direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×C23.D7, C24.61D14, (C2×C14)⋊3C42, C14.83(C4×D4), (C23×C4).3D7, C28⋊8(C22⋊C4), (C22×C28)⋊17C4, (C2×C28).500D4, C22⋊2(C4×Dic7), (C22×C4)⋊6Dic7, C23.36(C4×D7), C14.28(C2×C42), (C23×C28).19C2, (C22×C4).405D14, C23.29(C2×Dic7), C14.C42⋊49C2, C22.60(C4○D28), (C23×C14).96C22, C23.300(C22×D7), C14.47(C42⋊C2), (C22×C28).481C22, (C22×C14).360C23, C22.26(C22×Dic7), C2.4(C23.21D14), (C22×Dic7).195C22, C7⋊4(C4×C22⋊C4), C2.4(C4×C7⋊D4), (C2×C4×Dic7)⋊26C2, C2.16(C2×C4×Dic7), C22.65(C2×C4×D7), (C2×C28).320(C2×C4), C2.3(C2×C23.D7), (C2×Dic7)⋊15(C2×C4), (C2×C14).546(C2×D4), C14.68(C2×C22⋊C4), (C2×C4).65(C2×Dic7), C22.84(C2×C7⋊D4), (C2×C14).88(C4○D4), (C2×C4).280(C7⋊D4), (C22×C14).98(C2×C4), (C2×C23.D7).24C2, (C2×C14).141(C22×C4), SmallGroup(448,743)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C23.D7
G = < a,b,c,d,e,f | a4=b2=c2=d2=e7=1, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 772 in 258 conjugacy classes, 119 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C42, C2×C22⋊C4, C23×C4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C4×C22⋊C4, C4×Dic7, C23.D7, C22×Dic7, C22×C28, C22×C28, C22×C28, C23×C14, C14.C42, C2×C4×Dic7, C2×C23.D7, C23×C28, C4×C23.D7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, Dic7, D14, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×D7, C2×Dic7, C7⋊D4, C22×D7, C4×C22⋊C4, C4×Dic7, C23.D7, C2×C4×D7, C4○D28, C22×Dic7, C2×C7⋊D4, C2×C4×Dic7, C23.21D14, C4×C7⋊D4, C2×C23.D7, C4×C23.D7
►On 224 points
Generators in S224
(1 71 15 57)(2 72 16 58)(3 73 17 59)(4 74 18 60)(5 75 19 61)(6 76 20 62)(7 77 21 63)(8 78 22 64)(9 79 23 65)(10 80 24 66)(11 81 25 67)(12 82 26 68)(13 83 27 69)(14 84 28 70)(29 99 43 85)(30 100 44 86)(31 101 45 87)(32 102 46 88)(33 103 47 89)(34 104 48 90)(35 105 49 91)(36 106 50 92)(37 107 51 93)(38 108 52 94)(39 109 53 95)(40 110 54 96)(41 111 55 97)(42 112 56 98)(113 183 127 169)(114 184 128 170)(115 185 129 171)(116 186 130 172)(117 187 131 173)(118 188 132 174)(119 189 133 175)(120 190 134 176)(121 191 135 177)(122 192 136 178)(123 193 137 179)(124 194 138 180)(125 195 139 181)(126 196 140 182)(141 211 155 197)(142 212 156 198)(143 213 157 199)(144 214 158 200)(145 215 159 201)(146 216 160 202)(147 217 161 203)(148 218 162 204)(149 219 163 205)(150 220 164 206)(151 221 165 207)(152 222 166 208)(153 223 167 209)(154 224 168 210)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(113 162)(114 163)(115 164)(116 165)(117 166)(118 167)(119 168)(120 155)(121 156)(122 157)(123 158)(124 159)(125 160)(126 161)(127 148)(128 149)(129 150)(130 151)(131 152)(132 153)(133 154)(134 141)(135 142)(136 143)(137 144)(138 145)(139 146)(140 147)(169 218)(170 219)(171 220)(172 221)(173 222)(174 223)(175 224)(176 211)(177 212)(178 213)(179 214)(180 215)(181 216)(182 217)(183 204)(184 205)(185 206)(186 207)(187 208)(188 209)(189 210)(190 197)(191 198)(192 199)(193 200)(194 201)(195 202)(196 203)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)(181 209)(182 210)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 219)(192 220)(193 221)(194 222)(195 223)(196 224)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(133 140)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(155 162)(156 163)(157 164)(158 165)(159 166)(160 167)(161 168)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)(217 224)
(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)
(1 145 29 117)(2 144 30 116)(3 143 31 115)(4 142 32 114)(5 141 33 113)(6 147 34 119)(7 146 35 118)(8 152 36 124)(9 151 37 123)(10 150 38 122)(11 149 39 121)(12 148 40 120)(13 154 41 126)(14 153 42 125)(15 159 43 131)(16 158 44 130)(17 157 45 129)(18 156 46 128)(19 155 47 127)(20 161 48 133)(21 160 49 132)(22 166 50 138)(23 165 51 137)(24 164 52 136)(25 163 53 135)(26 162 54 134)(27 168 55 140)(28 167 56 139)(57 201 85 173)(58 200 86 172)(59 199 87 171)(60 198 88 170)(61 197 89 169)(62 203 90 175)(63 202 91 174)(64 208 92 180)(65 207 93 179)(66 206 94 178)(67 205 95 177)(68 204 96 176)(69 210 97 182)(70 209 98 181)(71 215 99 187)(72 214 100 186)(73 213 101 185)(74 212 102 184)(75 211 103 183)(76 217 104 189)(77 216 105 188)(78 222 106 194)(79 221 107 193)(80 220 108 192)(81 219 109 191)(82 218 110 190)(83 224 111 196)(84 223 112 195)
G:=sub<Sym(224)| (1,71,15,57)(2,72,16,58)(3,73,17,59)(4,74,18,60)(5,75,19,61)(6,76,20,62)(7,77,21,63)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,99,43,85)(30,100,44,86)(31,101,45,87)(32,102,46,88)(33,103,47,89)(34,104,48,90)(35,105,49,91)(36,106,50,92)(37,107,51,93)(38,108,52,94)(39,109,53,95)(40,110,54,96)(41,111,55,97)(42,112,56,98)(113,183,127,169)(114,184,128,170)(115,185,129,171)(116,186,130,172)(117,187,131,173)(118,188,132,174)(119,189,133,175)(120,190,134,176)(121,191,135,177)(122,192,136,178)(123,193,137,179)(124,194,138,180)(125,195,139,181)(126,196,140,182)(141,211,155,197)(142,212,156,198)(143,213,157,199)(144,214,158,200)(145,215,159,201)(146,216,160,202)(147,217,161,203)(148,218,162,204)(149,219,163,205)(150,220,164,206)(151,221,165,207)(152,222,166,208)(153,223,167,209)(154,224,168,210), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)(120,155)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)(140,147)(169,218)(170,219)(171,220)(172,221)(173,222)(174,223)(175,224)(176,211)(177,212)(178,213)(179,214)(180,215)(181,216)(182,217)(183,204)(184,205)(185,206)(186,207)(187,208)(188,209)(189,210)(190,197)(191,198)(192,199)(193,200)(194,201)(195,202)(196,203), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (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), (1,145,29,117)(2,144,30,116)(3,143,31,115)(4,142,32,114)(5,141,33,113)(6,147,34,119)(7,146,35,118)(8,152,36,124)(9,151,37,123)(10,150,38,122)(11,149,39,121)(12,148,40,120)(13,154,41,126)(14,153,42,125)(15,159,43,131)(16,158,44,130)(17,157,45,129)(18,156,46,128)(19,155,47,127)(20,161,48,133)(21,160,49,132)(22,166,50,138)(23,165,51,137)(24,164,52,136)(25,163,53,135)(26,162,54,134)(27,168,55,140)(28,167,56,139)(57,201,85,173)(58,200,86,172)(59,199,87,171)(60,198,88,170)(61,197,89,169)(62,203,90,175)(63,202,91,174)(64,208,92,180)(65,207,93,179)(66,206,94,178)(67,205,95,177)(68,204,96,176)(69,210,97,182)(70,209,98,181)(71,215,99,187)(72,214,100,186)(73,213,101,185)(74,212,102,184)(75,211,103,183)(76,217,104,189)(77,216,105,188)(78,222,106,194)(79,221,107,193)(80,220,108,192)(81,219,109,191)(82,218,110,190)(83,224,111,196)(84,223,112,195)>;
G:=Group( (1,71,15,57)(2,72,16,58)(3,73,17,59)(4,74,18,60)(5,75,19,61)(6,76,20,62)(7,77,21,63)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,99,43,85)(30,100,44,86)(31,101,45,87)(32,102,46,88)(33,103,47,89)(34,104,48,90)(35,105,49,91)(36,106,50,92)(37,107,51,93)(38,108,52,94)(39,109,53,95)(40,110,54,96)(41,111,55,97)(42,112,56,98)(113,183,127,169)(114,184,128,170)(115,185,129,171)(116,186,130,172)(117,187,131,173)(118,188,132,174)(119,189,133,175)(120,190,134,176)(121,191,135,177)(122,192,136,178)(123,193,137,179)(124,194,138,180)(125,195,139,181)(126,196,140,182)(141,211,155,197)(142,212,156,198)(143,213,157,199)(144,214,158,200)(145,215,159,201)(146,216,160,202)(147,217,161,203)(148,218,162,204)(149,219,163,205)(150,220,164,206)(151,221,165,207)(152,222,166,208)(153,223,167,209)(154,224,168,210), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)(120,155)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)(140,147)(169,218)(170,219)(171,220)(172,221)(173,222)(174,223)(175,224)(176,211)(177,212)(178,213)(179,214)(180,215)(181,216)(182,217)(183,204)(184,205)(185,206)(186,207)(187,208)(188,209)(189,210)(190,197)(191,198)(192,199)(193,200)(194,201)(195,202)(196,203), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (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), (1,145,29,117)(2,144,30,116)(3,143,31,115)(4,142,32,114)(5,141,33,113)(6,147,34,119)(7,146,35,118)(8,152,36,124)(9,151,37,123)(10,150,38,122)(11,149,39,121)(12,148,40,120)(13,154,41,126)(14,153,42,125)(15,159,43,131)(16,158,44,130)(17,157,45,129)(18,156,46,128)(19,155,47,127)(20,161,48,133)(21,160,49,132)(22,166,50,138)(23,165,51,137)(24,164,52,136)(25,163,53,135)(26,162,54,134)(27,168,55,140)(28,167,56,139)(57,201,85,173)(58,200,86,172)(59,199,87,171)(60,198,88,170)(61,197,89,169)(62,203,90,175)(63,202,91,174)(64,208,92,180)(65,207,93,179)(66,206,94,178)(67,205,95,177)(68,204,96,176)(69,210,97,182)(70,209,98,181)(71,215,99,187)(72,214,100,186)(73,213,101,185)(74,212,102,184)(75,211,103,183)(76,217,104,189)(77,216,105,188)(78,222,106,194)(79,221,107,193)(80,220,108,192)(81,219,109,191)(82,218,110,190)(83,224,111,196)(84,223,112,195) );
G=PermutationGroup([[(1,71,15,57),(2,72,16,58),(3,73,17,59),(4,74,18,60),(5,75,19,61),(6,76,20,62),(7,77,21,63),(8,78,22,64),(9,79,23,65),(10,80,24,66),(11,81,25,67),(12,82,26,68),(13,83,27,69),(14,84,28,70),(29,99,43,85),(30,100,44,86),(31,101,45,87),(32,102,46,88),(33,103,47,89),(34,104,48,90),(35,105,49,91),(36,106,50,92),(37,107,51,93),(38,108,52,94),(39,109,53,95),(40,110,54,96),(41,111,55,97),(42,112,56,98),(113,183,127,169),(114,184,128,170),(115,185,129,171),(116,186,130,172),(117,187,131,173),(118,188,132,174),(119,189,133,175),(120,190,134,176),(121,191,135,177),(122,192,136,178),(123,193,137,179),(124,194,138,180),(125,195,139,181),(126,196,140,182),(141,211,155,197),(142,212,156,198),(143,213,157,199),(144,214,158,200),(145,215,159,201),(146,216,160,202),(147,217,161,203),(148,218,162,204),(149,219,163,205),(150,220,164,206),(151,221,165,207),(152,222,166,208),(153,223,167,209),(154,224,168,210)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(27,41),(28,42),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,105),(64,106),(65,107),(66,108),(67,109),(68,110),(69,111),(70,112),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,97),(84,98),(113,162),(114,163),(115,164),(116,165),(117,166),(118,167),(119,168),(120,155),(121,156),(122,157),(123,158),(124,159),(125,160),(126,161),(127,148),(128,149),(129,150),(130,151),(131,152),(132,153),(133,154),(134,141),(135,142),(136,143),(137,144),(138,145),(139,146),(140,147),(169,218),(170,219),(171,220),(172,221),(173,222),(174,223),(175,224),(176,211),(177,212),(178,213),(179,214),(180,215),(181,216),(182,217),(183,204),(184,205),(185,206),(186,207),(187,208),(188,209),(189,210),(190,197),(191,198),(192,199),(193,200),(194,201),(195,202),(196,203)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158),(131,159),(132,160),(133,161),(134,162),(135,163),(136,164),(137,165),(138,166),(139,167),(140,168),(169,197),(170,198),(171,199),(172,200),(173,201),(174,202),(175,203),(176,204),(177,205),(178,206),(179,207),(180,208),(181,209),(182,210),(183,211),(184,212),(185,213),(186,214),(187,215),(188,216),(189,217),(190,218),(191,219),(192,220),(193,221),(194,222),(195,223),(196,224)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(113,120),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(127,134),(128,135),(129,136),(130,137),(131,138),(132,139),(133,140),(141,148),(142,149),(143,150),(144,151),(145,152),(146,153),(147,154),(155,162),(156,163),(157,164),(158,165),(159,166),(160,167),(161,168),(169,176),(170,177),(171,178),(172,179),(173,180),(174,181),(175,182),(183,190),(184,191),(185,192),(186,193),(187,194),(188,195),(189,196),(197,204),(198,205),(199,206),(200,207),(201,208),(202,209),(203,210),(211,218),(212,219),(213,220),(214,221),(215,222),(216,223),(217,224)], [(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)], [(1,145,29,117),(2,144,30,116),(3,143,31,115),(4,142,32,114),(5,141,33,113),(6,147,34,119),(7,146,35,118),(8,152,36,124),(9,151,37,123),(10,150,38,122),(11,149,39,121),(12,148,40,120),(13,154,41,126),(14,153,42,125),(15,159,43,131),(16,158,44,130),(17,157,45,129),(18,156,46,128),(19,155,47,127),(20,161,48,133),(21,160,49,132),(22,166,50,138),(23,165,51,137),(24,164,52,136),(25,163,53,135),(26,162,54,134),(27,168,55,140),(28,167,56,139),(57,201,85,173),(58,200,86,172),(59,199,87,171),(60,198,88,170),(61,197,89,169),(62,203,90,175),(63,202,91,174),(64,208,92,180),(65,207,93,179),(66,206,94,178),(67,205,95,177),(68,204,96,176),(69,210,97,182),(70,209,98,181),(71,215,99,187),(72,214,100,186),(73,213,101,185),(74,212,102,184),(75,211,103,183),(76,217,104,189),(77,216,105,188),(78,222,106,194),(79,221,107,193),(80,220,108,192),(81,219,109,191),(82,218,110,190),(83,224,111,196),(84,223,112,195)]])
136 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4AB | 7A | 7B | 7C | 14A | ··· | 14AS | 28A | ··· | 28AV |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D7 | C4○D4 | Dic7 | D14 | D14 | C7⋊D4 | C4×D7 | C4○D28 |
| kernel | C4×C23.D7 | C14.C42 | C2×C4×Dic7 | C2×C23.D7 | C23×C28 | C23.D7 | C22×C28 | C2×C28 | C23×C4 | C2×C14 | C22×C4 | C22×C4 | C24 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 16 | 8 | 4 | 3 | 4 | 12 | 6 | 3 | 24 | 24 | 24 |
Matrix representation of C4×C23.D7 ►in GL4(𝔽29) generated by
| 12 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 25 |
| 28 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 25 |
| 0 | 0 | 7 | 0 |
G:=sub<GL(4,GF(29))| [12,0,0,0,0,17,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,28],[1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,1,0,0,0,0,7,0,0,0,0,25],[28,0,0,0,0,12,0,0,0,0,0,7,0,0,25,0] >;
C4×C23.D7 in GAP, Magma, Sage, TeX
C_4\times C_2^3.D_7
% in TeX
G:=Group("C4xC2^3.D7"); // GroupNames label
G:=SmallGroup(448,743);
// by ID
G=gap.SmallGroup(448,743);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,477,100,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^4=b^2=c^2=d^2=e^7=1,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=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^-1=e^-1>;
// generators/relations