(177 195)(178 196)(179 190)(180 191)(181 192)(182 193)(183 201)(184 202)(185 203)(186 197)(187 198)(188 199)(189 200)
(1 76)(2 77)(3 71)(4 72)(5 73)(6 74)(7 75)(8 213)(9 214)(10 215)(11 216)(12 217)(13 211)(14 212)(15 33)(16 34)(17 35)(18 29)(19 30)(20 31)(21 32)(22 40)(23 41)(24 42)(25 36)(26 37)(27 38)(28 39)(43 220)(44 221)(45 222)(46 223)(47 224)(48 218)(49 219)(50 95)(51 96)(52 97)(53 98)(54 92)(55 93)(56 94)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 82)(65 83)(66 84)(67 78)(68 79)(69 80)(70 81)(99 117)(100 118)(101 119)(102 113)(103 114)(104 115)(105 116)(106 151)(107 152)(108 153)(109 154)(110 148)(111 149)(112 150)(120 138)(121 139)(122 140)(123 134)(124 135)(125 136)(126 137)(127 145)(128 146)(129 147)(130 141)(131 142)(132 143)(133 144)(155 173)(156 174)(157 175)(158 169)(159 170)(160 171)(161 172)(162 207)(163 208)(164 209)(165 210)(166 204)(167 205)(168 206)(176 194)(177 195)(178 196)(179 190)(180 191)(181 192)(182 193)(183 201)(184 202)(185 203)(186 197)(187 198)(188 199)(189 200)
(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)
(1 106 50 143)(2 107 51 144)(3 108 52 145)(4 109 53 146)(5 110 54 147)(6 111 55 141)(7 112 56 142)(8 203 35 166)(9 197 29 167)(10 198 30 168)(11 199 31 162)(12 200 32 163)(13 201 33 164)(14 202 34 165)(15 209 211 183)(16 210 212 184)(17 204 213 185)(18 205 214 186)(19 206 215 187)(20 207 216 188)(21 208 217 189)(22 160 48 179)(23 161 49 180)(24 155 43 181)(25 156 44 182)(26 157 45 176)(27 158 46 177)(28 159 47 178)(36 174 221 193)(37 175 222 194)(38 169 223 195)(39 170 224 196)(40 171 218 190)(41 172 219 191)(42 173 220 192)(57 139 83 113)(58 140 84 114)(59 134 78 115)(60 135 79 116)(61 136 80 117)(62 137 81 118)(63 138 82 119)(64 101 91 120)(65 102 85 121)(66 103 86 122)(67 104 87 123)(68 105 88 124)(69 99 89 125)(70 100 90 126)(71 153 97 127)(72 154 98 128)(73 148 92 129)(74 149 93 130)(75 150 94 131)(76 151 95 132)(77 152 96 133)
(1 11)(2 12)(3 13)(4 14)(5 8)(6 9)(7 10)(15 97)(16 98)(17 92)(18 93)(19 94)(20 95)(21 96)(22 87)(23 88)(24 89)(25 90)(26 91)(27 85)(28 86)(29 55)(30 56)(31 50)(32 51)(33 52)(34 53)(35 54)(36 62)(37 63)(38 57)(39 58)(40 59)(41 60)(42 61)(43 69)(44 70)(45 64)(46 65)(47 66)(48 67)(49 68)(71 211)(72 212)(73 213)(74 214)(75 215)(76 216)(77 217)(78 218)(79 219)(80 220)(81 221)(82 222)(83 223)(84 224)(99 209)(100 210)(101 204)(102 205)(103 206)(104 207)(105 208)(106 171)(107 172)(108 173)(109 174)(110 175)(111 169)(112 170)(113 167)(114 168)(115 162)(116 163)(117 164)(118 165)(119 166)(120 185)(121 186)(122 187)(123 188)(124 189)(125 183)(126 184)(127 181)(128 182)(129 176)(130 177)(131 178)(132 179)(133 180)(134 199)(135 200)(136 201)(137 202)(138 203)(139 197)(140 198)(141 195)(142 196)(143 190)(144 191)(145 192)(146 193)(147 194)(148 157)(149 158)(150 159)(151 160)(152 161)(153 155)(154 156)
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), (8,213)(9,214)(10,215)(11,216)(12,217)(13,211)(14,212)(15,33)(16,34)(17,35)(18,29)(19,30)(20,31)(21,32)(22,40)(23,41)(24,42)(25,36)(26,37)(27,38)(28,39)(43,220)(44,221)(45,222)(46,223)(47,224)(48,218)(49,219)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(176,194)(177,195)(178,196)(179,190)(180,191)(181,192)(182,193)(183,201)(184,202)(185,203)(186,197)(187,198)(188,199)(189,200), (1,76)(2,77)(3,71)(4,72)(5,73)(6,74)(7,75)(8,213)(9,214)(10,215)(11,216)(12,217)(13,211)(14,212)(15,33)(16,34)(17,35)(18,29)(19,30)(20,31)(21,32)(22,40)(23,41)(24,42)(25,36)(26,37)(27,38)(28,39)(43,220)(44,221)(45,222)(46,223)(47,224)(48,218)(49,219)(50,95)(51,96)(52,97)(53,98)(54,92)(55,93)(56,94)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,82)(65,83)(66,84)(67,78)(68,79)(69,80)(70,81)(99,117)(100,118)(101,119)(102,113)(103,114)(104,115)(105,116)(106,151)(107,152)(108,153)(109,154)(110,148)(111,149)(112,150)(120,138)(121,139)(122,140)(123,134)(124,135)(125,136)(126,137)(127,145)(128,146)(129,147)(130,141)(131,142)(132,143)(133,144)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(176,194)(177,195)(178,196)(179,190)(180,191)(181,192)(182,193)(183,201)(184,202)(185,203)(186,197)(187,198)(188,199)(189,200), (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), (1,106,50,143)(2,107,51,144)(3,108,52,145)(4,109,53,146)(5,110,54,147)(6,111,55,141)(7,112,56,142)(8,203,35,166)(9,197,29,167)(10,198,30,168)(11,199,31,162)(12,200,32,163)(13,201,33,164)(14,202,34,165)(15,209,211,183)(16,210,212,184)(17,204,213,185)(18,205,214,186)(19,206,215,187)(20,207,216,188)(21,208,217,189)(22,160,48,179)(23,161,49,180)(24,155,43,181)(25,156,44,182)(26,157,45,176)(27,158,46,177)(28,159,47,178)(36,174,221,193)(37,175,222,194)(38,169,223,195)(39,170,224,196)(40,171,218,190)(41,172,219,191)(42,173,220,192)(57,139,83,113)(58,140,84,114)(59,134,78,115)(60,135,79,116)(61,136,80,117)(62,137,81,118)(63,138,82,119)(64,101,91,120)(65,102,85,121)(66,103,86,122)(67,104,87,123)(68,105,88,124)(69,99,89,125)(70,100,90,126)(71,153,97,127)(72,154,98,128)(73,148,92,129)(74,149,93,130)(75,150,94,131)(76,151,95,132)(77,152,96,133), (1,11)(2,12)(3,13)(4,14)(5,8)(6,9)(7,10)(15,97)(16,98)(17,92)(18,93)(19,94)(20,95)(21,96)(22,87)(23,88)(24,89)(25,90)(26,91)(27,85)(28,86)(29,55)(30,56)(31,50)(32,51)(33,52)(34,53)(35,54)(36,62)(37,63)(38,57)(39,58)(40,59)(41,60)(42,61)(43,69)(44,70)(45,64)(46,65)(47,66)(48,67)(49,68)(71,211)(72,212)(73,213)(74,214)(75,215)(76,216)(77,217)(78,218)(79,219)(80,220)(81,221)(82,222)(83,223)(84,224)(99,209)(100,210)(101,204)(102,205)(103,206)(104,207)(105,208)(106,171)(107,172)(108,173)(109,174)(110,175)(111,169)(112,170)(113,167)(114,168)(115,162)(116,163)(117,164)(118,165)(119,166)(120,185)(121,186)(122,187)(123,188)(124,189)(125,183)(126,184)(127,181)(128,182)(129,176)(130,177)(131,178)(132,179)(133,180)(134,199)(135,200)(136,201)(137,202)(138,203)(139,197)(140,198)(141,195)(142,196)(143,190)(144,191)(145,192)(146,193)(147,194)(148,157)(149,158)(150,159)(151,160)(152,161)(153,155)(154,156)>;
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), (8,213)(9,214)(10,215)(11,216)(12,217)(13,211)(14,212)(15,33)(16,34)(17,35)(18,29)(19,30)(20,31)(21,32)(22,40)(23,41)(24,42)(25,36)(26,37)(27,38)(28,39)(43,220)(44,221)(45,222)(46,223)(47,224)(48,218)(49,219)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(176,194)(177,195)(178,196)(179,190)(180,191)(181,192)(182,193)(183,201)(184,202)(185,203)(186,197)(187,198)(188,199)(189,200), (1,76)(2,77)(3,71)(4,72)(5,73)(6,74)(7,75)(8,213)(9,214)(10,215)(11,216)(12,217)(13,211)(14,212)(15,33)(16,34)(17,35)(18,29)(19,30)(20,31)(21,32)(22,40)(23,41)(24,42)(25,36)(26,37)(27,38)(28,39)(43,220)(44,221)(45,222)(46,223)(47,224)(48,218)(49,219)(50,95)(51,96)(52,97)(53,98)(54,92)(55,93)(56,94)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,82)(65,83)(66,84)(67,78)(68,79)(69,80)(70,81)(99,117)(100,118)(101,119)(102,113)(103,114)(104,115)(105,116)(106,151)(107,152)(108,153)(109,154)(110,148)(111,149)(112,150)(120,138)(121,139)(122,140)(123,134)(124,135)(125,136)(126,137)(127,145)(128,146)(129,147)(130,141)(131,142)(132,143)(133,144)(155,173)(156,174)(157,175)(158,169)(159,170)(160,171)(161,172)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(176,194)(177,195)(178,196)(179,190)(180,191)(181,192)(182,193)(183,201)(184,202)(185,203)(186,197)(187,198)(188,199)(189,200), (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), (1,106,50,143)(2,107,51,144)(3,108,52,145)(4,109,53,146)(5,110,54,147)(6,111,55,141)(7,112,56,142)(8,203,35,166)(9,197,29,167)(10,198,30,168)(11,199,31,162)(12,200,32,163)(13,201,33,164)(14,202,34,165)(15,209,211,183)(16,210,212,184)(17,204,213,185)(18,205,214,186)(19,206,215,187)(20,207,216,188)(21,208,217,189)(22,160,48,179)(23,161,49,180)(24,155,43,181)(25,156,44,182)(26,157,45,176)(27,158,46,177)(28,159,47,178)(36,174,221,193)(37,175,222,194)(38,169,223,195)(39,170,224,196)(40,171,218,190)(41,172,219,191)(42,173,220,192)(57,139,83,113)(58,140,84,114)(59,134,78,115)(60,135,79,116)(61,136,80,117)(62,137,81,118)(63,138,82,119)(64,101,91,120)(65,102,85,121)(66,103,86,122)(67,104,87,123)(68,105,88,124)(69,99,89,125)(70,100,90,126)(71,153,97,127)(72,154,98,128)(73,148,92,129)(74,149,93,130)(75,150,94,131)(76,151,95,132)(77,152,96,133), (1,11)(2,12)(3,13)(4,14)(5,8)(6,9)(7,10)(15,97)(16,98)(17,92)(18,93)(19,94)(20,95)(21,96)(22,87)(23,88)(24,89)(25,90)(26,91)(27,85)(28,86)(29,55)(30,56)(31,50)(32,51)(33,52)(34,53)(35,54)(36,62)(37,63)(38,57)(39,58)(40,59)(41,60)(42,61)(43,69)(44,70)(45,64)(46,65)(47,66)(48,67)(49,68)(71,211)(72,212)(73,213)(74,214)(75,215)(76,216)(77,217)(78,218)(79,219)(80,220)(81,221)(82,222)(83,223)(84,224)(99,209)(100,210)(101,204)(102,205)(103,206)(104,207)(105,208)(106,171)(107,172)(108,173)(109,174)(110,175)(111,169)(112,170)(113,167)(114,168)(115,162)(116,163)(117,164)(118,165)(119,166)(120,185)(121,186)(122,187)(123,188)(124,189)(125,183)(126,184)(127,181)(128,182)(129,176)(130,177)(131,178)(132,179)(133,180)(134,199)(135,200)(136,201)(137,202)(138,203)(139,197)(140,198)(141,195)(142,196)(143,190)(144,191)(145,192)(146,193)(147,194)(148,157)(149,158)(150,159)(151,160)(152,161)(153,155)(154,156) );
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)], [(8,213),(9,214),(10,215),(11,216),(12,217),(13,211),(14,212),(15,33),(16,34),(17,35),(18,29),(19,30),(20,31),(21,32),(22,40),(23,41),(24,42),(25,36),(26,37),(27,38),(28,39),(43,220),(44,221),(45,222),(46,223),(47,224),(48,218),(49,219),(155,173),(156,174),(157,175),(158,169),(159,170),(160,171),(161,172),(162,207),(163,208),(164,209),(165,210),(166,204),(167,205),(168,206),(176,194),(177,195),(178,196),(179,190),(180,191),(181,192),(182,193),(183,201),(184,202),(185,203),(186,197),(187,198),(188,199),(189,200)], [(1,76),(2,77),(3,71),(4,72),(5,73),(6,74),(7,75),(8,213),(9,214),(10,215),(11,216),(12,217),(13,211),(14,212),(15,33),(16,34),(17,35),(18,29),(19,30),(20,31),(21,32),(22,40),(23,41),(24,42),(25,36),(26,37),(27,38),(28,39),(43,220),(44,221),(45,222),(46,223),(47,224),(48,218),(49,219),(50,95),(51,96),(52,97),(53,98),(54,92),(55,93),(56,94),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,82),(65,83),(66,84),(67,78),(68,79),(69,80),(70,81),(99,117),(100,118),(101,119),(102,113),(103,114),(104,115),(105,116),(106,151),(107,152),(108,153),(109,154),(110,148),(111,149),(112,150),(120,138),(121,139),(122,140),(123,134),(124,135),(125,136),(126,137),(127,145),(128,146),(129,147),(130,141),(131,142),(132,143),(133,144),(155,173),(156,174),(157,175),(158,169),(159,170),(160,171),(161,172),(162,207),(163,208),(164,209),(165,210),(166,204),(167,205),(168,206),(176,194),(177,195),(178,196),(179,190),(180,191),(181,192),(182,193),(183,201),(184,202),(185,203),(186,197),(187,198),(188,199),(189,200)], [(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)], [(1,106,50,143),(2,107,51,144),(3,108,52,145),(4,109,53,146),(5,110,54,147),(6,111,55,141),(7,112,56,142),(8,203,35,166),(9,197,29,167),(10,198,30,168),(11,199,31,162),(12,200,32,163),(13,201,33,164),(14,202,34,165),(15,209,211,183),(16,210,212,184),(17,204,213,185),(18,205,214,186),(19,206,215,187),(20,207,216,188),(21,208,217,189),(22,160,48,179),(23,161,49,180),(24,155,43,181),(25,156,44,182),(26,157,45,176),(27,158,46,177),(28,159,47,178),(36,174,221,193),(37,175,222,194),(38,169,223,195),(39,170,224,196),(40,171,218,190),(41,172,219,191),(42,173,220,192),(57,139,83,113),(58,140,84,114),(59,134,78,115),(60,135,79,116),(61,136,80,117),(62,137,81,118),(63,138,82,119),(64,101,91,120),(65,102,85,121),(66,103,86,122),(67,104,87,123),(68,105,88,124),(69,99,89,125),(70,100,90,126),(71,153,97,127),(72,154,98,128),(73,148,92,129),(74,149,93,130),(75,150,94,131),(76,151,95,132),(77,152,96,133)], [(1,11),(2,12),(3,13),(4,14),(5,8),(6,9),(7,10),(15,97),(16,98),(17,92),(18,93),(19,94),(20,95),(21,96),(22,87),(23,88),(24,89),(25,90),(26,91),(27,85),(28,86),(29,55),(30,56),(31,50),(32,51),(33,52),(34,53),(35,54),(36,62),(37,63),(38,57),(39,58),(40,59),(41,60),(42,61),(43,69),(44,70),(45,64),(46,65),(47,66),(48,67),(49,68),(71,211),(72,212),(73,213),(74,214),(75,215),(76,216),(77,217),(78,218),(79,219),(80,220),(81,221),(82,222),(83,223),(84,224),(99,209),(100,210),(101,204),(102,205),(103,206),(104,207),(105,208),(106,171),(107,172),(108,173),(109,174),(110,175),(111,169),(112,170),(113,167),(114,168),(115,162),(116,163),(117,164),(118,165),(119,166),(120,185),(121,186),(122,187),(123,188),(124,189),(125,183),(126,184),(127,181),(128,182),(129,176),(130,177),(131,178),(132,179),(133,180),(134,199),(135,200),(136,201),(137,202),(138,203),(139,197),(140,198),(141,195),(142,196),(143,190),(144,191),(145,192),(146,193),(147,194),(148,157),(149,158),(150,159),(151,160),(152,161),(153,155),(154,156)]])
196 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4N | 7A | ··· | 7F | 14A | ··· | 14AP | 14AQ | ··· | 14BN | 14BO | ··· | 14BZ | 28A | ··· | 28AV | 28AW | ··· | 28CF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | D4 | D4 | C4○D4 | C7×D4 | C7×D4 | C7×C4○D4 |
| kernel | C7×C23.23D4 | C7×C2.C42 | C14×C22⋊C4 | C23×C28 | D4×C2×C14 | D4×C14 | C23.23D4 | C2.C42 | C2×C22⋊C4 | C23×C4 | C22×D4 | C2×D4 | C2×C28 | C22×C14 | C2×C14 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 3 | 1 | 1 | 8 | 6 | 12 | 18 | 6 | 6 | 48 | 4 | 4 | 4 | 24 | 24 | 24 |
Matrix representation of C7×C23.23D4 ►in GL5(𝔽29)
| 25 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 0 | 23 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 6 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 5 |
| 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 17 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 |
| 0 | 15 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 14 | 5 | 0 | 0 |
| 0 | 19 | 15 | 0 | 0 |
| 0 | 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 27 | 24 |
G:=sub<GL(5,GF(29))| [25,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,23,0,0,0,0,0,23],[28,0,0,0,0,0,1,6,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,5,28],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[17,0,0,0,0,0,17,15,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,14,19,0,0,0,5,15,0,0,0,0,0,5,27,0,0,0,12,24] >;
C7×C23.23D4 in GAP, Magma, Sage, TeX
C_7\times C_2^3._{23}D_4 % in TeX
G:=Group("C7xC2^3.23D4"); // GroupNames label
G:=SmallGroup(448,794);
// by ID
G=gap.SmallGroup(448,794);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,1968,2438]);
// 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,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=d*e^-1>;
// generators/relations