G = C14.352+ 1+4 order 448 = 26·7
35th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.352+ 1+4, C14.702- 1+4, C4⋊C4.91D14, C4⋊D4.8D7, (D4×Dic7)⋊18C2, C22⋊C4.6D14, (C2×D4).154D14, Dic7.Q8⋊12C2, C28.48D4⋊43C2, (C2×C28).625C23, (C2×C14).146C24, (C22×C4).221D14, C4⋊Dic7.45C22, C2.37(D4⋊6D14), C23.12(C22×D7), C22⋊Dic14⋊17C2, (D4×C14).120C22, C23.D14⋊16C2, C22.6(D4⋊2D7), (C22×C14).17C23, (C4×Dic7).93C22, (C2×Dic7).67C23, C22.167(C23×D7), C23.D7.23C22, C23.18D14⋊21C2, Dic7⋊C4.159C22, (C22×C28).311C22, C7⋊3(C22.33C24), (C2×Dic14).33C22, C2.28(D4.10D14), (C22×Dic7).107C22, C14.82(C2×C4○D4), (C7×C4⋊D4).8C2, (C2×Dic7⋊C4)⋊29C2, C2.34(C2×D4⋊2D7), (C2×C14).22(C4○D4), (C7×C4⋊C4).142C22, (C2×C4).174(C22×D7), (C7×C22⋊C4).11C22, SmallGroup(448,1055)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.352+ 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=ebe-1=a7b, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 844 in 218 conjugacy classes, 95 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic7, C28, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22.33C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C22×Dic7, C22×Dic7, C22×C28, D4×C14, D4×C14, C22⋊Dic14, C23.D14, Dic7.Q8, C2×Dic7⋊C4, C28.48D4, D4×Dic7, C23.18D14, C7×C4⋊D4, C14.352+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D7, C22.33C24, D4⋊2D7, C23×D7, C2×D4⋊2D7, D4⋊6D14, D4.10D14, C14.352+ 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 58 16 80)(2 59 17 81)(3 60 18 82)(4 61 19 83)(5 62 20 84)(6 63 21 71)(7 64 22 72)(8 65 23 73)(9 66 24 74)(10 67 25 75)(11 68 26 76)(12 69 27 77)(13 70 28 78)(14 57 15 79)(29 89 55 103)(30 90 56 104)(31 91 43 105)(32 92 44 106)(33 93 45 107)(34 94 46 108)(35 95 47 109)(36 96 48 110)(37 97 49 111)(38 98 50 112)(39 85 51 99)(40 86 52 100)(41 87 53 101)(42 88 54 102)(113 173 130 188)(114 174 131 189)(115 175 132 190)(116 176 133 191)(117 177 134 192)(118 178 135 193)(119 179 136 194)(120 180 137 195)(121 181 138 196)(122 182 139 183)(123 169 140 184)(124 170 127 185)(125 171 128 186)(126 172 129 187)(141 200 157 212)(142 201 158 213)(143 202 159 214)(144 203 160 215)(145 204 161 216)(146 205 162 217)(147 206 163 218)(148 207 164 219)(149 208 165 220)(150 209 166 221)(151 210 167 222)(152 197 168 223)(153 198 155 224)(154 199 156 211)
(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 71)(64 72)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 190)(176 191)(177 192)(178 193)(179 194)(180 195)(181 196)(182 183)(197 223)(198 224)(199 211)(200 212)(201 213)(202 214)(203 215)(204 216)(205 217)(206 218)(207 219)(208 220)(209 221)(210 222)
(1 151 16 167)(2 150 17 166)(3 149 18 165)(4 148 19 164)(5 147 20 163)(6 146 21 162)(7 145 22 161)(8 144 23 160)(9 143 24 159)(10 142 25 158)(11 141 26 157)(12 154 27 156)(13 153 28 155)(14 152 15 168)(29 115 55 132)(30 114 56 131)(31 113 43 130)(32 126 44 129)(33 125 45 128)(34 124 46 127)(35 123 47 140)(36 122 48 139)(37 121 49 138)(38 120 50 137)(39 119 51 136)(40 118 52 135)(41 117 53 134)(42 116 54 133)(57 204 79 216)(58 203 80 215)(59 202 81 214)(60 201 82 213)(61 200 83 212)(62 199 84 211)(63 198 71 224)(64 197 72 223)(65 210 73 222)(66 209 74 221)(67 208 75 220)(68 207 76 219)(69 206 77 218)(70 205 78 217)(85 172 99 187)(86 171 100 186)(87 170 101 185)(88 169 102 184)(89 182 103 183)(90 181 104 196)(91 180 105 195)(92 179 106 194)(93 178 107 193)(94 177 108 192)(95 176 109 191)(96 175 110 190)(97 174 111 189)(98 173 112 188)
(1 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 47 22 54)(16 48 23 55)(17 49 24 56)(18 50 25 43)(19 51 26 44)(20 52 27 45)(21 53 28 46)(57 88 64 95)(58 89 65 96)(59 90 66 97)(60 91 67 98)(61 92 68 85)(62 93 69 86)(63 94 70 87)(71 108 78 101)(72 109 79 102)(73 110 80 103)(74 111 81 104)(75 112 82 105)(76 99 83 106)(77 100 84 107)(113 165 120 158)(114 166 121 159)(115 167 122 160)(116 168 123 161)(117 155 124 162)(118 156 125 163)(119 157 126 164)(127 146 134 153)(128 147 135 154)(129 148 136 141)(130 149 137 142)(131 150 138 143)(132 151 139 144)(133 152 140 145)(169 223 176 216)(170 224 177 217)(171 211 178 218)(172 212 179 219)(173 213 180 220)(174 214 181 221)(175 215 182 222)(183 210 190 203)(184 197 191 204)(185 198 192 205)(186 199 193 206)(187 200 194 207)(188 201 195 208)(189 202 196 209)
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), (1,58,16,80)(2,59,17,81)(3,60,18,82)(4,61,19,83)(5,62,20,84)(6,63,21,71)(7,64,22,72)(8,65,23,73)(9,66,24,74)(10,67,25,75)(11,68,26,76)(12,69,27,77)(13,70,28,78)(14,57,15,79)(29,89,55,103)(30,90,56,104)(31,91,43,105)(32,92,44,106)(33,93,45,107)(34,94,46,108)(35,95,47,109)(36,96,48,110)(37,97,49,111)(38,98,50,112)(39,85,51,99)(40,86,52,100)(41,87,53,101)(42,88,54,102)(113,173,130,188)(114,174,131,189)(115,175,132,190)(116,176,133,191)(117,177,134,192)(118,178,135,193)(119,179,136,194)(120,180,137,195)(121,181,138,196)(122,182,139,183)(123,169,140,184)(124,170,127,185)(125,171,128,186)(126,172,129,187)(141,200,157,212)(142,201,158,213)(143,202,159,214)(144,203,160,215)(145,204,161,216)(146,205,162,217)(147,206,163,218)(148,207,164,219)(149,208,165,220)(150,209,166,221)(151,210,167,222)(152,197,168,223)(153,198,155,224)(154,199,156,211), (57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,71)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,190)(176,191)(177,192)(178,193)(179,194)(180,195)(181,196)(182,183)(197,223)(198,224)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)(205,217)(206,218)(207,219)(208,220)(209,221)(210,222), (1,151,16,167)(2,150,17,166)(3,149,18,165)(4,148,19,164)(5,147,20,163)(6,146,21,162)(7,145,22,161)(8,144,23,160)(9,143,24,159)(10,142,25,158)(11,141,26,157)(12,154,27,156)(13,153,28,155)(14,152,15,168)(29,115,55,132)(30,114,56,131)(31,113,43,130)(32,126,44,129)(33,125,45,128)(34,124,46,127)(35,123,47,140)(36,122,48,139)(37,121,49,138)(38,120,50,137)(39,119,51,136)(40,118,52,135)(41,117,53,134)(42,116,54,133)(57,204,79,216)(58,203,80,215)(59,202,81,214)(60,201,82,213)(61,200,83,212)(62,199,84,211)(63,198,71,224)(64,197,72,223)(65,210,73,222)(66,209,74,221)(67,208,75,220)(68,207,76,219)(69,206,77,218)(70,205,78,217)(85,172,99,187)(86,171,100,186)(87,170,101,185)(88,169,102,184)(89,182,103,183)(90,181,104,196)(91,180,105,195)(92,179,106,194)(93,178,107,193)(94,177,108,192)(95,176,109,191)(96,175,110,190)(97,174,111,189)(98,173,112,188), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,47,22,54)(16,48,23,55)(17,49,24,56)(18,50,25,43)(19,51,26,44)(20,52,27,45)(21,53,28,46)(57,88,64,95)(58,89,65,96)(59,90,66,97)(60,91,67,98)(61,92,68,85)(62,93,69,86)(63,94,70,87)(71,108,78,101)(72,109,79,102)(73,110,80,103)(74,111,81,104)(75,112,82,105)(76,99,83,106)(77,100,84,107)(113,165,120,158)(114,166,121,159)(115,167,122,160)(116,168,123,161)(117,155,124,162)(118,156,125,163)(119,157,126,164)(127,146,134,153)(128,147,135,154)(129,148,136,141)(130,149,137,142)(131,150,138,143)(132,151,139,144)(133,152,140,145)(169,223,176,216)(170,224,177,217)(171,211,178,218)(172,212,179,219)(173,213,180,220)(174,214,181,221)(175,215,182,222)(183,210,190,203)(184,197,191,204)(185,198,192,205)(186,199,193,206)(187,200,194,207)(188,201,195,208)(189,202,196,209)>;
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), (1,58,16,80)(2,59,17,81)(3,60,18,82)(4,61,19,83)(5,62,20,84)(6,63,21,71)(7,64,22,72)(8,65,23,73)(9,66,24,74)(10,67,25,75)(11,68,26,76)(12,69,27,77)(13,70,28,78)(14,57,15,79)(29,89,55,103)(30,90,56,104)(31,91,43,105)(32,92,44,106)(33,93,45,107)(34,94,46,108)(35,95,47,109)(36,96,48,110)(37,97,49,111)(38,98,50,112)(39,85,51,99)(40,86,52,100)(41,87,53,101)(42,88,54,102)(113,173,130,188)(114,174,131,189)(115,175,132,190)(116,176,133,191)(117,177,134,192)(118,178,135,193)(119,179,136,194)(120,180,137,195)(121,181,138,196)(122,182,139,183)(123,169,140,184)(124,170,127,185)(125,171,128,186)(126,172,129,187)(141,200,157,212)(142,201,158,213)(143,202,159,214)(144,203,160,215)(145,204,161,216)(146,205,162,217)(147,206,163,218)(148,207,164,219)(149,208,165,220)(150,209,166,221)(151,210,167,222)(152,197,168,223)(153,198,155,224)(154,199,156,211), (57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,71)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,190)(176,191)(177,192)(178,193)(179,194)(180,195)(181,196)(182,183)(197,223)(198,224)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216)(205,217)(206,218)(207,219)(208,220)(209,221)(210,222), (1,151,16,167)(2,150,17,166)(3,149,18,165)(4,148,19,164)(5,147,20,163)(6,146,21,162)(7,145,22,161)(8,144,23,160)(9,143,24,159)(10,142,25,158)(11,141,26,157)(12,154,27,156)(13,153,28,155)(14,152,15,168)(29,115,55,132)(30,114,56,131)(31,113,43,130)(32,126,44,129)(33,125,45,128)(34,124,46,127)(35,123,47,140)(36,122,48,139)(37,121,49,138)(38,120,50,137)(39,119,51,136)(40,118,52,135)(41,117,53,134)(42,116,54,133)(57,204,79,216)(58,203,80,215)(59,202,81,214)(60,201,82,213)(61,200,83,212)(62,199,84,211)(63,198,71,224)(64,197,72,223)(65,210,73,222)(66,209,74,221)(67,208,75,220)(68,207,76,219)(69,206,77,218)(70,205,78,217)(85,172,99,187)(86,171,100,186)(87,170,101,185)(88,169,102,184)(89,182,103,183)(90,181,104,196)(91,180,105,195)(92,179,106,194)(93,178,107,193)(94,177,108,192)(95,176,109,191)(96,175,110,190)(97,174,111,189)(98,173,112,188), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,47,22,54)(16,48,23,55)(17,49,24,56)(18,50,25,43)(19,51,26,44)(20,52,27,45)(21,53,28,46)(57,88,64,95)(58,89,65,96)(59,90,66,97)(60,91,67,98)(61,92,68,85)(62,93,69,86)(63,94,70,87)(71,108,78,101)(72,109,79,102)(73,110,80,103)(74,111,81,104)(75,112,82,105)(76,99,83,106)(77,100,84,107)(113,165,120,158)(114,166,121,159)(115,167,122,160)(116,168,123,161)(117,155,124,162)(118,156,125,163)(119,157,126,164)(127,146,134,153)(128,147,135,154)(129,148,136,141)(130,149,137,142)(131,150,138,143)(132,151,139,144)(133,152,140,145)(169,223,176,216)(170,224,177,217)(171,211,178,218)(172,212,179,219)(173,213,180,220)(174,214,181,221)(175,215,182,222)(183,210,190,203)(184,197,191,204)(185,198,192,205)(186,199,193,206)(187,200,194,207)(188,201,195,208)(189,202,196,209) );
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)], [(1,58,16,80),(2,59,17,81),(3,60,18,82),(4,61,19,83),(5,62,20,84),(6,63,21,71),(7,64,22,72),(8,65,23,73),(9,66,24,74),(10,67,25,75),(11,68,26,76),(12,69,27,77),(13,70,28,78),(14,57,15,79),(29,89,55,103),(30,90,56,104),(31,91,43,105),(32,92,44,106),(33,93,45,107),(34,94,46,108),(35,95,47,109),(36,96,48,110),(37,97,49,111),(38,98,50,112),(39,85,51,99),(40,86,52,100),(41,87,53,101),(42,88,54,102),(113,173,130,188),(114,174,131,189),(115,175,132,190),(116,176,133,191),(117,177,134,192),(118,178,135,193),(119,179,136,194),(120,180,137,195),(121,181,138,196),(122,182,139,183),(123,169,140,184),(124,170,127,185),(125,171,128,186),(126,172,129,187),(141,200,157,212),(142,201,158,213),(143,202,159,214),(144,203,160,215),(145,204,161,216),(146,205,162,217),(147,206,163,218),(148,207,164,219),(149,208,165,220),(150,209,166,221),(151,210,167,222),(152,197,168,223),(153,198,155,224),(154,199,156,211)], [(57,79),(58,80),(59,81),(60,82),(61,83),(62,84),(63,71),(64,72),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(169,184),(170,185),(171,186),(172,187),(173,188),(174,189),(175,190),(176,191),(177,192),(178,193),(179,194),(180,195),(181,196),(182,183),(197,223),(198,224),(199,211),(200,212),(201,213),(202,214),(203,215),(204,216),(205,217),(206,218),(207,219),(208,220),(209,221),(210,222)], [(1,151,16,167),(2,150,17,166),(3,149,18,165),(4,148,19,164),(5,147,20,163),(6,146,21,162),(7,145,22,161),(8,144,23,160),(9,143,24,159),(10,142,25,158),(11,141,26,157),(12,154,27,156),(13,153,28,155),(14,152,15,168),(29,115,55,132),(30,114,56,131),(31,113,43,130),(32,126,44,129),(33,125,45,128),(34,124,46,127),(35,123,47,140),(36,122,48,139),(37,121,49,138),(38,120,50,137),(39,119,51,136),(40,118,52,135),(41,117,53,134),(42,116,54,133),(57,204,79,216),(58,203,80,215),(59,202,81,214),(60,201,82,213),(61,200,83,212),(62,199,84,211),(63,198,71,224),(64,197,72,223),(65,210,73,222),(66,209,74,221),(67,208,75,220),(68,207,76,219),(69,206,77,218),(70,205,78,217),(85,172,99,187),(86,171,100,186),(87,170,101,185),(88,169,102,184),(89,182,103,183),(90,181,104,196),(91,180,105,195),(92,179,106,194),(93,178,107,193),(94,177,108,192),(95,176,109,191),(96,175,110,190),(97,174,111,189),(98,173,112,188)], [(1,36,8,29),(2,37,9,30),(3,38,10,31),(4,39,11,32),(5,40,12,33),(6,41,13,34),(7,42,14,35),(15,47,22,54),(16,48,23,55),(17,49,24,56),(18,50,25,43),(19,51,26,44),(20,52,27,45),(21,53,28,46),(57,88,64,95),(58,89,65,96),(59,90,66,97),(60,91,67,98),(61,92,68,85),(62,93,69,86),(63,94,70,87),(71,108,78,101),(72,109,79,102),(73,110,80,103),(74,111,81,104),(75,112,82,105),(76,99,83,106),(77,100,84,107),(113,165,120,158),(114,166,121,159),(115,167,122,160),(116,168,123,161),(117,155,124,162),(118,156,125,163),(119,157,126,164),(127,146,134,153),(128,147,135,154),(129,148,136,141),(130,149,137,142),(131,150,138,143),(132,151,139,144),(133,152,140,145),(169,223,176,216),(170,224,177,217),(171,211,178,218),(172,212,179,219),(173,213,180,220),(174,214,181,221),(175,215,182,222),(183,210,190,203),(184,197,191,204),(185,198,192,205),(186,199,193,206),(187,200,194,207),(188,201,195,208),(189,202,196,209)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 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 | 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 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 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 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | 2- 1+4 | D4⋊2D7 | D4⋊6D14 | D4.10D14 |
| kernel | C14.352+ 1+4 | C22⋊Dic14 | C23.D14 | Dic7.Q8 | C2×Dic7⋊C4 | C28.48D4 | D4×Dic7 | C23.18D14 | C7×C4⋊D4 | C4⋊D4 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 4 | 1 | 3 | 4 | 6 | 3 | 3 | 9 | 1 | 1 | 6 | 6 | 6 |
Matrix representation of C14.352+ 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 25 | 0 | 0 |
| 0 | 0 | 4 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 21 |
| 0 | 0 | 0 | 0 | 4 | 12 |
| 13 | 8 | 0 | 0 | 0 | 0 |
| 8 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 14 | 4 |
| 0 | 0 | 0 | 28 | 23 | 27 |
| 0 | 0 | 28 | 27 | 1 | 0 |
| 0 | 0 | 3 | 7 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 15 | 25 |
| 0 | 0 | 0 | 1 | 6 | 2 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 19 | 0 | 0 |
| 0 | 0 | 13 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 15 |
| 0 | 0 | 0 | 0 | 13 | 6 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 2 | 0 | 0 |
| 0 | 0 | 27 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 4 |
| 0 | 0 | 0 | 0 | 27 | 3 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,11,4,0,0,0,0,25,25,0,0,0,0,0,0,24,4,0,0,0,0,21,12],[13,8,0,0,0,0,8,16,0,0,0,0,0,0,28,0,28,3,0,0,0,28,27,7,0,0,14,23,1,0,0,0,4,27,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,15,6,28,0,0,0,25,2,0,28],[0,17,0,0,0,0,12,0,0,0,0,0,0,0,10,13,0,0,0,0,19,19,0,0,0,0,0,0,23,13,0,0,0,0,15,6],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,18,27,0,0,0,0,2,11,0,0,0,0,0,0,26,27,0,0,0,0,4,3] >;
C14.352+ 1+4 in GAP, Magma, Sage, TeX
C_{14}._{35}2_+^{1+4} % in TeX
G:=Group("C14.35ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1055);
// by ID
G=gap.SmallGroup(448,1055);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,219,675,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=b^2,e^2=a^7,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=a^7*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations