G = C14.862- 1+4 order 448 = 26·7
41st non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.862- 1+4, C14.692+ 1+4, C28⋊7D4⋊15C2, C28⋊2D4⋊32C2, C4⋊C4.109D14, D14⋊D4⋊36C2, D14⋊2Q8⋊34C2, D14⋊Q8⋊33C2, (C2×D4).106D14, C22⋊C4.32D14, C28.3Q8⋊31C2, Dic7⋊D4⋊24C2, D14.5D4⋊32C2, D14.D4⋊39C2, C28.48D4⋊15C2, (C2×C28).185C23, (C2×C14).211C24, D14⋊C4.35C22, (C2×D28).33C22, (C22×C4).263D14, C22.D4⋊16D7, C2.71(D4⋊6D14), C2.47(D4⋊8D14), C22⋊Dic14⋊36C2, Dic7.D4⋊36C2, (D4×C14).149C22, C22.D28⋊24C2, Dic7⋊C4.46C22, C4⋊Dic7.231C22, (C22×C28).88C22, (C22×D7).92C23, C22.232(C23×D7), C23.132(C22×D7), C23.D7.49C22, (C22×C14).225C23, C7⋊3(C22.56C24), (C2×Dic7).110C23, (C4×Dic7).129C22, C2.47(D4.10D14), (C2×Dic14).172C22, (C22×Dic7).136C22, (C2×C4×D7).117C22, (C2×C4).72(C22×D7), (C7×C4⋊C4).183C22, (C2×C7⋊D4).54C22, (C7×C22.D4)⋊19C2, (C7×C22⋊C4).58C22, SmallGroup(448,1120)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.862- 1+4
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=a7b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, ebe=a7b, dcd-1=ece=a7c, ede=a7b2d >
Subgroups: 1100 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, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C42.C2, Dic14, C4×D7, D28, 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, C2×D28, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C22⋊Dic14, D14.D4, D14⋊D4, Dic7.D4, C22.D28, C28.3Q8, D14.5D4, D14⋊Q8, D14⋊2Q8, C28.48D4, C28⋊7D4, C28⋊2D4, Dic7⋊D4, C7×C22.D4, C14.862- 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⋊8D14, D4.10D14, C14.862- 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)
(1 112 27 92)(2 99 28 93)(3 100 15 94)(4 101 16 95)(5 102 17 96)(6 103 18 97)(7 104 19 98)(8 105 20 85)(9 106 21 86)(10 107 22 87)(11 108 23 88)(12 109 24 89)(13 110 25 90)(14 111 26 91)(29 73 43 57)(30 74 44 58)(31 75 45 59)(32 76 46 60)(33 77 47 61)(34 78 48 62)(35 79 49 63)(36 80 50 64)(37 81 51 65)(38 82 52 66)(39 83 53 67)(40 84 54 68)(41 71 55 69)(42 72 56 70)(113 212 140 202)(114 213 127 203)(115 214 128 204)(116 215 129 205)(117 216 130 206)(118 217 131 207)(119 218 132 208)(120 219 133 209)(121 220 134 210)(122 221 135 197)(123 222 136 198)(124 223 137 199)(125 224 138 200)(126 211 139 201)(141 186 155 176)(142 187 156 177)(143 188 157 178)(144 189 158 179)(145 190 159 180)(146 191 160 181)(147 192 161 182)(148 193 162 169)(149 194 163 170)(150 195 164 171)(151 196 165 172)(152 183 166 173)(153 184 167 174)(154 185 168 175)
(1 68 8 61)(2 69 9 62)(3 70 10 63)(4 57 11 64)(5 58 12 65)(6 59 13 66)(7 60 14 67)(15 72 22 79)(16 73 23 80)(17 74 24 81)(18 75 25 82)(19 76 26 83)(20 77 27 84)(21 78 28 71)(29 88 36 95)(30 89 37 96)(31 90 38 97)(32 91 39 98)(33 92 40 85)(34 93 41 86)(35 94 42 87)(43 108 50 101)(44 109 51 102)(45 110 52 103)(46 111 53 104)(47 112 54 105)(48 99 55 106)(49 100 56 107)(113 179 120 172)(114 180 121 173)(115 181 122 174)(116 182 123 175)(117 169 124 176)(118 170 125 177)(119 171 126 178)(127 190 134 183)(128 191 135 184)(129 192 136 185)(130 193 137 186)(131 194 138 187)(132 195 139 188)(133 196 140 189)(141 206 148 199)(142 207 149 200)(143 208 150 201)(144 209 151 202)(145 210 152 203)(146 197 153 204)(147 198 154 205)(155 216 162 223)(156 217 163 224)(157 218 164 211)(158 219 165 212)(159 220 166 213)(160 221 167 214)(161 222 168 215)
(1 122 20 128)(2 121 21 127)(3 120 22 140)(4 119 23 139)(5 118 24 138)(6 117 25 137)(7 116 26 136)(8 115 27 135)(9 114 28 134)(10 113 15 133)(11 126 16 132)(12 125 17 131)(13 124 18 130)(14 123 19 129)(29 143 50 164)(30 142 51 163)(31 141 52 162)(32 154 53 161)(33 153 54 160)(34 152 55 159)(35 151 56 158)(36 150 43 157)(37 149 44 156)(38 148 45 155)(39 147 46 168)(40 146 47 167)(41 145 48 166)(42 144 49 165)(57 178 80 195)(58 177 81 194)(59 176 82 193)(60 175 83 192)(61 174 84 191)(62 173 71 190)(63 172 72 189)(64 171 73 188)(65 170 74 187)(66 169 75 186)(67 182 76 185)(68 181 77 184)(69 180 78 183)(70 179 79 196)(85 204 112 221)(86 203 99 220)(87 202 100 219)(88 201 101 218)(89 200 102 217)(90 199 103 216)(91 198 104 215)(92 197 105 214)(93 210 106 213)(94 209 107 212)(95 208 108 211)(96 207 109 224)(97 206 110 223)(98 205 111 222)
(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(113 133)(114 134)(115 135)(116 136)(117 137)(118 138)(119 139)(120 140)(121 127)(122 128)(123 129)(124 130)(125 131)(126 132)(141 162)(142 163)(143 164)(144 165)(145 166)(146 167)(147 168)(148 155)(149 156)(150 157)(151 158)(152 159)(153 160)(154 161)(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)(197 221)(198 222)(199 223)(200 224)(201 211)(202 212)(203 213)(204 214)(205 215)(206 216)(207 217)(208 218)(209 219)(210 220)
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), (1,112,27,92)(2,99,28,93)(3,100,15,94)(4,101,16,95)(5,102,17,96)(6,103,18,97)(7,104,19,98)(8,105,20,85)(9,106,21,86)(10,107,22,87)(11,108,23,88)(12,109,24,89)(13,110,25,90)(14,111,26,91)(29,73,43,57)(30,74,44,58)(31,75,45,59)(32,76,46,60)(33,77,47,61)(34,78,48,62)(35,79,49,63)(36,80,50,64)(37,81,51,65)(38,82,52,66)(39,83,53,67)(40,84,54,68)(41,71,55,69)(42,72,56,70)(113,212,140,202)(114,213,127,203)(115,214,128,204)(116,215,129,205)(117,216,130,206)(118,217,131,207)(119,218,132,208)(120,219,133,209)(121,220,134,210)(122,221,135,197)(123,222,136,198)(124,223,137,199)(125,224,138,200)(126,211,139,201)(141,186,155,176)(142,187,156,177)(143,188,157,178)(144,189,158,179)(145,190,159,180)(146,191,160,181)(147,192,161,182)(148,193,162,169)(149,194,163,170)(150,195,164,171)(151,196,165,172)(152,183,166,173)(153,184,167,174)(154,185,168,175), (1,68,8,61)(2,69,9,62)(3,70,10,63)(4,57,11,64)(5,58,12,65)(6,59,13,66)(7,60,14,67)(15,72,22,79)(16,73,23,80)(17,74,24,81)(18,75,25,82)(19,76,26,83)(20,77,27,84)(21,78,28,71)(29,88,36,95)(30,89,37,96)(31,90,38,97)(32,91,39,98)(33,92,40,85)(34,93,41,86)(35,94,42,87)(43,108,50,101)(44,109,51,102)(45,110,52,103)(46,111,53,104)(47,112,54,105)(48,99,55,106)(49,100,56,107)(113,179,120,172)(114,180,121,173)(115,181,122,174)(116,182,123,175)(117,169,124,176)(118,170,125,177)(119,171,126,178)(127,190,134,183)(128,191,135,184)(129,192,136,185)(130,193,137,186)(131,194,138,187)(132,195,139,188)(133,196,140,189)(141,206,148,199)(142,207,149,200)(143,208,150,201)(144,209,151,202)(145,210,152,203)(146,197,153,204)(147,198,154,205)(155,216,162,223)(156,217,163,224)(157,218,164,211)(158,219,165,212)(159,220,166,213)(160,221,167,214)(161,222,168,215), (1,122,20,128)(2,121,21,127)(3,120,22,140)(4,119,23,139)(5,118,24,138)(6,117,25,137)(7,116,26,136)(8,115,27,135)(9,114,28,134)(10,113,15,133)(11,126,16,132)(12,125,17,131)(13,124,18,130)(14,123,19,129)(29,143,50,164)(30,142,51,163)(31,141,52,162)(32,154,53,161)(33,153,54,160)(34,152,55,159)(35,151,56,158)(36,150,43,157)(37,149,44,156)(38,148,45,155)(39,147,46,168)(40,146,47,167)(41,145,48,166)(42,144,49,165)(57,178,80,195)(58,177,81,194)(59,176,82,193)(60,175,83,192)(61,174,84,191)(62,173,71,190)(63,172,72,189)(64,171,73,188)(65,170,74,187)(66,169,75,186)(67,182,76,185)(68,181,77,184)(69,180,78,183)(70,179,79,196)(85,204,112,221)(86,203,99,220)(87,202,100,219)(88,201,101,218)(89,200,102,217)(90,199,103,216)(91,198,104,215)(92,197,105,214)(93,210,106,213)(94,209,107,212)(95,208,108,211)(96,207,109,224)(97,206,110,223)(98,205,111,222), (57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161)(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)(197,221)(198,222)(199,223)(200,224)(201,211)(202,212)(203,213)(204,214)(205,215)(206,216)(207,217)(208,218)(209,219)(210,220)>;
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), (1,112,27,92)(2,99,28,93)(3,100,15,94)(4,101,16,95)(5,102,17,96)(6,103,18,97)(7,104,19,98)(8,105,20,85)(9,106,21,86)(10,107,22,87)(11,108,23,88)(12,109,24,89)(13,110,25,90)(14,111,26,91)(29,73,43,57)(30,74,44,58)(31,75,45,59)(32,76,46,60)(33,77,47,61)(34,78,48,62)(35,79,49,63)(36,80,50,64)(37,81,51,65)(38,82,52,66)(39,83,53,67)(40,84,54,68)(41,71,55,69)(42,72,56,70)(113,212,140,202)(114,213,127,203)(115,214,128,204)(116,215,129,205)(117,216,130,206)(118,217,131,207)(119,218,132,208)(120,219,133,209)(121,220,134,210)(122,221,135,197)(123,222,136,198)(124,223,137,199)(125,224,138,200)(126,211,139,201)(141,186,155,176)(142,187,156,177)(143,188,157,178)(144,189,158,179)(145,190,159,180)(146,191,160,181)(147,192,161,182)(148,193,162,169)(149,194,163,170)(150,195,164,171)(151,196,165,172)(152,183,166,173)(153,184,167,174)(154,185,168,175), (1,68,8,61)(2,69,9,62)(3,70,10,63)(4,57,11,64)(5,58,12,65)(6,59,13,66)(7,60,14,67)(15,72,22,79)(16,73,23,80)(17,74,24,81)(18,75,25,82)(19,76,26,83)(20,77,27,84)(21,78,28,71)(29,88,36,95)(30,89,37,96)(31,90,38,97)(32,91,39,98)(33,92,40,85)(34,93,41,86)(35,94,42,87)(43,108,50,101)(44,109,51,102)(45,110,52,103)(46,111,53,104)(47,112,54,105)(48,99,55,106)(49,100,56,107)(113,179,120,172)(114,180,121,173)(115,181,122,174)(116,182,123,175)(117,169,124,176)(118,170,125,177)(119,171,126,178)(127,190,134,183)(128,191,135,184)(129,192,136,185)(130,193,137,186)(131,194,138,187)(132,195,139,188)(133,196,140,189)(141,206,148,199)(142,207,149,200)(143,208,150,201)(144,209,151,202)(145,210,152,203)(146,197,153,204)(147,198,154,205)(155,216,162,223)(156,217,163,224)(157,218,164,211)(158,219,165,212)(159,220,166,213)(160,221,167,214)(161,222,168,215), (1,122,20,128)(2,121,21,127)(3,120,22,140)(4,119,23,139)(5,118,24,138)(6,117,25,137)(7,116,26,136)(8,115,27,135)(9,114,28,134)(10,113,15,133)(11,126,16,132)(12,125,17,131)(13,124,18,130)(14,123,19,129)(29,143,50,164)(30,142,51,163)(31,141,52,162)(32,154,53,161)(33,153,54,160)(34,152,55,159)(35,151,56,158)(36,150,43,157)(37,149,44,156)(38,148,45,155)(39,147,46,168)(40,146,47,167)(41,145,48,166)(42,144,49,165)(57,178,80,195)(58,177,81,194)(59,176,82,193)(60,175,83,192)(61,174,84,191)(62,173,71,190)(63,172,72,189)(64,171,73,188)(65,170,74,187)(66,169,75,186)(67,182,76,185)(68,181,77,184)(69,180,78,183)(70,179,79,196)(85,204,112,221)(86,203,99,220)(87,202,100,219)(88,201,101,218)(89,200,102,217)(90,199,103,216)(91,198,104,215)(92,197,105,214)(93,210,106,213)(94,209,107,212)(95,208,108,211)(96,207,109,224)(97,206,110,223)(98,205,111,222), (57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132)(141,162)(142,163)(143,164)(144,165)(145,166)(146,167)(147,168)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161)(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)(197,221)(198,222)(199,223)(200,224)(201,211)(202,212)(203,213)(204,214)(205,215)(206,216)(207,217)(208,218)(209,219)(210,220) );
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)], [(1,112,27,92),(2,99,28,93),(3,100,15,94),(4,101,16,95),(5,102,17,96),(6,103,18,97),(7,104,19,98),(8,105,20,85),(9,106,21,86),(10,107,22,87),(11,108,23,88),(12,109,24,89),(13,110,25,90),(14,111,26,91),(29,73,43,57),(30,74,44,58),(31,75,45,59),(32,76,46,60),(33,77,47,61),(34,78,48,62),(35,79,49,63),(36,80,50,64),(37,81,51,65),(38,82,52,66),(39,83,53,67),(40,84,54,68),(41,71,55,69),(42,72,56,70),(113,212,140,202),(114,213,127,203),(115,214,128,204),(116,215,129,205),(117,216,130,206),(118,217,131,207),(119,218,132,208),(120,219,133,209),(121,220,134,210),(122,221,135,197),(123,222,136,198),(124,223,137,199),(125,224,138,200),(126,211,139,201),(141,186,155,176),(142,187,156,177),(143,188,157,178),(144,189,158,179),(145,190,159,180),(146,191,160,181),(147,192,161,182),(148,193,162,169),(149,194,163,170),(150,195,164,171),(151,196,165,172),(152,183,166,173),(153,184,167,174),(154,185,168,175)], [(1,68,8,61),(2,69,9,62),(3,70,10,63),(4,57,11,64),(5,58,12,65),(6,59,13,66),(7,60,14,67),(15,72,22,79),(16,73,23,80),(17,74,24,81),(18,75,25,82),(19,76,26,83),(20,77,27,84),(21,78,28,71),(29,88,36,95),(30,89,37,96),(31,90,38,97),(32,91,39,98),(33,92,40,85),(34,93,41,86),(35,94,42,87),(43,108,50,101),(44,109,51,102),(45,110,52,103),(46,111,53,104),(47,112,54,105),(48,99,55,106),(49,100,56,107),(113,179,120,172),(114,180,121,173),(115,181,122,174),(116,182,123,175),(117,169,124,176),(118,170,125,177),(119,171,126,178),(127,190,134,183),(128,191,135,184),(129,192,136,185),(130,193,137,186),(131,194,138,187),(132,195,139,188),(133,196,140,189),(141,206,148,199),(142,207,149,200),(143,208,150,201),(144,209,151,202),(145,210,152,203),(146,197,153,204),(147,198,154,205),(155,216,162,223),(156,217,163,224),(157,218,164,211),(158,219,165,212),(159,220,166,213),(160,221,167,214),(161,222,168,215)], [(1,122,20,128),(2,121,21,127),(3,120,22,140),(4,119,23,139),(5,118,24,138),(6,117,25,137),(7,116,26,136),(8,115,27,135),(9,114,28,134),(10,113,15,133),(11,126,16,132),(12,125,17,131),(13,124,18,130),(14,123,19,129),(29,143,50,164),(30,142,51,163),(31,141,52,162),(32,154,53,161),(33,153,54,160),(34,152,55,159),(35,151,56,158),(36,150,43,157),(37,149,44,156),(38,148,45,155),(39,147,46,168),(40,146,47,167),(41,145,48,166),(42,144,49,165),(57,178,80,195),(58,177,81,194),(59,176,82,193),(60,175,83,192),(61,174,84,191),(62,173,71,190),(63,172,72,189),(64,171,73,188),(65,170,74,187),(66,169,75,186),(67,182,76,185),(68,181,77,184),(69,180,78,183),(70,179,79,196),(85,204,112,221),(86,203,99,220),(87,202,100,219),(88,201,101,218),(89,200,102,217),(90,199,103,216),(91,198,104,215),(92,197,105,214),(93,210,106,213),(94,209,107,212),(95,208,108,211),(96,207,109,224),(97,206,110,223),(98,205,111,222)], [(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(113,133),(114,134),(115,135),(116,136),(117,137),(118,138),(119,139),(120,140),(121,127),(122,128),(123,129),(124,130),(125,131),(126,132),(141,162),(142,163),(143,164),(144,165),(145,166),(146,167),(147,168),(148,155),(149,156),(150,157),(151,158),(152,159),(153,160),(154,161),(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),(197,221),(198,222),(199,223),(200,224),(201,211),(202,212),(203,213),(204,214),(205,215),(206,216),(207,217),(208,218),(209,219),(210,220)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4E | 4F | ··· | 4K | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 28 | 28 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 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 | C2 | D7 | D14 | D14 | D14 | D14 | 2+ 1+4 | 2- 1+4 | D4⋊6D14 | D4⋊8D14 | D4.10D14 |
| kernel | C14.862- 1+4 | C22⋊Dic14 | D14.D4 | D14⋊D4 | Dic7.D4 | C22.D28 | C28.3Q8 | D14.5D4 | D14⋊Q8 | D14⋊2Q8 | C28.48D4 | C28⋊7D4 | C28⋊2D4 | Dic7⋊D4 | C7×C22.D4 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 9 | 6 | 3 | 3 | 2 | 1 | 6 | 6 | 6 |
Matrix representation of C14.862- 1+4 ►in GL8(𝔽29)
| 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 21 | 16 | 0 | 0 | 0 | 0 | 0 |
| 21 | 3 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 7 | 9 |
| 18 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 26 | 14 | 18 | 21 | 0 | 0 | 0 | 0 |
| 15 | 3 | 8 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 7 | 14 | 12 | 24 |
| 0 | 0 | 0 | 0 | 24 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 21 | 10 | 7 | 15 |
| 8 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 18 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 15 | 3 | 8 | 11 | 0 | 0 | 0 | 0 |
| 26 | 14 | 18 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 0 | 24 | 0 |
| 0 | 0 | 0 | 0 | 14 | 23 | 8 | 27 |
| 0 | 0 | 0 | 0 | 2 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 | 4 | 11 | 6 |
| 5 | 6 | 3 | 0 | 0 | 0 | 0 | 0 |
| 6 | 5 | 0 | 3 | 0 | 0 | 0 | 0 |
| 18 | 9 | 24 | 23 | 0 | 0 | 0 | 0 |
| 9 | 18 | 23 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 17 | 7 | 25 |
| 0 | 0 | 0 | 0 | 9 | 5 | 2 | 21 |
| 0 | 0 | 0 | 0 | 0 | 1 | 19 | 14 |
| 0 | 0 | 0 | 0 | 18 | 19 | 3 | 23 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 25 | 28 | 0 | 0 | 0 | 0 | 0 |
| 25 | 16 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 15 | 23 | 0 | 28 |
G:=sub<GL(8,GF(29))| [20,0,3,21,0,0,0,0,0,20,21,3,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,13,8,0,16,0,0,0,0,0,9,0,0,0,0,0,0,0,0,13,7,0,0,0,0,0,0,0,9],[18,8,26,15,0,0,0,0,21,11,14,3,0,0,0,0,0,0,18,8,0,0,0,0,0,0,21,11,0,0,0,0,0,0,0,0,22,7,24,21,0,0,0,0,0,14,0,10,0,0,0,0,27,12,7,7,0,0,0,0,0,24,0,15],[8,18,15,26,0,0,0,0,11,21,3,14,0,0,0,0,0,0,8,18,0,0,0,0,0,0,11,21,0,0,0,0,0,0,0,0,26,14,2,3,0,0,0,0,0,23,0,4,0,0,0,0,24,8,3,11,0,0,0,0,0,27,0,6],[5,6,18,9,0,0,0,0,6,5,9,18,0,0,0,0,3,0,24,23,0,0,0,0,0,3,23,24,0,0,0,0,0,0,0,0,11,9,0,18,0,0,0,0,17,5,1,19,0,0,0,0,7,2,19,3,0,0,0,0,25,21,14,23],[1,0,16,25,0,0,0,0,0,1,25,16,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,22,15,0,0,0,0,0,1,0,23,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28] >;
C14.862- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{86}2_-^{1+4} % in TeX
G:=Group("C14.86ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1120);
// by ID
G=gap.SmallGroup(448,1120);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,184,1571,570,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=a^7*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e=a^7*b,d*c*d^-1=e*c*e=a^7*c,e*d*e=a^7*b^2*d>;
// generators/relations