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, (C2×C28).502D4, C28.263(C2×D4), Q8⋊D7⋊21C22, C4.34(C23×D7), (C2×D4).232D14, C4○D28⋊20C22, D4.D7⋊20C22, (C2×Q8).190D14, D4.22(C22×D7), C7⋊Q16⋊20C22, (C7×D4).22C23, Q8.22(C22×D7), (C7×Q8).22C23, (C2×C28).556C23, (C22×C14).123D4, C14.159(C22×D4), (C22×C4).387D14, 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, (C2×C7⋊C8)⋊42C22, (C22×C7⋊C8)⋊15C2, (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
Subgroups: 1044 in 266 conjugacy classes, 111 normal (35 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], C7, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23, C23 [×2], D7 [×2], C14, C14 [×2], C14 [×4], C2×C8 [×6], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, C4○D4 [×4], C4○D4 [×8], Dic7 [×2], C28 [×2], C28 [×2], C28 [×2], D14 [×4], C2×C14, C2×C14 [×2], C2×C14 [×6], C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4, C2×C4○D4, C7⋊C8 [×4], Dic14 [×2], Dic14, C4×D7 [×4], D28 [×2], D28, C2×Dic7, C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×4], C2×C28 [×5], C7×D4 [×2], C7×D4 [×5], C7×Q8 [×2], C7×Q8, C22×D7, C22×C14, C22×C14, C2×C4○D8, C2×C7⋊C8 [×2], C2×C7⋊C8 [×4], D4⋊D7 [×4], D4.D7 [×4], Q8⋊D7 [×4], C7⋊Q16 [×4], C2×Dic14, C2×C4×D7, C2×D28, C4○D28 [×4], C4○D28 [×2], C2×C7⋊D4, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4 [×4], C7×C4○D4 [×2], C22×C7⋊C8, C2×D4⋊D7, C2×D4.D7, C2×Q8⋊D7, C2×C7⋊Q16, D4.8D14 [×8], C2×C4○D28, C14×C4○D4, C2×D4.8D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C4○D8 [×2], C22×D4, C7⋊D4 [×4], C22×D7 [×7], C2×C4○D8, C2×C7⋊D4 [×6], C23×D7, D4.8D14 [×2], C22×C7⋊D4, C2×D4.8D14
Generators and relations
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 >
►On 224 points
Generators in S224
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 57)(26 58)(27 59)(28 60)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(113 174)(114 175)(115 176)(116 177)(117 178)(118 179)(119 180)(120 181)(121 182)(122 183)(123 184)(124 185)(125 186)(126 187)(127 188)(128 189)(129 190)(130 191)(131 192)(132 193)(133 194)(134 195)(135 196)(136 169)(137 170)(138 171)(139 172)(140 173)(141 212)(142 213)(143 214)(144 215)(145 216)(146 217)(147 218)(148 219)(149 220)(150 221)(151 222)(152 223)(153 224)(154 197)(155 198)(156 199)(157 200)(158 201)(159 202)(160 203)(161 204)(162 205)(163 206)(164 207)(165 208)(166 209)(167 210)(168 211)
(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 53)(2 54)(3 55)(4 56)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(57 105)(58 106)(59 107)(60 108)(61 109)(62 110)(63 111)(64 112)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 91)(72 92)(73 93)(74 94)(75 95)(76 96)(77 97)(78 98)(79 99)(80 100)(81 101)(82 102)(83 103)(84 104)(113 149)(114 150)(115 151)(116 152)(117 153)(118 154)(119 155)(120 156)(121 157)(122 158)(123 159)(124 160)(125 161)(126 162)(127 163)(128 164)(129 165)(130 166)(131 167)(132 168)(133 141)(134 142)(135 143)(136 144)(137 145)(138 146)(139 147)(140 148)(169 215)(170 216)(171 217)(172 218)(173 219)(174 220)(175 221)(176 222)(177 223)(178 224)(179 197)(180 198)(181 199)(182 200)(183 201)(184 202)(185 203)(186 204)(187 205)(188 206)(189 207)(190 208)(191 209)(192 210)(193 211)(194 212)(195 213)(196 214)
(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)
(1 137 15 123)(2 122 16 136)(3 135 17 121)(4 120 18 134)(5 133 19 119)(6 118 20 132)(7 131 21 117)(8 116 22 130)(9 129 23 115)(10 114 24 128)(11 127 25 113)(12 140 26 126)(13 125 27 139)(14 138 28 124)(29 162 43 148)(30 147 44 161)(31 160 45 146)(32 145 46 159)(33 158 47 144)(34 143 48 157)(35 156 49 142)(36 141 50 155)(37 154 51 168)(38 167 52 153)(39 152 53 166)(40 165 54 151)(41 150 55 164)(42 163 56 149)(57 174 71 188)(58 187 72 173)(59 172 73 186)(60 185 74 171)(61 170 75 184)(62 183 76 169)(63 196 77 182)(64 181 78 195)(65 194 79 180)(66 179 80 193)(67 192 81 178)(68 177 82 191)(69 190 83 176)(70 175 84 189)(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)| (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,57)(26,58)(27,59)(28,60)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(113,174)(114,175)(115,176)(116,177)(117,178)(118,179)(119,180)(120,181)(121,182)(122,183)(123,184)(124,185)(125,186)(126,187)(127,188)(128,189)(129,190)(130,191)(131,192)(132,193)(133,194)(134,195)(135,196)(136,169)(137,170)(138,171)(139,172)(140,173)(141,212)(142,213)(143,214)(144,215)(145,216)(146,217)(147,218)(148,219)(149,220)(150,221)(151,222)(152,223)(153,224)(154,197)(155,198)(156,199)(157,200)(158,201)(159,202)(160,203)(161,204)(162,205)(163,206)(164,207)(165,208)(166,209)(167,210)(168,211), (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), (1,53)(2,54)(3,55)(4,56)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104)(113,149)(114,150)(115,151)(116,152)(117,153)(118,154)(119,155)(120,156)(121,157)(122,158)(123,159)(124,160)(125,161)(126,162)(127,163)(128,164)(129,165)(130,166)(131,167)(132,168)(133,141)(134,142)(135,143)(136,144)(137,145)(138,146)(139,147)(140,148)(169,215)(170,216)(171,217)(172,218)(173,219)(174,220)(175,221)(176,222)(177,223)(178,224)(179,197)(180,198)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)(187,205)(188,206)(189,207)(190,208)(191,209)(192,210)(193,211)(194,212)(195,213)(196,214), (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), (1,137,15,123)(2,122,16,136)(3,135,17,121)(4,120,18,134)(5,133,19,119)(6,118,20,132)(7,131,21,117)(8,116,22,130)(9,129,23,115)(10,114,24,128)(11,127,25,113)(12,140,26,126)(13,125,27,139)(14,138,28,124)(29,162,43,148)(30,147,44,161)(31,160,45,146)(32,145,46,159)(33,158,47,144)(34,143,48,157)(35,156,49,142)(36,141,50,155)(37,154,51,168)(38,167,52,153)(39,152,53,166)(40,165,54,151)(41,150,55,164)(42,163,56,149)(57,174,71,188)(58,187,72,173)(59,172,73,186)(60,185,74,171)(61,170,75,184)(62,183,76,169)(63,196,77,182)(64,181,78,195)(65,194,79,180)(66,179,80,193)(67,192,81,178)(68,177,82,191)(69,190,83,176)(70,175,84,189)(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( (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,57)(26,58)(27,59)(28,60)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(113,174)(114,175)(115,176)(116,177)(117,178)(118,179)(119,180)(120,181)(121,182)(122,183)(123,184)(124,185)(125,186)(126,187)(127,188)(128,189)(129,190)(130,191)(131,192)(132,193)(133,194)(134,195)(135,196)(136,169)(137,170)(138,171)(139,172)(140,173)(141,212)(142,213)(143,214)(144,215)(145,216)(146,217)(147,218)(148,219)(149,220)(150,221)(151,222)(152,223)(153,224)(154,197)(155,198)(156,199)(157,200)(158,201)(159,202)(160,203)(161,204)(162,205)(163,206)(164,207)(165,208)(166,209)(167,210)(168,211), (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), (1,53)(2,54)(3,55)(4,56)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,91)(72,92)(73,93)(74,94)(75,95)(76,96)(77,97)(78,98)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104)(113,149)(114,150)(115,151)(116,152)(117,153)(118,154)(119,155)(120,156)(121,157)(122,158)(123,159)(124,160)(125,161)(126,162)(127,163)(128,164)(129,165)(130,166)(131,167)(132,168)(133,141)(134,142)(135,143)(136,144)(137,145)(138,146)(139,147)(140,148)(169,215)(170,216)(171,217)(172,218)(173,219)(174,220)(175,221)(176,222)(177,223)(178,224)(179,197)(180,198)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)(187,205)(188,206)(189,207)(190,208)(191,209)(192,210)(193,211)(194,212)(195,213)(196,214), (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), (1,137,15,123)(2,122,16,136)(3,135,17,121)(4,120,18,134)(5,133,19,119)(6,118,20,132)(7,131,21,117)(8,116,22,130)(9,129,23,115)(10,114,24,128)(11,127,25,113)(12,140,26,126)(13,125,27,139)(14,138,28,124)(29,162,43,148)(30,147,44,161)(31,160,45,146)(32,145,46,159)(33,158,47,144)(34,143,48,157)(35,156,49,142)(36,141,50,155)(37,154,51,168)(38,167,52,153)(39,152,53,166)(40,165,54,151)(41,150,55,164)(42,163,56,149)(57,174,71,188)(58,187,72,173)(59,172,73,186)(60,185,74,171)(61,170,75,184)(62,183,76,169)(63,196,77,182)(64,181,78,195)(65,194,79,180)(66,179,80,193)(67,192,81,178)(68,177,82,191)(69,190,83,176)(70,175,84,189)(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([(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,57),(26,58),(27,59),(28,60),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(113,174),(114,175),(115,176),(116,177),(117,178),(118,179),(119,180),(120,181),(121,182),(122,183),(123,184),(124,185),(125,186),(126,187),(127,188),(128,189),(129,190),(130,191),(131,192),(132,193),(133,194),(134,195),(135,196),(136,169),(137,170),(138,171),(139,172),(140,173),(141,212),(142,213),(143,214),(144,215),(145,216),(146,217),(147,218),(148,219),(149,220),(150,221),(151,222),(152,223),(153,224),(154,197),(155,198),(156,199),(157,200),(158,201),(159,202),(160,203),(161,204),(162,205),(163,206),(164,207),(165,208),(166,209),(167,210),(168,211)], [(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)], [(1,53),(2,54),(3,55),(4,56),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,51),(28,52),(57,105),(58,106),(59,107),(60,108),(61,109),(62,110),(63,111),(64,112),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,91),(72,92),(73,93),(74,94),(75,95),(76,96),(77,97),(78,98),(79,99),(80,100),(81,101),(82,102),(83,103),(84,104),(113,149),(114,150),(115,151),(116,152),(117,153),(118,154),(119,155),(120,156),(121,157),(122,158),(123,159),(124,160),(125,161),(126,162),(127,163),(128,164),(129,165),(130,166),(131,167),(132,168),(133,141),(134,142),(135,143),(136,144),(137,145),(138,146),(139,147),(140,148),(169,215),(170,216),(171,217),(172,218),(173,219),(174,220),(175,221),(176,222),(177,223),(178,224),(179,197),(180,198),(181,199),(182,200),(183,201),(184,202),(185,203),(186,204),(187,205),(188,206),(189,207),(190,208),(191,209),(192,210),(193,211),(194,212),(195,213),(196,214)], [(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)], [(1,137,15,123),(2,122,16,136),(3,135,17,121),(4,120,18,134),(5,133,19,119),(6,118,20,132),(7,131,21,117),(8,116,22,130),(9,129,23,115),(10,114,24,128),(11,127,25,113),(12,140,26,126),(13,125,27,139),(14,138,28,124),(29,162,43,148),(30,147,44,161),(31,160,45,146),(32,145,46,159),(33,158,47,144),(34,143,48,157),(35,156,49,142),(36,141,50,155),(37,154,51,168),(38,167,52,153),(39,152,53,166),(40,165,54,151),(41,150,55,164),(42,163,56,149),(57,174,71,188),(58,187,72,173),(59,172,73,186),(60,185,74,171),(61,170,75,184),(62,183,76,169),(63,196,77,182),(64,181,78,195),(65,194,79,180),(66,179,80,193),(67,192,81,178),(68,177,82,191),(69,190,83,176),(70,175,84,189),(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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