G = C14.492+ 1+4 order 448 = 26·7
49th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.492+ 1+4, C14.742- 1+4, C4⋊D4⋊24D7, C4⋊C4.93D14, C28⋊2D4⋊29C2, (C2×D4).96D14, D14⋊Q8⋊16C2, Dic7.Q8⋊14C2, (C2×C28).47C23, C22⋊C4.12D14, Dic7⋊D4⋊19C2, D14.D4⋊23C2, C28.17D4⋊20C2, C28.48D4⋊45C2, (C2×C14).165C24, D14⋊C4.18C22, (C22×C4).231D14, C4⋊Dic7.46C22, C2.51(D4⋊6D14), C22⋊Dic14⋊21C2, Dic7.D4⋊23C2, (D4×C14).129C22, Dic7⋊C4.79C22, (C2×Dic7).82C23, (C22×D7).72C23, C22.186(C23×D7), C23.115(C22×D7), C23.D7.29C22, C23.18D14⋊23C2, C23.23D14⋊13C2, (C22×C28).313C22, (C22×C14).193C23, C7⋊1(C22.56C24), (C2×Dic14).34C22, (C4×Dic7).100C22, C2.32(D4.10D14), (C22×Dic7).116C22, (C7×C4⋊D4)⋊27C2, (C2×C4×D7).90C22, (C2×C4).43(C22×D7), (C7×C4⋊C4).151C22, (C2×C7⋊D4).36C22, (C7×C22⋊C4).20C22, SmallGroup(448,1074)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.492+ 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=e2=b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, bd=db, ebe-1=a7b, dcd-1=a7c, ce=ec, ede-1=a7b2d >
Subgroups: 1004 in 220 conjugacy classes, 91 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic7, C28, D14, C2×C14, C2×C14, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C22.56C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C22⋊Dic14, D14.D4, Dic7.D4, Dic7.Q8, D14⋊Q8, C28.48D4, C23.23D14, C23.18D14, C28.17D4, C28⋊2D4, Dic7⋊D4, C7×C4⋊D4, C14.492+ 1+4
Quotients: C1, C2, C22, C23, D7, C24, D14, 2+ 1+4, 2- 1+4, C22×D7, C22.56C24, C23×D7, D4⋊6D14, D4.10D14, C14.492+ 1+4
►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)
(1 184 53 32)(2 183 54 31)(3 196 55 30)(4 195 56 29)(5 194 43 42)(6 193 44 41)(7 192 45 40)(8 191 46 39)(9 190 47 38)(10 189 48 37)(11 188 49 36)(12 187 50 35)(13 186 51 34)(14 185 52 33)(15 108 165 146)(16 107 166 145)(17 106 167 144)(18 105 168 143)(19 104 155 142)(20 103 156 141)(21 102 157 154)(22 101 158 153)(23 100 159 152)(24 99 160 151)(25 112 161 150)(26 111 162 149)(27 110 163 148)(28 109 164 147)(57 177 127 77)(58 176 128 76)(59 175 129 75)(60 174 130 74)(61 173 131 73)(62 172 132 72)(63 171 133 71)(64 170 134 84)(65 169 135 83)(66 182 136 82)(67 181 137 81)(68 180 138 80)(69 179 139 79)(70 178 140 78)(85 202 113 216)(86 201 114 215)(87 200 115 214)(88 199 116 213)(89 198 117 212)(90 197 118 211)(91 210 119 224)(92 209 120 223)(93 208 121 222)(94 207 122 221)(95 206 123 220)(96 205 124 219)(97 204 125 218)(98 203 126 217)
(1 130)(2 131)(3 132)(4 133)(5 134)(6 135)(7 136)(8 137)(9 138)(10 139)(11 140)(12 127)(13 128)(14 129)(15 220)(16 221)(17 222)(18 223)(19 224)(20 211)(21 212)(22 213)(23 214)(24 215)(25 216)(26 217)(27 218)(28 219)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)(85 150)(86 151)(87 152)(88 153)(89 154)(90 141)(91 142)(92 143)(93 144)(94 145)(95 146)(96 147)(97 148)(98 149)(99 114)(100 115)(101 116)(102 117)(103 118)(104 119)(105 120)(106 121)(107 122)(108 123)(109 124)(110 125)(111 126)(112 113)(155 210)(156 197)(157 198)(158 199)(159 200)(160 201)(161 202)(162 203)(163 204)(164 205)(165 206)(166 207)(167 208)(168 209)(169 193)(170 194)(171 195)(172 196)(173 183)(174 184)(175 185)(176 186)(177 187)(178 188)(179 189)(180 190)(181 191)(182 192)
(1 184 53 32)(2 183 54 31)(3 196 55 30)(4 195 56 29)(5 194 43 42)(6 193 44 41)(7 192 45 40)(8 191 46 39)(9 190 47 38)(10 189 48 37)(11 188 49 36)(12 187 50 35)(13 186 51 34)(14 185 52 33)(15 101 165 153)(16 100 166 152)(17 99 167 151)(18 112 168 150)(19 111 155 149)(20 110 156 148)(21 109 157 147)(22 108 158 146)(23 107 159 145)(24 106 160 144)(25 105 161 143)(26 104 162 142)(27 103 163 141)(28 102 164 154)(57 84 127 170)(58 83 128 169)(59 82 129 182)(60 81 130 181)(61 80 131 180)(62 79 132 179)(63 78 133 178)(64 77 134 177)(65 76 135 176)(66 75 136 175)(67 74 137 174)(68 73 138 173)(69 72 139 172)(70 71 140 171)(85 216 113 202)(86 215 114 201)(87 214 115 200)(88 213 116 199)(89 212 117 198)(90 211 118 197)(91 224 119 210)(92 223 120 209)(93 222 121 208)(94 221 122 207)(95 220 123 206)(96 219 124 205)(97 218 125 204)(98 217 126 203)
(1 124 53 96)(2 125 54 97)(3 126 55 98)(4 113 56 85)(5 114 43 86)(6 115 44 87)(7 116 45 88)(8 117 46 89)(9 118 47 90)(10 119 48 91)(11 120 49 92)(12 121 50 93)(13 122 51 94)(14 123 52 95)(15 82 165 182)(16 83 166 169)(17 84 167 170)(18 71 168 171)(19 72 155 172)(20 73 156 173)(21 74 157 174)(22 75 158 175)(23 76 159 176)(24 77 160 177)(25 78 161 178)(26 79 162 179)(27 80 163 180)(28 81 164 181)(29 209 195 223)(30 210 196 224)(31 197 183 211)(32 198 184 212)(33 199 185 213)(34 200 186 214)(35 201 187 215)(36 202 188 216)(37 203 189 217)(38 204 190 218)(39 205 191 219)(40 206 192 220)(41 207 193 221)(42 208 194 222)(57 144 127 106)(58 145 128 107)(59 146 129 108)(60 147 130 109)(61 148 131 110)(62 149 132 111)(63 150 133 112)(64 151 134 99)(65 152 135 100)(66 153 136 101)(67 154 137 102)(68 141 138 103)(69 142 139 104)(70 143 140 105)
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), (1,184,53,32)(2,183,54,31)(3,196,55,30)(4,195,56,29)(5,194,43,42)(6,193,44,41)(7,192,45,40)(8,191,46,39)(9,190,47,38)(10,189,48,37)(11,188,49,36)(12,187,50,35)(13,186,51,34)(14,185,52,33)(15,108,165,146)(16,107,166,145)(17,106,167,144)(18,105,168,143)(19,104,155,142)(20,103,156,141)(21,102,157,154)(22,101,158,153)(23,100,159,152)(24,99,160,151)(25,112,161,150)(26,111,162,149)(27,110,163,148)(28,109,164,147)(57,177,127,77)(58,176,128,76)(59,175,129,75)(60,174,130,74)(61,173,131,73)(62,172,132,72)(63,171,133,71)(64,170,134,84)(65,169,135,83)(66,182,136,82)(67,181,137,81)(68,180,138,80)(69,179,139,79)(70,178,140,78)(85,202,113,216)(86,201,114,215)(87,200,115,214)(88,199,116,213)(89,198,117,212)(90,197,118,211)(91,210,119,224)(92,209,120,223)(93,208,121,222)(94,207,122,221)(95,206,123,220)(96,205,124,219)(97,204,125,218)(98,203,126,217), (1,130)(2,131)(3,132)(4,133)(5,134)(6,135)(7,136)(8,137)(9,138)(10,139)(11,140)(12,127)(13,128)(14,129)(15,220)(16,221)(17,222)(18,223)(19,224)(20,211)(21,212)(22,213)(23,214)(24,215)(25,216)(26,217)(27,218)(28,219)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(85,150)(86,151)(87,152)(88,153)(89,154)(90,141)(91,142)(92,143)(93,144)(94,145)(95,146)(96,147)(97,148)(98,149)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(106,121)(107,122)(108,123)(109,124)(110,125)(111,126)(112,113)(155,210)(156,197)(157,198)(158,199)(159,200)(160,201)(161,202)(162,203)(163,204)(164,205)(165,206)(166,207)(167,208)(168,209)(169,193)(170,194)(171,195)(172,196)(173,183)(174,184)(175,185)(176,186)(177,187)(178,188)(179,189)(180,190)(181,191)(182,192), (1,184,53,32)(2,183,54,31)(3,196,55,30)(4,195,56,29)(5,194,43,42)(6,193,44,41)(7,192,45,40)(8,191,46,39)(9,190,47,38)(10,189,48,37)(11,188,49,36)(12,187,50,35)(13,186,51,34)(14,185,52,33)(15,101,165,153)(16,100,166,152)(17,99,167,151)(18,112,168,150)(19,111,155,149)(20,110,156,148)(21,109,157,147)(22,108,158,146)(23,107,159,145)(24,106,160,144)(25,105,161,143)(26,104,162,142)(27,103,163,141)(28,102,164,154)(57,84,127,170)(58,83,128,169)(59,82,129,182)(60,81,130,181)(61,80,131,180)(62,79,132,179)(63,78,133,178)(64,77,134,177)(65,76,135,176)(66,75,136,175)(67,74,137,174)(68,73,138,173)(69,72,139,172)(70,71,140,171)(85,216,113,202)(86,215,114,201)(87,214,115,200)(88,213,116,199)(89,212,117,198)(90,211,118,197)(91,224,119,210)(92,223,120,209)(93,222,121,208)(94,221,122,207)(95,220,123,206)(96,219,124,205)(97,218,125,204)(98,217,126,203), (1,124,53,96)(2,125,54,97)(3,126,55,98)(4,113,56,85)(5,114,43,86)(6,115,44,87)(7,116,45,88)(8,117,46,89)(9,118,47,90)(10,119,48,91)(11,120,49,92)(12,121,50,93)(13,122,51,94)(14,123,52,95)(15,82,165,182)(16,83,166,169)(17,84,167,170)(18,71,168,171)(19,72,155,172)(20,73,156,173)(21,74,157,174)(22,75,158,175)(23,76,159,176)(24,77,160,177)(25,78,161,178)(26,79,162,179)(27,80,163,180)(28,81,164,181)(29,209,195,223)(30,210,196,224)(31,197,183,211)(32,198,184,212)(33,199,185,213)(34,200,186,214)(35,201,187,215)(36,202,188,216)(37,203,189,217)(38,204,190,218)(39,205,191,219)(40,206,192,220)(41,207,193,221)(42,208,194,222)(57,144,127,106)(58,145,128,107)(59,146,129,108)(60,147,130,109)(61,148,131,110)(62,149,132,111)(63,150,133,112)(64,151,134,99)(65,152,135,100)(66,153,136,101)(67,154,137,102)(68,141,138,103)(69,142,139,104)(70,143,140,105)>;
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), (1,184,53,32)(2,183,54,31)(3,196,55,30)(4,195,56,29)(5,194,43,42)(6,193,44,41)(7,192,45,40)(8,191,46,39)(9,190,47,38)(10,189,48,37)(11,188,49,36)(12,187,50,35)(13,186,51,34)(14,185,52,33)(15,108,165,146)(16,107,166,145)(17,106,167,144)(18,105,168,143)(19,104,155,142)(20,103,156,141)(21,102,157,154)(22,101,158,153)(23,100,159,152)(24,99,160,151)(25,112,161,150)(26,111,162,149)(27,110,163,148)(28,109,164,147)(57,177,127,77)(58,176,128,76)(59,175,129,75)(60,174,130,74)(61,173,131,73)(62,172,132,72)(63,171,133,71)(64,170,134,84)(65,169,135,83)(66,182,136,82)(67,181,137,81)(68,180,138,80)(69,179,139,79)(70,178,140,78)(85,202,113,216)(86,201,114,215)(87,200,115,214)(88,199,116,213)(89,198,117,212)(90,197,118,211)(91,210,119,224)(92,209,120,223)(93,208,121,222)(94,207,122,221)(95,206,123,220)(96,205,124,219)(97,204,125,218)(98,203,126,217), (1,130)(2,131)(3,132)(4,133)(5,134)(6,135)(7,136)(8,137)(9,138)(10,139)(11,140)(12,127)(13,128)(14,129)(15,220)(16,221)(17,222)(18,223)(19,224)(20,211)(21,212)(22,213)(23,214)(24,215)(25,216)(26,217)(27,218)(28,219)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)(85,150)(86,151)(87,152)(88,153)(89,154)(90,141)(91,142)(92,143)(93,144)(94,145)(95,146)(96,147)(97,148)(98,149)(99,114)(100,115)(101,116)(102,117)(103,118)(104,119)(105,120)(106,121)(107,122)(108,123)(109,124)(110,125)(111,126)(112,113)(155,210)(156,197)(157,198)(158,199)(159,200)(160,201)(161,202)(162,203)(163,204)(164,205)(165,206)(166,207)(167,208)(168,209)(169,193)(170,194)(171,195)(172,196)(173,183)(174,184)(175,185)(176,186)(177,187)(178,188)(179,189)(180,190)(181,191)(182,192), (1,184,53,32)(2,183,54,31)(3,196,55,30)(4,195,56,29)(5,194,43,42)(6,193,44,41)(7,192,45,40)(8,191,46,39)(9,190,47,38)(10,189,48,37)(11,188,49,36)(12,187,50,35)(13,186,51,34)(14,185,52,33)(15,101,165,153)(16,100,166,152)(17,99,167,151)(18,112,168,150)(19,111,155,149)(20,110,156,148)(21,109,157,147)(22,108,158,146)(23,107,159,145)(24,106,160,144)(25,105,161,143)(26,104,162,142)(27,103,163,141)(28,102,164,154)(57,84,127,170)(58,83,128,169)(59,82,129,182)(60,81,130,181)(61,80,131,180)(62,79,132,179)(63,78,133,178)(64,77,134,177)(65,76,135,176)(66,75,136,175)(67,74,137,174)(68,73,138,173)(69,72,139,172)(70,71,140,171)(85,216,113,202)(86,215,114,201)(87,214,115,200)(88,213,116,199)(89,212,117,198)(90,211,118,197)(91,224,119,210)(92,223,120,209)(93,222,121,208)(94,221,122,207)(95,220,123,206)(96,219,124,205)(97,218,125,204)(98,217,126,203), (1,124,53,96)(2,125,54,97)(3,126,55,98)(4,113,56,85)(5,114,43,86)(6,115,44,87)(7,116,45,88)(8,117,46,89)(9,118,47,90)(10,119,48,91)(11,120,49,92)(12,121,50,93)(13,122,51,94)(14,123,52,95)(15,82,165,182)(16,83,166,169)(17,84,167,170)(18,71,168,171)(19,72,155,172)(20,73,156,173)(21,74,157,174)(22,75,158,175)(23,76,159,176)(24,77,160,177)(25,78,161,178)(26,79,162,179)(27,80,163,180)(28,81,164,181)(29,209,195,223)(30,210,196,224)(31,197,183,211)(32,198,184,212)(33,199,185,213)(34,200,186,214)(35,201,187,215)(36,202,188,216)(37,203,189,217)(38,204,190,218)(39,205,191,219)(40,206,192,220)(41,207,193,221)(42,208,194,222)(57,144,127,106)(58,145,128,107)(59,146,129,108)(60,147,130,109)(61,148,131,110)(62,149,132,111)(63,150,133,112)(64,151,134,99)(65,152,135,100)(66,153,136,101)(67,154,137,102)(68,141,138,103)(69,142,139,104)(70,143,140,105) );
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)], [(1,184,53,32),(2,183,54,31),(3,196,55,30),(4,195,56,29),(5,194,43,42),(6,193,44,41),(7,192,45,40),(8,191,46,39),(9,190,47,38),(10,189,48,37),(11,188,49,36),(12,187,50,35),(13,186,51,34),(14,185,52,33),(15,108,165,146),(16,107,166,145),(17,106,167,144),(18,105,168,143),(19,104,155,142),(20,103,156,141),(21,102,157,154),(22,101,158,153),(23,100,159,152),(24,99,160,151),(25,112,161,150),(26,111,162,149),(27,110,163,148),(28,109,164,147),(57,177,127,77),(58,176,128,76),(59,175,129,75),(60,174,130,74),(61,173,131,73),(62,172,132,72),(63,171,133,71),(64,170,134,84),(65,169,135,83),(66,182,136,82),(67,181,137,81),(68,180,138,80),(69,179,139,79),(70,178,140,78),(85,202,113,216),(86,201,114,215),(87,200,115,214),(88,199,116,213),(89,198,117,212),(90,197,118,211),(91,210,119,224),(92,209,120,223),(93,208,121,222),(94,207,122,221),(95,206,123,220),(96,205,124,219),(97,204,125,218),(98,203,126,217)], [(1,130),(2,131),(3,132),(4,133),(5,134),(6,135),(7,136),(8,137),(9,138),(10,139),(11,140),(12,127),(13,128),(14,129),(15,220),(16,221),(17,222),(18,223),(19,224),(20,211),(21,212),(22,213),(23,214),(24,215),(25,216),(26,217),(27,218),(28,219),(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,64),(44,65),(45,66),(46,67),(47,68),(48,69),(49,70),(50,57),(51,58),(52,59),(53,60),(54,61),(55,62),(56,63),(85,150),(86,151),(87,152),(88,153),(89,154),(90,141),(91,142),(92,143),(93,144),(94,145),(95,146),(96,147),(97,148),(98,149),(99,114),(100,115),(101,116),(102,117),(103,118),(104,119),(105,120),(106,121),(107,122),(108,123),(109,124),(110,125),(111,126),(112,113),(155,210),(156,197),(157,198),(158,199),(159,200),(160,201),(161,202),(162,203),(163,204),(164,205),(165,206),(166,207),(167,208),(168,209),(169,193),(170,194),(171,195),(172,196),(173,183),(174,184),(175,185),(176,186),(177,187),(178,188),(179,189),(180,190),(181,191),(182,192)], [(1,184,53,32),(2,183,54,31),(3,196,55,30),(4,195,56,29),(5,194,43,42),(6,193,44,41),(7,192,45,40),(8,191,46,39),(9,190,47,38),(10,189,48,37),(11,188,49,36),(12,187,50,35),(13,186,51,34),(14,185,52,33),(15,101,165,153),(16,100,166,152),(17,99,167,151),(18,112,168,150),(19,111,155,149),(20,110,156,148),(21,109,157,147),(22,108,158,146),(23,107,159,145),(24,106,160,144),(25,105,161,143),(26,104,162,142),(27,103,163,141),(28,102,164,154),(57,84,127,170),(58,83,128,169),(59,82,129,182),(60,81,130,181),(61,80,131,180),(62,79,132,179),(63,78,133,178),(64,77,134,177),(65,76,135,176),(66,75,136,175),(67,74,137,174),(68,73,138,173),(69,72,139,172),(70,71,140,171),(85,216,113,202),(86,215,114,201),(87,214,115,200),(88,213,116,199),(89,212,117,198),(90,211,118,197),(91,224,119,210),(92,223,120,209),(93,222,121,208),(94,221,122,207),(95,220,123,206),(96,219,124,205),(97,218,125,204),(98,217,126,203)], [(1,124,53,96),(2,125,54,97),(3,126,55,98),(4,113,56,85),(5,114,43,86),(6,115,44,87),(7,116,45,88),(8,117,46,89),(9,118,47,90),(10,119,48,91),(11,120,49,92),(12,121,50,93),(13,122,51,94),(14,123,52,95),(15,82,165,182),(16,83,166,169),(17,84,167,170),(18,71,168,171),(19,72,155,172),(20,73,156,173),(21,74,157,174),(22,75,158,175),(23,76,159,176),(24,77,160,177),(25,78,161,178),(26,79,162,179),(27,80,163,180),(28,81,164,181),(29,209,195,223),(30,210,196,224),(31,197,183,211),(32,198,184,212),(33,199,185,213),(34,200,186,214),(35,201,187,215),(36,202,188,216),(37,203,189,217),(38,204,190,218),(39,205,191,219),(40,206,192,220),(41,207,193,221),(42,208,194,222),(57,144,127,106),(58,145,128,107),(59,146,129,108),(60,147,130,109),(61,148,131,110),(62,149,132,111),(63,150,133,112),(64,151,134,99),(65,152,135,100),(66,153,136,101),(67,154,137,102),(68,141,138,103),(69,142,139,104),(70,143,140,105)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4K | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 28 | 4 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | D14 | 2+ 1+4 | 2- 1+4 | D4⋊6D14 | D4.10D14 |
| kernel | C14.492+ 1+4 | C22⋊Dic14 | D14.D4 | Dic7.D4 | Dic7.Q8 | D14⋊Q8 | C28.48D4 | C23.23D14 | C23.18D14 | C28.17D4 | C28⋊2D4 | Dic7⋊D4 | C7×C4⋊D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 3 | 6 | 3 | 3 | 9 | 2 | 1 | 12 | 6 |
Matrix representation of C14.492+ 1+4 ►in GL8(𝔽29)
| 19 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 14 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 25 |
| 19 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 | 1 | 4 |
| 0 | 0 | 0 | 0 | 17 | 0 | 8 | 27 |
| 0 | 0 | 0 | 0 | 0 | 0 | 19 | 23 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 10 |
| 1 | 0 | 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 21 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 18 |
| 19 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 19 | 19 | 10 | 10 | 0 | 0 | 0 | 0 |
| 7 | 10 | 22 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 | 3 | 17 |
| 0 | 0 | 0 | 0 | 17 | 0 | 0 | 26 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 19 |
| 20 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 1 | 27 | 8 |
| 0 | 0 | 0 | 0 | 28 | 12 | 14 | 0 |
| 0 | 0 | 0 | 0 | 25 | 27 | 21 | 13 |
| 0 | 0 | 0 | 0 | 1 | 21 | 1 | 6 |
G:=sub<GL(8,GF(29))| [19,10,0,0,0,0,0,0,19,7,0,0,0,0,0,0,0,0,19,10,0,0,0,0,0,0,19,7,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,19,19,0,0,0,0,0,0,0,0,1,10,0,0,0,0,0,0,14,25],[19,7,0,0,0,0,0,0,19,10,0,0,0,0,0,0,0,0,19,7,0,0,0,0,0,0,19,10,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,0,1,8,19,12,0,0,0,0,4,27,23,10],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,27,0,28,0,0,0,0,0,0,27,0,28,0,0,0,0,0,0,0,0,20,15,0,0,0,0,0,0,14,9,0,0,0,0,0,0,0,0,11,15,0,0,0,0,0,0,21,18],[19,7,19,7,0,0,0,0,19,10,19,10,0,0,0,0,0,0,10,22,0,0,0,0,0,0,10,19,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,0,3,0,10,17,0,0,0,0,17,26,6,19],[20,14,0,0,0,0,0,0,15,9,0,0,0,0,0,0,0,0,20,14,0,0,0,0,0,0,15,9,0,0,0,0,0,0,0,0,19,28,25,1,0,0,0,0,1,12,27,21,0,0,0,0,27,14,21,1,0,0,0,0,8,0,13,6] >;
C14.492+ 1+4 in GAP, Magma, Sage, TeX
C_{14}._{49}2_+^{1+4} % in TeX
G:=Group("C14.49ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1074);
// by ID
G=gap.SmallGroup(448,1074);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,1571,570,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=e^2=b^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a^7*b,d*c*d^-1=a^7*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations