G = C14.682- 1+4 order 448 = 26·7
23rd non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.682- 1+4, C14.322+ 1+4, C4⋊D4⋊5D7, C4⋊C4.89D14, (C2×Dic7)⋊9D4, C28⋊7D4⋊43C2, C22.2(D4×D7), (C2×D4).89D14, C22⋊C4.4D14, D14⋊D4⋊16C2, D14⋊Q8⋊13C2, Dic7⋊D4⋊8C2, Dic7.45(C2×D4), C14.60(C22×D4), C23.8(C22×D7), (C2×C28).172C23, (C2×C14).141C24, D14⋊C4.69C22, (C2×D28).30C22, (C22×C4).217D14, C4⋊Dic7.44C22, C2.34(D4⋊6D14), C22⋊Dic14⋊15C2, (D4×C14).115C22, (C22×C14).12C23, (C22×D7).60C23, C22.162(C23×D7), C23.D7.19C22, Dic7⋊C4.158C22, (C22×C28).310C22, C7⋊2(C22.31C24), (C2×Dic7).224C23, C2.26(D4.10D14), (C2×Dic14).151C22, (C22×Dic7).102C22, C2.33(C2×D4×D7), (C7×C4⋊D4)⋊6C2, (C2×C14).4(C2×D4), (C2×D4⋊2D7)⋊9C2, (C2×C4×D7).80C22, (C2×Dic7⋊C4)⋊28C2, (C2×C4).35(C22×D7), (C7×C4⋊C4).137C22, (C2×C7⋊D4).24C22, (C7×C22⋊C4).6C22, SmallGroup(448,1050)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.682- 1+4
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=b2, ab=ba, cac-1=eae=a-1, ad=da, cbc-1=a7b-1, dbd-1=a7b, be=eb, dcd-1=ece=a7c, ede=a7b2d >
Subgroups: 1420 in 294 conjugacy classes, 103 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D7, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C22×C14, C22.31C24, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, D4⋊2D7, C22×Dic7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C22⋊Dic14, D14⋊D4, D14⋊Q8, C2×Dic7⋊C4, C28⋊7D4, Dic7⋊D4, C7×C4⋊D4, C2×D4⋊2D7, C14.682- 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, 2- 1+4, C22×D7, C22.31C24, D4×D7, C23×D7, C2×D4×D7, D4⋊6D14, D4.10D14, C14.682- 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)
(1 178 65 37)(2 179 66 38)(3 180 67 39)(4 181 68 40)(5 182 69 41)(6 169 70 42)(7 170 57 29)(8 171 58 30)(9 172 59 31)(10 173 60 32)(11 174 61 33)(12 175 62 34)(13 176 63 35)(14 177 64 36)(15 146 76 123)(16 147 77 124)(17 148 78 125)(18 149 79 126)(19 150 80 113)(20 151 81 114)(21 152 82 115)(22 153 83 116)(23 154 84 117)(24 141 71 118)(25 142 72 119)(26 143 73 120)(27 144 74 121)(28 145 75 122)(43 167 136 198)(44 168 137 199)(45 155 138 200)(46 156 139 201)(47 157 140 202)(48 158 127 203)(49 159 128 204)(50 160 129 205)(51 161 130 206)(52 162 131 207)(53 163 132 208)(54 164 133 209)(55 165 134 210)(56 166 135 197)(85 106 192 213)(86 107 193 214)(87 108 194 215)(88 109 195 216)(89 110 196 217)(90 111 183 218)(91 112 184 219)(92 99 185 220)(93 100 186 221)(94 101 187 222)(95 102 188 223)(96 103 189 224)(97 104 190 211)(98 105 191 212)
(1 98 8 91)(2 97 9 90)(3 96 10 89)(4 95 11 88)(5 94 12 87)(6 93 13 86)(7 92 14 85)(15 200 22 207)(16 199 23 206)(17 198 24 205)(18 197 25 204)(19 210 26 203)(20 209 27 202)(21 208 28 201)(29 106 36 99)(30 105 37 112)(31 104 38 111)(32 103 39 110)(33 102 40 109)(34 101 41 108)(35 100 42 107)(43 125 50 118)(44 124 51 117)(45 123 52 116)(46 122 53 115)(47 121 54 114)(48 120 55 113)(49 119 56 126)(57 185 64 192)(58 184 65 191)(59 183 66 190)(60 196 67 189)(61 195 68 188)(62 194 69 187)(63 193 70 186)(71 160 78 167)(72 159 79 166)(73 158 80 165)(74 157 81 164)(75 156 82 163)(76 155 83 162)(77 168 84 161)(127 143 134 150)(128 142 135 149)(129 141 136 148)(130 154 137 147)(131 153 138 146)(132 152 139 145)(133 151 140 144)(169 214 176 221)(170 213 177 220)(171 212 178 219)(172 211 179 218)(173 224 180 217)(174 223 181 216)(175 222 182 215)
(1 142 65 119)(2 143 66 120)(3 144 67 121)(4 145 68 122)(5 146 69 123)(6 147 70 124)(7 148 57 125)(8 149 58 126)(9 150 59 113)(10 151 60 114)(11 152 61 115)(12 153 62 116)(13 154 63 117)(14 141 64 118)(15 175 76 34)(16 176 77 35)(17 177 78 36)(18 178 79 37)(19 179 80 38)(20 180 81 39)(21 181 82 40)(22 182 83 41)(23 169 84 42)(24 170 71 29)(25 171 72 30)(26 172 73 31)(27 173 74 32)(28 174 75 33)(43 92 136 185)(44 93 137 186)(45 94 138 187)(46 95 139 188)(47 96 140 189)(48 97 127 190)(49 98 128 191)(50 85 129 192)(51 86 130 193)(52 87 131 194)(53 88 132 195)(54 89 133 196)(55 90 134 183)(56 91 135 184)(99 205 220 160)(100 206 221 161)(101 207 222 162)(102 208 223 163)(103 209 224 164)(104 210 211 165)(105 197 212 166)(106 198 213 167)(107 199 214 168)(108 200 215 155)(109 201 216 156)(110 202 217 157)(111 203 218 158)(112 204 219 159)
(1 135)(2 134)(3 133)(4 132)(5 131)(6 130)(7 129)(8 128)(9 127)(10 140)(11 139)(12 138)(13 137)(14 136)(15 222)(16 221)(17 220)(18 219)(19 218)(20 217)(21 216)(22 215)(23 214)(24 213)(25 212)(26 211)(27 224)(28 223)(29 160)(30 159)(31 158)(32 157)(33 156)(34 155)(35 168)(36 167)(37 166)(38 165)(39 164)(40 163)(41 162)(42 161)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(71 106)(72 105)(73 104)(74 103)(75 102)(76 101)(77 100)(78 99)(79 112)(80 111)(81 110)(82 109)(83 108)(84 107)(85 141)(86 154)(87 153)(88 152)(89 151)(90 150)(91 149)(92 148)(93 147)(94 146)(95 145)(96 144)(97 143)(98 142)(113 183)(114 196)(115 195)(116 194)(117 193)(118 192)(119 191)(120 190)(121 189)(122 188)(123 187)(124 186)(125 185)(126 184)(169 206)(170 205)(171 204)(172 203)(173 202)(174 201)(175 200)(176 199)(177 198)(178 197)(179 210)(180 209)(181 208)(182 207)
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), (1,178,65,37)(2,179,66,38)(3,180,67,39)(4,181,68,40)(5,182,69,41)(6,169,70,42)(7,170,57,29)(8,171,58,30)(9,172,59,31)(10,173,60,32)(11,174,61,33)(12,175,62,34)(13,176,63,35)(14,177,64,36)(15,146,76,123)(16,147,77,124)(17,148,78,125)(18,149,79,126)(19,150,80,113)(20,151,81,114)(21,152,82,115)(22,153,83,116)(23,154,84,117)(24,141,71,118)(25,142,72,119)(26,143,73,120)(27,144,74,121)(28,145,75,122)(43,167,136,198)(44,168,137,199)(45,155,138,200)(46,156,139,201)(47,157,140,202)(48,158,127,203)(49,159,128,204)(50,160,129,205)(51,161,130,206)(52,162,131,207)(53,163,132,208)(54,164,133,209)(55,165,134,210)(56,166,135,197)(85,106,192,213)(86,107,193,214)(87,108,194,215)(88,109,195,216)(89,110,196,217)(90,111,183,218)(91,112,184,219)(92,99,185,220)(93,100,186,221)(94,101,187,222)(95,102,188,223)(96,103,189,224)(97,104,190,211)(98,105,191,212), (1,98,8,91)(2,97,9,90)(3,96,10,89)(4,95,11,88)(5,94,12,87)(6,93,13,86)(7,92,14,85)(15,200,22,207)(16,199,23,206)(17,198,24,205)(18,197,25,204)(19,210,26,203)(20,209,27,202)(21,208,28,201)(29,106,36,99)(30,105,37,112)(31,104,38,111)(32,103,39,110)(33,102,40,109)(34,101,41,108)(35,100,42,107)(43,125,50,118)(44,124,51,117)(45,123,52,116)(46,122,53,115)(47,121,54,114)(48,120,55,113)(49,119,56,126)(57,185,64,192)(58,184,65,191)(59,183,66,190)(60,196,67,189)(61,195,68,188)(62,194,69,187)(63,193,70,186)(71,160,78,167)(72,159,79,166)(73,158,80,165)(74,157,81,164)(75,156,82,163)(76,155,83,162)(77,168,84,161)(127,143,134,150)(128,142,135,149)(129,141,136,148)(130,154,137,147)(131,153,138,146)(132,152,139,145)(133,151,140,144)(169,214,176,221)(170,213,177,220)(171,212,178,219)(172,211,179,218)(173,224,180,217)(174,223,181,216)(175,222,182,215), (1,142,65,119)(2,143,66,120)(3,144,67,121)(4,145,68,122)(5,146,69,123)(6,147,70,124)(7,148,57,125)(8,149,58,126)(9,150,59,113)(10,151,60,114)(11,152,61,115)(12,153,62,116)(13,154,63,117)(14,141,64,118)(15,175,76,34)(16,176,77,35)(17,177,78,36)(18,178,79,37)(19,179,80,38)(20,180,81,39)(21,181,82,40)(22,182,83,41)(23,169,84,42)(24,170,71,29)(25,171,72,30)(26,172,73,31)(27,173,74,32)(28,174,75,33)(43,92,136,185)(44,93,137,186)(45,94,138,187)(46,95,139,188)(47,96,140,189)(48,97,127,190)(49,98,128,191)(50,85,129,192)(51,86,130,193)(52,87,131,194)(53,88,132,195)(54,89,133,196)(55,90,134,183)(56,91,135,184)(99,205,220,160)(100,206,221,161)(101,207,222,162)(102,208,223,163)(103,209,224,164)(104,210,211,165)(105,197,212,166)(106,198,213,167)(107,199,214,168)(108,200,215,155)(109,201,216,156)(110,202,217,157)(111,203,218,158)(112,204,219,159), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,140)(11,139)(12,138)(13,137)(14,136)(15,222)(16,221)(17,220)(18,219)(19,218)(20,217)(21,216)(22,215)(23,214)(24,213)(25,212)(26,211)(27,224)(28,223)(29,160)(30,159)(31,158)(32,157)(33,156)(34,155)(35,168)(36,167)(37,166)(38,165)(39,164)(40,163)(41,162)(42,161)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(85,141)(86,154)(87,153)(88,152)(89,151)(90,150)(91,149)(92,148)(93,147)(94,146)(95,145)(96,144)(97,143)(98,142)(113,183)(114,196)(115,195)(116,194)(117,193)(118,192)(119,191)(120,190)(121,189)(122,188)(123,187)(124,186)(125,185)(126,184)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)(175,200)(176,199)(177,198)(178,197)(179,210)(180,209)(181,208)(182,207)>;
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), (1,178,65,37)(2,179,66,38)(3,180,67,39)(4,181,68,40)(5,182,69,41)(6,169,70,42)(7,170,57,29)(8,171,58,30)(9,172,59,31)(10,173,60,32)(11,174,61,33)(12,175,62,34)(13,176,63,35)(14,177,64,36)(15,146,76,123)(16,147,77,124)(17,148,78,125)(18,149,79,126)(19,150,80,113)(20,151,81,114)(21,152,82,115)(22,153,83,116)(23,154,84,117)(24,141,71,118)(25,142,72,119)(26,143,73,120)(27,144,74,121)(28,145,75,122)(43,167,136,198)(44,168,137,199)(45,155,138,200)(46,156,139,201)(47,157,140,202)(48,158,127,203)(49,159,128,204)(50,160,129,205)(51,161,130,206)(52,162,131,207)(53,163,132,208)(54,164,133,209)(55,165,134,210)(56,166,135,197)(85,106,192,213)(86,107,193,214)(87,108,194,215)(88,109,195,216)(89,110,196,217)(90,111,183,218)(91,112,184,219)(92,99,185,220)(93,100,186,221)(94,101,187,222)(95,102,188,223)(96,103,189,224)(97,104,190,211)(98,105,191,212), (1,98,8,91)(2,97,9,90)(3,96,10,89)(4,95,11,88)(5,94,12,87)(6,93,13,86)(7,92,14,85)(15,200,22,207)(16,199,23,206)(17,198,24,205)(18,197,25,204)(19,210,26,203)(20,209,27,202)(21,208,28,201)(29,106,36,99)(30,105,37,112)(31,104,38,111)(32,103,39,110)(33,102,40,109)(34,101,41,108)(35,100,42,107)(43,125,50,118)(44,124,51,117)(45,123,52,116)(46,122,53,115)(47,121,54,114)(48,120,55,113)(49,119,56,126)(57,185,64,192)(58,184,65,191)(59,183,66,190)(60,196,67,189)(61,195,68,188)(62,194,69,187)(63,193,70,186)(71,160,78,167)(72,159,79,166)(73,158,80,165)(74,157,81,164)(75,156,82,163)(76,155,83,162)(77,168,84,161)(127,143,134,150)(128,142,135,149)(129,141,136,148)(130,154,137,147)(131,153,138,146)(132,152,139,145)(133,151,140,144)(169,214,176,221)(170,213,177,220)(171,212,178,219)(172,211,179,218)(173,224,180,217)(174,223,181,216)(175,222,182,215), (1,142,65,119)(2,143,66,120)(3,144,67,121)(4,145,68,122)(5,146,69,123)(6,147,70,124)(7,148,57,125)(8,149,58,126)(9,150,59,113)(10,151,60,114)(11,152,61,115)(12,153,62,116)(13,154,63,117)(14,141,64,118)(15,175,76,34)(16,176,77,35)(17,177,78,36)(18,178,79,37)(19,179,80,38)(20,180,81,39)(21,181,82,40)(22,182,83,41)(23,169,84,42)(24,170,71,29)(25,171,72,30)(26,172,73,31)(27,173,74,32)(28,174,75,33)(43,92,136,185)(44,93,137,186)(45,94,138,187)(46,95,139,188)(47,96,140,189)(48,97,127,190)(49,98,128,191)(50,85,129,192)(51,86,130,193)(52,87,131,194)(53,88,132,195)(54,89,133,196)(55,90,134,183)(56,91,135,184)(99,205,220,160)(100,206,221,161)(101,207,222,162)(102,208,223,163)(103,209,224,164)(104,210,211,165)(105,197,212,166)(106,198,213,167)(107,199,214,168)(108,200,215,155)(109,201,216,156)(110,202,217,157)(111,203,218,158)(112,204,219,159), (1,135)(2,134)(3,133)(4,132)(5,131)(6,130)(7,129)(8,128)(9,127)(10,140)(11,139)(12,138)(13,137)(14,136)(15,222)(16,221)(17,220)(18,219)(19,218)(20,217)(21,216)(22,215)(23,214)(24,213)(25,212)(26,211)(27,224)(28,223)(29,160)(30,159)(31,158)(32,157)(33,156)(34,155)(35,168)(36,167)(37,166)(38,165)(39,164)(40,163)(41,162)(42,161)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(85,141)(86,154)(87,153)(88,152)(89,151)(90,150)(91,149)(92,148)(93,147)(94,146)(95,145)(96,144)(97,143)(98,142)(113,183)(114,196)(115,195)(116,194)(117,193)(118,192)(119,191)(120,190)(121,189)(122,188)(123,187)(124,186)(125,185)(126,184)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)(175,200)(176,199)(177,198)(178,197)(179,210)(180,209)(181,208)(182,207) );
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)], [(1,178,65,37),(2,179,66,38),(3,180,67,39),(4,181,68,40),(5,182,69,41),(6,169,70,42),(7,170,57,29),(8,171,58,30),(9,172,59,31),(10,173,60,32),(11,174,61,33),(12,175,62,34),(13,176,63,35),(14,177,64,36),(15,146,76,123),(16,147,77,124),(17,148,78,125),(18,149,79,126),(19,150,80,113),(20,151,81,114),(21,152,82,115),(22,153,83,116),(23,154,84,117),(24,141,71,118),(25,142,72,119),(26,143,73,120),(27,144,74,121),(28,145,75,122),(43,167,136,198),(44,168,137,199),(45,155,138,200),(46,156,139,201),(47,157,140,202),(48,158,127,203),(49,159,128,204),(50,160,129,205),(51,161,130,206),(52,162,131,207),(53,163,132,208),(54,164,133,209),(55,165,134,210),(56,166,135,197),(85,106,192,213),(86,107,193,214),(87,108,194,215),(88,109,195,216),(89,110,196,217),(90,111,183,218),(91,112,184,219),(92,99,185,220),(93,100,186,221),(94,101,187,222),(95,102,188,223),(96,103,189,224),(97,104,190,211),(98,105,191,212)], [(1,98,8,91),(2,97,9,90),(3,96,10,89),(4,95,11,88),(5,94,12,87),(6,93,13,86),(7,92,14,85),(15,200,22,207),(16,199,23,206),(17,198,24,205),(18,197,25,204),(19,210,26,203),(20,209,27,202),(21,208,28,201),(29,106,36,99),(30,105,37,112),(31,104,38,111),(32,103,39,110),(33,102,40,109),(34,101,41,108),(35,100,42,107),(43,125,50,118),(44,124,51,117),(45,123,52,116),(46,122,53,115),(47,121,54,114),(48,120,55,113),(49,119,56,126),(57,185,64,192),(58,184,65,191),(59,183,66,190),(60,196,67,189),(61,195,68,188),(62,194,69,187),(63,193,70,186),(71,160,78,167),(72,159,79,166),(73,158,80,165),(74,157,81,164),(75,156,82,163),(76,155,83,162),(77,168,84,161),(127,143,134,150),(128,142,135,149),(129,141,136,148),(130,154,137,147),(131,153,138,146),(132,152,139,145),(133,151,140,144),(169,214,176,221),(170,213,177,220),(171,212,178,219),(172,211,179,218),(173,224,180,217),(174,223,181,216),(175,222,182,215)], [(1,142,65,119),(2,143,66,120),(3,144,67,121),(4,145,68,122),(5,146,69,123),(6,147,70,124),(7,148,57,125),(8,149,58,126),(9,150,59,113),(10,151,60,114),(11,152,61,115),(12,153,62,116),(13,154,63,117),(14,141,64,118),(15,175,76,34),(16,176,77,35),(17,177,78,36),(18,178,79,37),(19,179,80,38),(20,180,81,39),(21,181,82,40),(22,182,83,41),(23,169,84,42),(24,170,71,29),(25,171,72,30),(26,172,73,31),(27,173,74,32),(28,174,75,33),(43,92,136,185),(44,93,137,186),(45,94,138,187),(46,95,139,188),(47,96,140,189),(48,97,127,190),(49,98,128,191),(50,85,129,192),(51,86,130,193),(52,87,131,194),(53,88,132,195),(54,89,133,196),(55,90,134,183),(56,91,135,184),(99,205,220,160),(100,206,221,161),(101,207,222,162),(102,208,223,163),(103,209,224,164),(104,210,211,165),(105,197,212,166),(106,198,213,167),(107,199,214,168),(108,200,215,155),(109,201,216,156),(110,202,217,157),(111,203,218,158),(112,204,219,159)], [(1,135),(2,134),(3,133),(4,132),(5,131),(6,130),(7,129),(8,128),(9,127),(10,140),(11,139),(12,138),(13,137),(14,136),(15,222),(16,221),(17,220),(18,219),(19,218),(20,217),(21,216),(22,215),(23,214),(24,213),(25,212),(26,211),(27,224),(28,223),(29,160),(30,159),(31,158),(32,157),(33,156),(34,155),(35,168),(36,167),(37,166),(38,165),(39,164),(40,163),(41,162),(42,161),(43,64),(44,63),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(71,106),(72,105),(73,104),(74,103),(75,102),(76,101),(77,100),(78,99),(79,112),(80,111),(81,110),(82,109),(83,108),(84,107),(85,141),(86,154),(87,153),(88,152),(89,151),(90,150),(91,149),(92,148),(93,147),(94,146),(95,145),(96,144),(97,143),(98,142),(113,183),(114,196),(115,195),(116,194),(117,193),(118,192),(119,191),(120,190),(121,189),(122,188),(123,187),(124,186),(125,185),(126,184),(169,206),(170,205),(171,204),(172,203),(173,202),(174,201),(175,200),(176,199),(177,198),(178,197),(179,210),(180,209),(181,208),(182,207)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 28 | 28 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | D14 | 2+ 1+4 | 2- 1+4 | D4×D7 | D4⋊6D14 | D4.10D14 |
| kernel | C14.682- 1+4 | C22⋊Dic14 | D14⋊D4 | D14⋊Q8 | C2×Dic7⋊C4 | C28⋊7D4 | Dic7⋊D4 | C7×C4⋊D4 | C2×D4⋊2D7 | C2×Dic7 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 4 | 1 | 2 | 4 | 3 | 6 | 3 | 3 | 9 | 1 | 1 | 6 | 6 | 6 |
Matrix representation of C14.682- 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 8 | 0 | 0 |
| 0 | 0 | 21 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 8 |
| 0 | 0 | 0 | 0 | 12 | 10 |
| 8 | 11 | 0 | 0 | 0 | 0 |
| 18 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 18 | 7 | 8 |
| 0 | 0 | 11 | 2 | 3 | 14 |
| 0 | 0 | 0 | 0 | 21 | 11 |
| 0 | 0 | 0 | 0 | 2 | 8 |
| 11 | 8 | 0 | 0 | 0 | 0 |
| 21 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 16 | 0 | 0 |
| 0 | 0 | 2 | 5 | 0 | 0 |
| 0 | 0 | 28 | 12 | 15 | 2 |
| 0 | 0 | 22 | 4 | 3 | 14 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 11 | 4 |
| 0 | 0 | 0 | 28 | 25 | 0 |
| 0 | 0 | 0 | 15 | 1 | 0 |
| 0 | 0 | 14 | 24 | 0 | 1 |
| 21 | 18 | 0 | 0 | 0 | 0 |
| 11 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 16 | 3 | 20 |
| 0 | 0 | 2 | 5 | 27 | 21 |
| 0 | 0 | 28 | 12 | 15 | 2 |
| 0 | 0 | 22 | 4 | 3 | 14 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,3,21,0,0,0,0,8,8,0,0,0,0,0,0,1,12,0,0,0,0,8,10],[8,18,0,0,0,0,11,21,0,0,0,0,0,0,27,11,0,0,0,0,18,2,0,0,0,0,7,3,21,2,0,0,8,14,11,8],[11,21,0,0,0,0,8,18,0,0,0,0,0,0,24,2,28,22,0,0,16,5,12,4,0,0,0,0,15,3,0,0,0,0,2,14],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,14,0,0,0,28,15,24,0,0,11,25,1,0,0,0,4,0,0,1],[21,11,0,0,0,0,18,8,0,0,0,0,0,0,24,2,28,22,0,0,16,5,12,4,0,0,3,27,15,3,0,0,20,21,2,14] >;
C14.682- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{68}2_-^{1+4} % in TeX
G:=Group("C14.68ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1050);
// by ID
G=gap.SmallGroup(448,1050);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,387,1123,570,185,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=b^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1,a*d=d*a,c*b*c^-1=a^7*b^-1,d*b*d^-1=a^7*b,b*e=e*b,d*c*d^-1=e*c*e=a^7*c,e*d*e=a^7*b^2*d>;
// generators/relations