direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.8D14, C28.34C24, D28.30C23, Dic14.29C23, C4○D4⋊15D14, C14⋊5(C4○D8), C7⋊C8.30C23, D4⋊D7⋊23C22, C28.263(C2×D4), (C2×C28).502D4, Q8⋊D7⋊21C22, C4.34(C23×D7), (C2×D4).232D14, C4○D28⋊20C22, D4.D7⋊20C22, (C2×Q8).190D14, (C7×D4).22C23, D4.22(C22×D7), C7⋊Q16⋊20C22, (C7×Q8).22C23, Q8.22(C22×D7), (C2×C28).556C23, (C22×C4).387D14, (C22×C14).123D4, C14.159(C22×D4), C23.45(C7⋊D4), (D4×C14).272C22, (C2×D28).280C22, (Q8×C14).237C22, (C22×C28).291C22, (C2×Dic14).309C22, C7⋊6(C2×C4○D8), (C2×C4○D4)⋊3D7, (C2×D4⋊D7)⋊33C2, (C14×C4○D4)⋊3C2, (C22×C7⋊C8)⋊15C2, (C2×C7⋊C8)⋊42C22, (C2×Q8⋊D7)⋊33C2, C4.30(C2×C7⋊D4), (C2×C4○D28)⋊30C2, (C2×D4.D7)⋊33C2, (C2×C7⋊Q16)⋊33C2, (C2×C14).76(C2×D4), (C7×C4○D4)⋊17C22, C2.32(C22×C7⋊D4), (C2×C4).158(C7⋊D4), (C2×C4).636(C22×D7), C22.119(C2×C7⋊D4), SmallGroup(448,1274)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4.8D14
G = < a,b,c,d,e | a2=b4=c2=1, d14=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d13 >
Subgroups: 1044 in 266 conjugacy classes, 111 normal (35 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D7, C14, C14, C14, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, C28, D14, C2×C14, C2×C14, C2×C14, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C4○D4, C7⋊C8, Dic14, Dic14, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×D7, C22×C14, C22×C14, C2×C4○D8, C2×C7⋊C8, C2×C7⋊C8, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C4○D28, C2×C7⋊D4, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C22×C7⋊C8, C2×D4⋊D7, C2×D4.D7, C2×Q8⋊D7, C2×C7⋊Q16, D4.8D14, C2×C4○D28, C14×C4○D4, C2×D4.8D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C4○D8, C22×D4, C7⋊D4, C22×D7, C2×C4○D8, C2×C7⋊D4, C23×D7, D4.8D14, C22×C7⋊D4, C2×D4.8D14
►On 224 points
Generators in S224
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 81)(8 82)(9 83)(10 84)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 105)(41 106)(42 107)(43 108)(44 109)(45 110)(46 111)(47 112)(48 85)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)(113 170)(114 171)(115 172)(116 173)(117 174)(118 175)(119 176)(120 177)(121 178)(122 179)(123 180)(124 181)(125 182)(126 183)(127 184)(128 185)(129 186)(130 187)(131 188)(132 189)(133 190)(134 191)(135 192)(136 193)(137 194)(138 195)(139 196)(140 169)(141 219)(142 220)(143 221)(144 222)(145 223)(146 224)(147 197)(148 198)(149 199)(150 200)(151 201)(152 202)(153 203)(154 204)(155 205)(156 206)(157 207)(158 208)(159 209)(160 210)(161 211)(162 212)(163 213)(164 214)(165 215)(166 216)(167 217)(168 218)
(1 8 15 22)(2 9 16 23)(3 10 17 24)(4 11 18 25)(5 12 19 26)(6 13 20 27)(7 14 21 28)(29 50 43 36)(30 51 44 37)(31 52 45 38)(32 53 46 39)(33 54 47 40)(34 55 48 41)(35 56 49 42)(57 64 71 78)(58 65 72 79)(59 66 73 80)(60 67 74 81)(61 68 75 82)(62 69 76 83)(63 70 77 84)(85 106 99 92)(86 107 100 93)(87 108 101 94)(88 109 102 95)(89 110 103 96)(90 111 104 97)(91 112 105 98)(113 134 127 120)(114 135 128 121)(115 136 129 122)(116 137 130 123)(117 138 131 124)(118 139 132 125)(119 140 133 126)(141 148 155 162)(142 149 156 163)(143 150 157 164)(144 151 158 165)(145 152 159 166)(146 153 160 167)(147 154 161 168)(169 190 183 176)(170 191 184 177)(171 192 185 178)(172 193 186 179)(173 194 187 180)(174 195 188 181)(175 196 189 182)(197 204 211 218)(198 205 212 219)(199 206 213 220)(200 207 214 221)(201 208 215 222)(202 209 216 223)(203 210 217 224)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 29)(24 30)(25 31)(26 32)(27 33)(28 34)(57 110)(58 111)(59 112)(60 85)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 97)(73 98)(74 99)(75 100)(76 101)(77 102)(78 103)(79 104)(80 105)(81 106)(82 107)(83 108)(84 109)(113 162)(114 163)(115 164)(116 165)(117 166)(118 167)(119 168)(120 141)(121 142)(122 143)(123 144)(124 145)(125 146)(126 147)(127 148)(128 149)(129 150)(130 151)(131 152)(132 153)(133 154)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 161)(169 211)(170 212)(171 213)(172 214)(173 215)(174 216)(175 217)(176 218)(177 219)(178 220)(179 221)(180 222)(181 223)(182 224)(183 197)(184 198)(185 199)(186 200)(187 201)(188 202)(189 203)(190 204)(191 205)(192 206)(193 207)(194 208)(195 209)(196 210)
(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)
(1 140 15 126)(2 125 16 139)(3 138 17 124)(4 123 18 137)(5 136 19 122)(6 121 20 135)(7 134 21 120)(8 119 22 133)(9 132 23 118)(10 117 24 131)(11 130 25 116)(12 115 26 129)(13 128 27 114)(14 113 28 127)(29 160 43 146)(30 145 44 159)(31 158 45 144)(32 143 46 157)(33 156 47 142)(34 141 48 155)(35 154 49 168)(36 167 50 153)(37 152 51 166)(38 165 52 151)(39 150 53 164)(40 163 54 149)(41 148 55 162)(42 161 56 147)(57 187 71 173)(58 172 72 186)(59 185 73 171)(60 170 74 184)(61 183 75 169)(62 196 76 182)(63 181 77 195)(64 194 78 180)(65 179 79 193)(66 192 80 178)(67 177 81 191)(68 190 82 176)(69 175 83 189)(70 188 84 174)(85 205 99 219)(86 218 100 204)(87 203 101 217)(88 216 102 202)(89 201 103 215)(90 214 104 200)(91 199 105 213)(92 212 106 198)(93 197 107 211)(94 210 108 224)(95 223 109 209)(96 208 110 222)(97 221 111 207)(98 206 112 220)
G:=sub<Sym(224)| (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(113,170)(114,171)(115,172)(116,173)(117,174)(118,175)(119,176)(120,177)(121,178)(122,179)(123,180)(124,181)(125,182)(126,183)(127,184)(128,185)(129,186)(130,187)(131,188)(132,189)(133,190)(134,191)(135,192)(136,193)(137,194)(138,195)(139,196)(140,169)(141,219)(142,220)(143,221)(144,222)(145,223)(146,224)(147,197)(148,198)(149,199)(150,200)(151,201)(152,202)(153,203)(154,204)(155,205)(156,206)(157,207)(158,208)(159,209)(160,210)(161,211)(162,212)(163,213)(164,214)(165,215)(166,216)(167,217)(168,218), (1,8,15,22)(2,9,16,23)(3,10,17,24)(4,11,18,25)(5,12,19,26)(6,13,20,27)(7,14,21,28)(29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42)(57,64,71,78)(58,65,72,79)(59,66,73,80)(60,67,74,81)(61,68,75,82)(62,69,76,83)(63,70,77,84)(85,106,99,92)(86,107,100,93)(87,108,101,94)(88,109,102,95)(89,110,103,96)(90,111,104,97)(91,112,105,98)(113,134,127,120)(114,135,128,121)(115,136,129,122)(116,137,130,123)(117,138,131,124)(118,139,132,125)(119,140,133,126)(141,148,155,162)(142,149,156,163)(143,150,157,164)(144,151,158,165)(145,152,159,166)(146,153,160,167)(147,154,161,168)(169,190,183,176)(170,191,184,177)(171,192,185,178)(172,193,186,179)(173,194,187,180)(174,195,188,181)(175,196,189,182)(197,204,211,218)(198,205,212,219)(199,206,213,220)(200,207,214,221)(201,208,215,222)(202,209,216,223)(203,210,217,224), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,29)(24,30)(25,31)(26,32)(27,33)(28,34)(57,110)(58,111)(59,112)(60,85)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)(120,141)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(169,211)(170,212)(171,213)(172,214)(173,215)(174,216)(175,217)(176,218)(177,219)(178,220)(179,221)(180,222)(181,223)(182,224)(183,197)(184,198)(185,199)(186,200)(187,201)(188,202)(189,203)(190,204)(191,205)(192,206)(193,207)(194,208)(195,209)(196,210), (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), (1,140,15,126)(2,125,16,139)(3,138,17,124)(4,123,18,137)(5,136,19,122)(6,121,20,135)(7,134,21,120)(8,119,22,133)(9,132,23,118)(10,117,24,131)(11,130,25,116)(12,115,26,129)(13,128,27,114)(14,113,28,127)(29,160,43,146)(30,145,44,159)(31,158,45,144)(32,143,46,157)(33,156,47,142)(34,141,48,155)(35,154,49,168)(36,167,50,153)(37,152,51,166)(38,165,52,151)(39,150,53,164)(40,163,54,149)(41,148,55,162)(42,161,56,147)(57,187,71,173)(58,172,72,186)(59,185,73,171)(60,170,74,184)(61,183,75,169)(62,196,76,182)(63,181,77,195)(64,194,78,180)(65,179,79,193)(66,192,80,178)(67,177,81,191)(68,190,82,176)(69,175,83,189)(70,188,84,174)(85,205,99,219)(86,218,100,204)(87,203,101,217)(88,216,102,202)(89,201,103,215)(90,214,104,200)(91,199,105,213)(92,212,106,198)(93,197,107,211)(94,210,108,224)(95,223,109,209)(96,208,110,222)(97,221,111,207)(98,206,112,220)>;
G:=Group( (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,81)(8,82)(9,83)(10,84)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,67)(22,68)(23,69)(24,70)(25,71)(26,72)(27,73)(28,74)(29,94)(30,95)(31,96)(32,97)(33,98)(34,99)(35,100)(36,101)(37,102)(38,103)(39,104)(40,105)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,85)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(113,170)(114,171)(115,172)(116,173)(117,174)(118,175)(119,176)(120,177)(121,178)(122,179)(123,180)(124,181)(125,182)(126,183)(127,184)(128,185)(129,186)(130,187)(131,188)(132,189)(133,190)(134,191)(135,192)(136,193)(137,194)(138,195)(139,196)(140,169)(141,219)(142,220)(143,221)(144,222)(145,223)(146,224)(147,197)(148,198)(149,199)(150,200)(151,201)(152,202)(153,203)(154,204)(155,205)(156,206)(157,207)(158,208)(159,209)(160,210)(161,211)(162,212)(163,213)(164,214)(165,215)(166,216)(167,217)(168,218), (1,8,15,22)(2,9,16,23)(3,10,17,24)(4,11,18,25)(5,12,19,26)(6,13,20,27)(7,14,21,28)(29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42)(57,64,71,78)(58,65,72,79)(59,66,73,80)(60,67,74,81)(61,68,75,82)(62,69,76,83)(63,70,77,84)(85,106,99,92)(86,107,100,93)(87,108,101,94)(88,109,102,95)(89,110,103,96)(90,111,104,97)(91,112,105,98)(113,134,127,120)(114,135,128,121)(115,136,129,122)(116,137,130,123)(117,138,131,124)(118,139,132,125)(119,140,133,126)(141,148,155,162)(142,149,156,163)(143,150,157,164)(144,151,158,165)(145,152,159,166)(146,153,160,167)(147,154,161,168)(169,190,183,176)(170,191,184,177)(171,192,185,178)(172,193,186,179)(173,194,187,180)(174,195,188,181)(175,196,189,182)(197,204,211,218)(198,205,212,219)(199,206,213,220)(200,207,214,221)(201,208,215,222)(202,209,216,223)(203,210,217,224), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,29)(24,30)(25,31)(26,32)(27,33)(28,34)(57,110)(58,111)(59,112)(60,85)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)(120,141)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(169,211)(170,212)(171,213)(172,214)(173,215)(174,216)(175,217)(176,218)(177,219)(178,220)(179,221)(180,222)(181,223)(182,224)(183,197)(184,198)(185,199)(186,200)(187,201)(188,202)(189,203)(190,204)(191,205)(192,206)(193,207)(194,208)(195,209)(196,210), (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), (1,140,15,126)(2,125,16,139)(3,138,17,124)(4,123,18,137)(5,136,19,122)(6,121,20,135)(7,134,21,120)(8,119,22,133)(9,132,23,118)(10,117,24,131)(11,130,25,116)(12,115,26,129)(13,128,27,114)(14,113,28,127)(29,160,43,146)(30,145,44,159)(31,158,45,144)(32,143,46,157)(33,156,47,142)(34,141,48,155)(35,154,49,168)(36,167,50,153)(37,152,51,166)(38,165,52,151)(39,150,53,164)(40,163,54,149)(41,148,55,162)(42,161,56,147)(57,187,71,173)(58,172,72,186)(59,185,73,171)(60,170,74,184)(61,183,75,169)(62,196,76,182)(63,181,77,195)(64,194,78,180)(65,179,79,193)(66,192,80,178)(67,177,81,191)(68,190,82,176)(69,175,83,189)(70,188,84,174)(85,205,99,219)(86,218,100,204)(87,203,101,217)(88,216,102,202)(89,201,103,215)(90,214,104,200)(91,199,105,213)(92,212,106,198)(93,197,107,211)(94,210,108,224)(95,223,109,209)(96,208,110,222)(97,221,111,207)(98,206,112,220) );
G=PermutationGroup([[(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,81),(8,82),(9,83),(10,84),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,67),(22,68),(23,69),(24,70),(25,71),(26,72),(27,73),(28,74),(29,94),(30,95),(31,96),(32,97),(33,98),(34,99),(35,100),(36,101),(37,102),(38,103),(39,104),(40,105),(41,106),(42,107),(43,108),(44,109),(45,110),(46,111),(47,112),(48,85),(49,86),(50,87),(51,88),(52,89),(53,90),(54,91),(55,92),(56,93),(113,170),(114,171),(115,172),(116,173),(117,174),(118,175),(119,176),(120,177),(121,178),(122,179),(123,180),(124,181),(125,182),(126,183),(127,184),(128,185),(129,186),(130,187),(131,188),(132,189),(133,190),(134,191),(135,192),(136,193),(137,194),(138,195),(139,196),(140,169),(141,219),(142,220),(143,221),(144,222),(145,223),(146,224),(147,197),(148,198),(149,199),(150,200),(151,201),(152,202),(153,203),(154,204),(155,205),(156,206),(157,207),(158,208),(159,209),(160,210),(161,211),(162,212),(163,213),(164,214),(165,215),(166,216),(167,217),(168,218)], [(1,8,15,22),(2,9,16,23),(3,10,17,24),(4,11,18,25),(5,12,19,26),(6,13,20,27),(7,14,21,28),(29,50,43,36),(30,51,44,37),(31,52,45,38),(32,53,46,39),(33,54,47,40),(34,55,48,41),(35,56,49,42),(57,64,71,78),(58,65,72,79),(59,66,73,80),(60,67,74,81),(61,68,75,82),(62,69,76,83),(63,70,77,84),(85,106,99,92),(86,107,100,93),(87,108,101,94),(88,109,102,95),(89,110,103,96),(90,111,104,97),(91,112,105,98),(113,134,127,120),(114,135,128,121),(115,136,129,122),(116,137,130,123),(117,138,131,124),(118,139,132,125),(119,140,133,126),(141,148,155,162),(142,149,156,163),(143,150,157,164),(144,151,158,165),(145,152,159,166),(146,153,160,167),(147,154,161,168),(169,190,183,176),(170,191,184,177),(171,192,185,178),(172,193,186,179),(173,194,187,180),(174,195,188,181),(175,196,189,182),(197,204,211,218),(198,205,212,219),(199,206,213,220),(200,207,214,221),(201,208,215,222),(202,209,216,223),(203,210,217,224)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,49),(16,50),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,29),(24,30),(25,31),(26,32),(27,33),(28,34),(57,110),(58,111),(59,112),(60,85),(61,86),(62,87),(63,88),(64,89),(65,90),(66,91),(67,92),(68,93),(69,94),(70,95),(71,96),(72,97),(73,98),(74,99),(75,100),(76,101),(77,102),(78,103),(79,104),(80,105),(81,106),(82,107),(83,108),(84,109),(113,162),(114,163),(115,164),(116,165),(117,166),(118,167),(119,168),(120,141),(121,142),(122,143),(123,144),(124,145),(125,146),(126,147),(127,148),(128,149),(129,150),(130,151),(131,152),(132,153),(133,154),(134,155),(135,156),(136,157),(137,158),(138,159),(139,160),(140,161),(169,211),(170,212),(171,213),(172,214),(173,215),(174,216),(175,217),(176,218),(177,219),(178,220),(179,221),(180,222),(181,223),(182,224),(183,197),(184,198),(185,199),(186,200),(187,201),(188,202),(189,203),(190,204),(191,205),(192,206),(193,207),(194,208),(195,209),(196,210)], [(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)], [(1,140,15,126),(2,125,16,139),(3,138,17,124),(4,123,18,137),(5,136,19,122),(6,121,20,135),(7,134,21,120),(8,119,22,133),(9,132,23,118),(10,117,24,131),(11,130,25,116),(12,115,26,129),(13,128,27,114),(14,113,28,127),(29,160,43,146),(30,145,44,159),(31,158,45,144),(32,143,46,157),(33,156,47,142),(34,141,48,155),(35,154,49,168),(36,167,50,153),(37,152,51,166),(38,165,52,151),(39,150,53,164),(40,163,54,149),(41,148,55,162),(42,161,56,147),(57,187,71,173),(58,172,72,186),(59,185,73,171),(60,170,74,184),(61,183,75,169),(62,196,76,182),(63,181,77,195),(64,194,78,180),(65,179,79,193),(66,192,80,178),(67,177,81,191),(68,190,82,176),(69,175,83,189),(70,188,84,174),(85,205,99,219),(86,218,100,204),(87,203,101,217),(88,216,102,202),(89,201,103,215),(90,214,104,200),(91,199,105,213),(92,212,106,198),(93,197,107,211),(94,210,108,224),(95,223,109,209),(96,208,110,222),(97,221,111,207),(98,206,112,220)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 28 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | C4○D8 | C7⋊D4 | C7⋊D4 | D4.8D14 |
| kernel | C2×D4.8D14 | C22×C7⋊C8 | C2×D4⋊D7 | C2×D4.D7 | C2×Q8⋊D7 | C2×C7⋊Q16 | D4.8D14 | C2×C4○D28 | C14×C4○D4 | C2×C28 | C22×C14 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C14 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 3 | 1 | 3 | 3 | 3 | 3 | 12 | 8 | 18 | 6 | 12 |
Matrix representation of C2×D4.8D14 ►in GL5(𝔽113)
| 112 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 15 | 53 |
| 0 | 0 | 0 | 0 | 98 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 84 | 101 | 0 | 0 |
| 0 | 70 | 29 | 0 | 0 |
| 0 | 0 | 0 | 83 | 75 |
| 0 | 0 | 0 | 98 | 30 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 1 | 88 | 0 | 0 |
| 0 | 14 | 103 | 0 | 0 |
| 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 0 | 98 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 46 | 54 | 0 | 0 |
| 0 | 55 | 67 | 0 | 0 |
| 0 | 0 | 0 | 88 | 32 |
| 0 | 0 | 0 | 44 | 25 |
G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,112,0,0,0,0,0,112],[1,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,15,0,0,0,0,53,98],[1,0,0,0,0,0,84,70,0,0,0,101,29,0,0,0,0,0,83,98,0,0,0,75,30],[112,0,0,0,0,0,1,14,0,0,0,88,103,0,0,0,0,0,98,0,0,0,0,0,98],[112,0,0,0,0,0,46,55,0,0,0,54,67,0,0,0,0,0,88,44,0,0,0,32,25] >;
C2×D4.8D14 in GAP, Magma, Sage, TeX
C_2\times D_4._8D_{14} % in TeX
G:=Group("C2xD4.8D14"); // GroupNames label
G:=SmallGroup(448,1274);
// by ID
G=gap.SmallGroup(448,1274);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,675,1684,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^14=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^13>;
// generators/relations