G = C14.582+ 1+4 order 448 = 26·7
58th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.582+ 1+4, C14.252- 1+4, C28⋊Q8⋊29C2, C22⋊Q8⋊23D7, C4⋊C4.101D14, (C2×Q8).79D14, D14⋊3Q8⋊25C2, D14⋊Q8⋊24C2, Dic7.Q8⋊22C2, D14⋊C4.8C22, C22⋊C4.23D14, C28.3Q8⋊27C2, Dic7⋊Q8⋊18C2, C28.48D4⋊47C2, (C2×C28).176C23, (C2×C14).190C24, D14.D4.3C2, (C22×C4).252D14, C2.60(D4⋊6D14), C23.D14⋊26C2, Dic7⋊C4.81C22, C4⋊Dic7.222C22, Dic7.D4.3C2, (Q8×C14).119C22, (C2×Dic7).96C23, (C22×D7).81C23, C23.126(C22×D7), C22.211(C23×D7), C23.D7.36C22, (C22×C14).218C23, (C22×C28).318C22, C7⋊1(C22.57C24), (C2×Dic14).35C22, (C4×Dic7).117C22, C23.23D14.2C2, C2.39(D4.10D14), C2.26(Q8.10D14), C4⋊C4⋊D7⋊25C2, (C7×C22⋊Q8)⋊26C2, (C2×C4×D7).106C22, (C7×C4⋊C4).170C22, (C2×C4).187(C22×D7), (C2×C7⋊D4).42C22, (C7×C22⋊C4).45C22, SmallGroup(448,1099)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.582+ 1+4
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=e2=b2, ab=ba, cac-1=eae-1=a-1, ad=da, cbc-1=a7b-1, bd=db, ebe-1=a7b, dcd-1=a7c, ce=ec, ede-1=b2d >
Subgroups: 828 in 196 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×Q8, C2×Q8, Dic7, C28, D14, C2×C14, C2×C14, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, Dic14, C4×D7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×Q8, C22×D7, C22×C14, C22.57C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×C7⋊D4, C22×C28, Q8×C14, C23.D14, D14.D4, Dic7.D4, C28⋊Q8, Dic7.Q8, C28.3Q8, D14⋊Q8, C4⋊C4⋊D7, C28.48D4, C23.23D14, Dic7⋊Q8, D14⋊3Q8, C7×C22⋊Q8, C14.582+ 1+4
Quotients: C1, C2, C22, C23, D7, C24, D14, 2+ 1+4, 2- 1+4, C22×D7, C22.57C24, C23×D7, D4⋊6D14, Q8.10D14, D4.10D14, C14.582+ 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 15 95 73)(2 16 96 74)(3 17 97 75)(4 18 98 76)(5 19 85 77)(6 20 86 78)(7 21 87 79)(8 22 88 80)(9 23 89 81)(10 24 90 82)(11 25 91 83)(12 26 92 84)(13 27 93 71)(14 28 94 72)(29 138 116 46)(30 139 117 47)(31 140 118 48)(32 127 119 49)(33 128 120 50)(34 129 121 51)(35 130 122 52)(36 131 123 53)(37 132 124 54)(38 133 125 55)(39 134 126 56)(40 135 113 43)(41 136 114 44)(42 137 115 45)(57 192 219 142)(58 193 220 143)(59 194 221 144)(60 195 222 145)(61 196 223 146)(62 183 224 147)(63 184 211 148)(64 185 212 149)(65 186 213 150)(66 187 214 151)(67 188 215 152)(68 189 216 153)(69 190 217 154)(70 191 218 141)(99 207 170 160)(100 208 171 161)(101 209 172 162)(102 210 173 163)(103 197 174 164)(104 198 175 165)(105 199 176 166)(106 200 177 167)(107 201 178 168)(108 202 179 155)(109 203 180 156)(110 204 181 157)(111 205 182 158)(112 206 169 159)
(1 58 8 65)(2 57 9 64)(3 70 10 63)(4 69 11 62)(5 68 12 61)(6 67 13 60)(7 66 14 59)(15 150 22 143)(16 149 23 142)(17 148 24 141)(18 147 25 154)(19 146 26 153)(20 145 27 152)(21 144 28 151)(29 103 36 110)(30 102 37 109)(31 101 38 108)(32 100 39 107)(33 99 40 106)(34 112 41 105)(35 111 42 104)(43 207 50 200)(44 206 51 199)(45 205 52 198)(46 204 53 197)(47 203 54 210)(48 202 55 209)(49 201 56 208)(71 188 78 195)(72 187 79 194)(73 186 80 193)(74 185 81 192)(75 184 82 191)(76 183 83 190)(77 196 84 189)(85 216 92 223)(86 215 93 222)(87 214 94 221)(88 213 95 220)(89 212 96 219)(90 211 97 218)(91 224 98 217)(113 177 120 170)(114 176 121 169)(115 175 122 182)(116 174 123 181)(117 173 124 180)(118 172 125 179)(119 171 126 178)(127 168 134 161)(128 167 135 160)(129 166 136 159)(130 165 137 158)(131 164 138 157)(132 163 139 156)(133 162 140 155)
(1 15 95 73)(2 16 96 74)(3 17 97 75)(4 18 98 76)(5 19 85 77)(6 20 86 78)(7 21 87 79)(8 22 88 80)(9 23 89 81)(10 24 90 82)(11 25 91 83)(12 26 92 84)(13 27 93 71)(14 28 94 72)(29 53 116 131)(30 54 117 132)(31 55 118 133)(32 56 119 134)(33 43 120 135)(34 44 121 136)(35 45 122 137)(36 46 123 138)(37 47 124 139)(38 48 125 140)(39 49 126 127)(40 50 113 128)(41 51 114 129)(42 52 115 130)(57 142 219 192)(58 143 220 193)(59 144 221 194)(60 145 222 195)(61 146 223 196)(62 147 224 183)(63 148 211 184)(64 149 212 185)(65 150 213 186)(66 151 214 187)(67 152 215 188)(68 153 216 189)(69 154 217 190)(70 141 218 191)(99 200 170 167)(100 201 171 168)(101 202 172 155)(102 203 173 156)(103 204 174 157)(104 205 175 158)(105 206 176 159)(106 207 177 160)(107 208 178 161)(108 209 179 162)(109 210 180 163)(110 197 181 164)(111 198 182 165)(112 199 169 166)
(1 116 95 29)(2 115 96 42)(3 114 97 41)(4 113 98 40)(5 126 85 39)(6 125 86 38)(7 124 87 37)(8 123 88 36)(9 122 89 35)(10 121 90 34)(11 120 91 33)(12 119 92 32)(13 118 93 31)(14 117 94 30)(15 53 73 131)(16 52 74 130)(17 51 75 129)(18 50 76 128)(19 49 77 127)(20 48 78 140)(21 47 79 139)(22 46 80 138)(23 45 81 137)(24 44 82 136)(25 43 83 135)(26 56 84 134)(27 55 71 133)(28 54 72 132)(57 175 219 104)(58 174 220 103)(59 173 221 102)(60 172 222 101)(61 171 223 100)(62 170 224 99)(63 169 211 112)(64 182 212 111)(65 181 213 110)(66 180 214 109)(67 179 215 108)(68 178 216 107)(69 177 217 106)(70 176 218 105)(141 206 191 159)(142 205 192 158)(143 204 193 157)(144 203 194 156)(145 202 195 155)(146 201 196 168)(147 200 183 167)(148 199 184 166)(149 198 185 165)(150 197 186 164)(151 210 187 163)(152 209 188 162)(153 208 189 161)(154 207 190 160)
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), (1,15,95,73)(2,16,96,74)(3,17,97,75)(4,18,98,76)(5,19,85,77)(6,20,86,78)(7,21,87,79)(8,22,88,80)(9,23,89,81)(10,24,90,82)(11,25,91,83)(12,26,92,84)(13,27,93,71)(14,28,94,72)(29,138,116,46)(30,139,117,47)(31,140,118,48)(32,127,119,49)(33,128,120,50)(34,129,121,51)(35,130,122,52)(36,131,123,53)(37,132,124,54)(38,133,125,55)(39,134,126,56)(40,135,113,43)(41,136,114,44)(42,137,115,45)(57,192,219,142)(58,193,220,143)(59,194,221,144)(60,195,222,145)(61,196,223,146)(62,183,224,147)(63,184,211,148)(64,185,212,149)(65,186,213,150)(66,187,214,151)(67,188,215,152)(68,189,216,153)(69,190,217,154)(70,191,218,141)(99,207,170,160)(100,208,171,161)(101,209,172,162)(102,210,173,163)(103,197,174,164)(104,198,175,165)(105,199,176,166)(106,200,177,167)(107,201,178,168)(108,202,179,155)(109,203,180,156)(110,204,181,157)(111,205,182,158)(112,206,169,159), (1,58,8,65)(2,57,9,64)(3,70,10,63)(4,69,11,62)(5,68,12,61)(6,67,13,60)(7,66,14,59)(15,150,22,143)(16,149,23,142)(17,148,24,141)(18,147,25,154)(19,146,26,153)(20,145,27,152)(21,144,28,151)(29,103,36,110)(30,102,37,109)(31,101,38,108)(32,100,39,107)(33,99,40,106)(34,112,41,105)(35,111,42,104)(43,207,50,200)(44,206,51,199)(45,205,52,198)(46,204,53,197)(47,203,54,210)(48,202,55,209)(49,201,56,208)(71,188,78,195)(72,187,79,194)(73,186,80,193)(74,185,81,192)(75,184,82,191)(76,183,83,190)(77,196,84,189)(85,216,92,223)(86,215,93,222)(87,214,94,221)(88,213,95,220)(89,212,96,219)(90,211,97,218)(91,224,98,217)(113,177,120,170)(114,176,121,169)(115,175,122,182)(116,174,123,181)(117,173,124,180)(118,172,125,179)(119,171,126,178)(127,168,134,161)(128,167,135,160)(129,166,136,159)(130,165,137,158)(131,164,138,157)(132,163,139,156)(133,162,140,155), (1,15,95,73)(2,16,96,74)(3,17,97,75)(4,18,98,76)(5,19,85,77)(6,20,86,78)(7,21,87,79)(8,22,88,80)(9,23,89,81)(10,24,90,82)(11,25,91,83)(12,26,92,84)(13,27,93,71)(14,28,94,72)(29,53,116,131)(30,54,117,132)(31,55,118,133)(32,56,119,134)(33,43,120,135)(34,44,121,136)(35,45,122,137)(36,46,123,138)(37,47,124,139)(38,48,125,140)(39,49,126,127)(40,50,113,128)(41,51,114,129)(42,52,115,130)(57,142,219,192)(58,143,220,193)(59,144,221,194)(60,145,222,195)(61,146,223,196)(62,147,224,183)(63,148,211,184)(64,149,212,185)(65,150,213,186)(66,151,214,187)(67,152,215,188)(68,153,216,189)(69,154,217,190)(70,141,218,191)(99,200,170,167)(100,201,171,168)(101,202,172,155)(102,203,173,156)(103,204,174,157)(104,205,175,158)(105,206,176,159)(106,207,177,160)(107,208,178,161)(108,209,179,162)(109,210,180,163)(110,197,181,164)(111,198,182,165)(112,199,169,166), (1,116,95,29)(2,115,96,42)(3,114,97,41)(4,113,98,40)(5,126,85,39)(6,125,86,38)(7,124,87,37)(8,123,88,36)(9,122,89,35)(10,121,90,34)(11,120,91,33)(12,119,92,32)(13,118,93,31)(14,117,94,30)(15,53,73,131)(16,52,74,130)(17,51,75,129)(18,50,76,128)(19,49,77,127)(20,48,78,140)(21,47,79,139)(22,46,80,138)(23,45,81,137)(24,44,82,136)(25,43,83,135)(26,56,84,134)(27,55,71,133)(28,54,72,132)(57,175,219,104)(58,174,220,103)(59,173,221,102)(60,172,222,101)(61,171,223,100)(62,170,224,99)(63,169,211,112)(64,182,212,111)(65,181,213,110)(66,180,214,109)(67,179,215,108)(68,178,216,107)(69,177,217,106)(70,176,218,105)(141,206,191,159)(142,205,192,158)(143,204,193,157)(144,203,194,156)(145,202,195,155)(146,201,196,168)(147,200,183,167)(148,199,184,166)(149,198,185,165)(150,197,186,164)(151,210,187,163)(152,209,188,162)(153,208,189,161)(154,207,190,160)>;
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), (1,15,95,73)(2,16,96,74)(3,17,97,75)(4,18,98,76)(5,19,85,77)(6,20,86,78)(7,21,87,79)(8,22,88,80)(9,23,89,81)(10,24,90,82)(11,25,91,83)(12,26,92,84)(13,27,93,71)(14,28,94,72)(29,138,116,46)(30,139,117,47)(31,140,118,48)(32,127,119,49)(33,128,120,50)(34,129,121,51)(35,130,122,52)(36,131,123,53)(37,132,124,54)(38,133,125,55)(39,134,126,56)(40,135,113,43)(41,136,114,44)(42,137,115,45)(57,192,219,142)(58,193,220,143)(59,194,221,144)(60,195,222,145)(61,196,223,146)(62,183,224,147)(63,184,211,148)(64,185,212,149)(65,186,213,150)(66,187,214,151)(67,188,215,152)(68,189,216,153)(69,190,217,154)(70,191,218,141)(99,207,170,160)(100,208,171,161)(101,209,172,162)(102,210,173,163)(103,197,174,164)(104,198,175,165)(105,199,176,166)(106,200,177,167)(107,201,178,168)(108,202,179,155)(109,203,180,156)(110,204,181,157)(111,205,182,158)(112,206,169,159), (1,58,8,65)(2,57,9,64)(3,70,10,63)(4,69,11,62)(5,68,12,61)(6,67,13,60)(7,66,14,59)(15,150,22,143)(16,149,23,142)(17,148,24,141)(18,147,25,154)(19,146,26,153)(20,145,27,152)(21,144,28,151)(29,103,36,110)(30,102,37,109)(31,101,38,108)(32,100,39,107)(33,99,40,106)(34,112,41,105)(35,111,42,104)(43,207,50,200)(44,206,51,199)(45,205,52,198)(46,204,53,197)(47,203,54,210)(48,202,55,209)(49,201,56,208)(71,188,78,195)(72,187,79,194)(73,186,80,193)(74,185,81,192)(75,184,82,191)(76,183,83,190)(77,196,84,189)(85,216,92,223)(86,215,93,222)(87,214,94,221)(88,213,95,220)(89,212,96,219)(90,211,97,218)(91,224,98,217)(113,177,120,170)(114,176,121,169)(115,175,122,182)(116,174,123,181)(117,173,124,180)(118,172,125,179)(119,171,126,178)(127,168,134,161)(128,167,135,160)(129,166,136,159)(130,165,137,158)(131,164,138,157)(132,163,139,156)(133,162,140,155), (1,15,95,73)(2,16,96,74)(3,17,97,75)(4,18,98,76)(5,19,85,77)(6,20,86,78)(7,21,87,79)(8,22,88,80)(9,23,89,81)(10,24,90,82)(11,25,91,83)(12,26,92,84)(13,27,93,71)(14,28,94,72)(29,53,116,131)(30,54,117,132)(31,55,118,133)(32,56,119,134)(33,43,120,135)(34,44,121,136)(35,45,122,137)(36,46,123,138)(37,47,124,139)(38,48,125,140)(39,49,126,127)(40,50,113,128)(41,51,114,129)(42,52,115,130)(57,142,219,192)(58,143,220,193)(59,144,221,194)(60,145,222,195)(61,146,223,196)(62,147,224,183)(63,148,211,184)(64,149,212,185)(65,150,213,186)(66,151,214,187)(67,152,215,188)(68,153,216,189)(69,154,217,190)(70,141,218,191)(99,200,170,167)(100,201,171,168)(101,202,172,155)(102,203,173,156)(103,204,174,157)(104,205,175,158)(105,206,176,159)(106,207,177,160)(107,208,178,161)(108,209,179,162)(109,210,180,163)(110,197,181,164)(111,198,182,165)(112,199,169,166), (1,116,95,29)(2,115,96,42)(3,114,97,41)(4,113,98,40)(5,126,85,39)(6,125,86,38)(7,124,87,37)(8,123,88,36)(9,122,89,35)(10,121,90,34)(11,120,91,33)(12,119,92,32)(13,118,93,31)(14,117,94,30)(15,53,73,131)(16,52,74,130)(17,51,75,129)(18,50,76,128)(19,49,77,127)(20,48,78,140)(21,47,79,139)(22,46,80,138)(23,45,81,137)(24,44,82,136)(25,43,83,135)(26,56,84,134)(27,55,71,133)(28,54,72,132)(57,175,219,104)(58,174,220,103)(59,173,221,102)(60,172,222,101)(61,171,223,100)(62,170,224,99)(63,169,211,112)(64,182,212,111)(65,181,213,110)(66,180,214,109)(67,179,215,108)(68,178,216,107)(69,177,217,106)(70,176,218,105)(141,206,191,159)(142,205,192,158)(143,204,193,157)(144,203,194,156)(145,202,195,155)(146,201,196,168)(147,200,183,167)(148,199,184,166)(149,198,185,165)(150,197,186,164)(151,210,187,163)(152,209,188,162)(153,208,189,161)(154,207,190,160) );
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)], [(1,15,95,73),(2,16,96,74),(3,17,97,75),(4,18,98,76),(5,19,85,77),(6,20,86,78),(7,21,87,79),(8,22,88,80),(9,23,89,81),(10,24,90,82),(11,25,91,83),(12,26,92,84),(13,27,93,71),(14,28,94,72),(29,138,116,46),(30,139,117,47),(31,140,118,48),(32,127,119,49),(33,128,120,50),(34,129,121,51),(35,130,122,52),(36,131,123,53),(37,132,124,54),(38,133,125,55),(39,134,126,56),(40,135,113,43),(41,136,114,44),(42,137,115,45),(57,192,219,142),(58,193,220,143),(59,194,221,144),(60,195,222,145),(61,196,223,146),(62,183,224,147),(63,184,211,148),(64,185,212,149),(65,186,213,150),(66,187,214,151),(67,188,215,152),(68,189,216,153),(69,190,217,154),(70,191,218,141),(99,207,170,160),(100,208,171,161),(101,209,172,162),(102,210,173,163),(103,197,174,164),(104,198,175,165),(105,199,176,166),(106,200,177,167),(107,201,178,168),(108,202,179,155),(109,203,180,156),(110,204,181,157),(111,205,182,158),(112,206,169,159)], [(1,58,8,65),(2,57,9,64),(3,70,10,63),(4,69,11,62),(5,68,12,61),(6,67,13,60),(7,66,14,59),(15,150,22,143),(16,149,23,142),(17,148,24,141),(18,147,25,154),(19,146,26,153),(20,145,27,152),(21,144,28,151),(29,103,36,110),(30,102,37,109),(31,101,38,108),(32,100,39,107),(33,99,40,106),(34,112,41,105),(35,111,42,104),(43,207,50,200),(44,206,51,199),(45,205,52,198),(46,204,53,197),(47,203,54,210),(48,202,55,209),(49,201,56,208),(71,188,78,195),(72,187,79,194),(73,186,80,193),(74,185,81,192),(75,184,82,191),(76,183,83,190),(77,196,84,189),(85,216,92,223),(86,215,93,222),(87,214,94,221),(88,213,95,220),(89,212,96,219),(90,211,97,218),(91,224,98,217),(113,177,120,170),(114,176,121,169),(115,175,122,182),(116,174,123,181),(117,173,124,180),(118,172,125,179),(119,171,126,178),(127,168,134,161),(128,167,135,160),(129,166,136,159),(130,165,137,158),(131,164,138,157),(132,163,139,156),(133,162,140,155)], [(1,15,95,73),(2,16,96,74),(3,17,97,75),(4,18,98,76),(5,19,85,77),(6,20,86,78),(7,21,87,79),(8,22,88,80),(9,23,89,81),(10,24,90,82),(11,25,91,83),(12,26,92,84),(13,27,93,71),(14,28,94,72),(29,53,116,131),(30,54,117,132),(31,55,118,133),(32,56,119,134),(33,43,120,135),(34,44,121,136),(35,45,122,137),(36,46,123,138),(37,47,124,139),(38,48,125,140),(39,49,126,127),(40,50,113,128),(41,51,114,129),(42,52,115,130),(57,142,219,192),(58,143,220,193),(59,144,221,194),(60,145,222,195),(61,146,223,196),(62,147,224,183),(63,148,211,184),(64,149,212,185),(65,150,213,186),(66,151,214,187),(67,152,215,188),(68,153,216,189),(69,154,217,190),(70,141,218,191),(99,200,170,167),(100,201,171,168),(101,202,172,155),(102,203,173,156),(103,204,174,157),(104,205,175,158),(105,206,176,159),(106,207,177,160),(107,208,178,161),(108,209,179,162),(109,210,180,163),(110,197,181,164),(111,198,182,165),(112,199,169,166)], [(1,116,95,29),(2,115,96,42),(3,114,97,41),(4,113,98,40),(5,126,85,39),(6,125,86,38),(7,124,87,37),(8,123,88,36),(9,122,89,35),(10,121,90,34),(11,120,91,33),(12,119,92,32),(13,118,93,31),(14,117,94,30),(15,53,73,131),(16,52,74,130),(17,51,75,129),(18,50,76,128),(19,49,77,127),(20,48,78,140),(21,47,79,139),(22,46,80,138),(23,45,81,137),(24,44,82,136),(25,43,83,135),(26,56,84,134),(27,55,71,133),(28,54,72,132),(57,175,219,104),(58,174,220,103),(59,173,221,102),(60,172,222,101),(61,171,223,100),(62,170,224,99),(63,169,211,112),(64,182,212,111),(65,181,213,110),(66,180,214,109),(67,179,215,108),(68,178,216,107),(69,177,217,106),(70,176,218,105),(141,206,191,159),(142,205,192,158),(143,204,193,157),(144,203,194,156),(145,202,195,155),(146,201,196,168),(147,200,183,167),(148,199,184,166),(149,198,185,165),(150,197,186,164),(151,210,187,163),(152,209,188,162),(153,208,189,161),(154,207,190,160)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | ··· | 4M | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 28 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | 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 | Q8.10D14 | D4.10D14 |
| kernel | C14.582+ 1+4 | C23.D14 | D14.D4 | Dic7.D4 | C28⋊Q8 | Dic7.Q8 | C28.3Q8 | D14⋊Q8 | C4⋊C4⋊D7 | C28.48D4 | C23.23D14 | Dic7⋊Q8 | D14⋊3Q8 | C7×C22⋊Q8 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C14 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 6 | 9 | 3 | 3 | 1 | 2 | 6 | 6 | 6 |
Matrix representation of C14.582+ 1+4 ►in GL8(𝔽29)
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 0 | 22 | 0 | 0 | 0 | 0 | 0 |
| 15 | 0 | 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 13 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 26 | 0 | 9 | 17 |
| 1 | 0 | 22 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 21 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 8 | 28 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 0 | 27 | 15 |
| 0 | 0 | 0 | 0 | 19 | 0 | 1 | 20 |
| 0 | 0 | 0 | 0 | 0 | 5 | 9 | 5 |
| 0 | 0 | 0 | 0 | 10 | 20 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 20 | 12 |
| 28 | 0 | 0 | 22 | 0 | 0 | 0 | 0 |
| 21 | 0 | 28 | 1 | 0 | 0 | 0 | 0 |
| 8 | 28 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 9 | 5 |
| 0 | 0 | 0 | 0 | 19 | 0 | 1 | 20 |
G:=sub<GL(8,GF(29))| [4,0,14,15,0,0,0,0,0,4,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,9,14,0,15,0,0,0,0,0,13,0,0,0,0,0,0,0,0,9,13,0,0,0,0,0,0,0,13],[1,0,21,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,17,26,21,26,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,9,0,0,0,0,0,0,0,17],[1,0,21,8,0,0,0,0,0,0,0,28,0,0,0,0,22,1,28,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,20,19,0,10,0,0,0,0,0,0,5,20,0,0,0,0,27,1,9,0,0,0,0,0,15,20,5,0],[1,8,21,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,17,26,0,3,0,0,0,0,0,12,0,0,0,0,0,0,0,0,17,20,0,0,0,0,0,0,0,12],[28,21,8,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,22,1,28,1,0,0,0,0,0,0,0,0,9,10,0,19,0,0,0,0,15,20,5,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,5,20] >;
C14.582+ 1+4 in GAP, Magma, Sage, TeX
C_{14}._{58}2_+^{1+4} % in TeX
G:=Group("C14.58ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1099);
// by ID
G=gap.SmallGroup(448,1099);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,1571,570,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=a^7,d^2=e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1,a*d=d*a,c*b*c^-1=a^7*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=b^2*d>;
// generators/relations