G = C14.692+ 1+4 order 448 = 26·7
69th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.692+ 1+4, C14.862- 1+4, C28⋊7D4⋊15C2, C28⋊2D4⋊32C2, C4⋊C4.109D14, D14⋊D4⋊36C2, D14⋊Q8⋊33C2, D14⋊2Q8⋊34C2, (C2×D4).106D14, C22⋊C4.32D14, C4.Dic14⋊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, C2.47(D4⋊8D14), C2.71(D4⋊6D14), Dic7.D4⋊36C2, C22⋊Dic14⋊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 C14.692+ 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 [×3], C2 [×4], C4 [×11], C22, C22 [×12], C7, C2×C4 [×5], C2×C4 [×10], D4 [×6], Q8 [×2], C23 [×2], C23 [×2], D7 [×2], C14 [×3], C14 [×2], C42, C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×5], C2×Q8 [×2], Dic7 [×6], C28 [×5], D14 [×6], C2×C14, C2×C14 [×6], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4, C22.D4 [×3], C4.4D4 [×2], C42.C2, Dic14 [×2], C4×D7 [×2], D28, C2×Dic7 [×6], C2×Dic7, C7⋊D4 [×4], C2×C28 [×5], C2×C28, C7×D4, C22×D7 [×2], C22×C14 [×2], C22.56C24, C4×Dic7, Dic7⋊C4 [×4], C4⋊Dic7 [×4], D14⋊C4 [×6], C23.D7 [×3], C7×C22⋊C4 [×3], C7×C4⋊C4 [×2], C2×Dic14 [×2], C2×C4×D7 [×2], C2×D28, C22×Dic7, C2×C7⋊D4 [×4], C22×C28, D4×C14, C22⋊Dic14, D14.D4, D14⋊D4, Dic7.D4 [×2], C22.D28, C4.Dic14, D14.5D4, D14⋊Q8, D14⋊2Q8, C28.48D4, C28⋊7D4, C28⋊2D4, Dic7⋊D4, C7×C22.D4, C14.692+ 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ 1+4 [×2], 2- 1+4, C22×D7 [×7], C22.56C24, C23×D7, D4⋊6D14, D4⋊8D14, D4.10D14, C14.692+ 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 16 98)(2 99 17 85)(3 100 18 86)(4 101 19 87)(5 102 20 88)(6 103 21 89)(7 104 22 90)(8 105 23 91)(9 106 24 92)(10 107 25 93)(11 108 26 94)(12 109 27 95)(13 110 28 96)(14 111 15 97)(29 84 43 60)(30 71 44 61)(31 72 45 62)(32 73 46 63)(33 74 47 64)(34 75 48 65)(35 76 49 66)(36 77 50 67)(37 78 51 68)(38 79 52 69)(39 80 53 70)(40 81 54 57)(41 82 55 58)(42 83 56 59)(113 220 138 206)(114 221 139 207)(115 222 140 208)(116 223 127 209)(117 224 128 210)(118 211 129 197)(119 212 130 198)(120 213 131 199)(121 214 132 200)(122 215 133 201)(123 216 134 202)(124 217 135 203)(125 218 136 204)(126 219 137 205)(141 189 157 174)(142 190 158 175)(143 191 159 176)(144 192 160 177)(145 193 161 178)(146 194 162 179)(147 195 163 180)(148 196 164 181)(149 183 165 182)(150 184 166 169)(151 185 167 170)(152 186 168 171)(153 187 155 172)(154 188 156 173)
(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 77 22 84)(16 78 23 71)(17 79 24 72)(18 80 25 73)(19 81 26 74)(20 82 27 75)(21 83 28 76)(29 97 36 90)(30 98 37 91)(31 85 38 92)(32 86 39 93)(33 87 40 94)(34 88 41 95)(35 89 42 96)(43 111 50 104)(44 112 51 105)(45 99 52 106)(46 100 53 107)(47 101 54 108)(48 102 55 109)(49 103 56 110)(113 171 120 178)(114 172 121 179)(115 173 122 180)(116 174 123 181)(117 175 124 182)(118 176 125 169)(119 177 126 170)(127 189 134 196)(128 190 135 183)(129 191 136 184)(130 192 137 185)(131 193 138 186)(132 194 139 187)(133 195 140 188)(141 202 148 209)(142 203 149 210)(143 204 150 197)(144 205 151 198)(145 206 152 199)(146 207 153 200)(147 208 154 201)(155 214 162 221)(156 215 163 222)(157 216 164 223)(158 217 165 224)(159 218 166 211)(160 219 167 212)(161 220 168 213)
(1 126 23 130)(2 125 24 129)(3 124 25 128)(4 123 26 127)(5 122 27 140)(6 121 28 139)(7 120 15 138)(8 119 16 137)(9 118 17 136)(10 117 18 135)(11 116 19 134)(12 115 20 133)(13 114 21 132)(14 113 22 131)(29 152 50 161)(30 151 51 160)(31 150 52 159)(32 149 53 158)(33 148 54 157)(34 147 55 156)(35 146 56 155)(36 145 43 168)(37 144 44 167)(38 143 45 166)(39 142 46 165)(40 141 47 164)(41 154 48 163)(42 153 49 162)(57 174 74 196)(58 173 75 195)(59 172 76 194)(60 171 77 193)(61 170 78 192)(62 169 79 191)(63 182 80 190)(64 181 81 189)(65 180 82 188)(66 179 83 187)(67 178 84 186)(68 177 71 185)(69 176 72 184)(70 175 73 183)(85 204 106 211)(86 203 107 224)(87 202 108 223)(88 201 109 222)(89 200 110 221)(90 199 111 220)(91 198 112 219)(92 197 99 218)(93 210 100 217)(94 209 101 216)(95 208 102 215)(96 207 103 214)(97 206 104 213)(98 205 105 212)
(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)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)(121 139)(122 140)(123 127)(124 128)(125 129)(126 130)(141 164)(142 165)(143 166)(144 167)(145 168)(146 155)(147 156)(148 157)(149 158)(150 159)(151 160)(152 161)(153 162)(154 163)(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)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)
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), (1,112,16,98)(2,99,17,85)(3,100,18,86)(4,101,19,87)(5,102,20,88)(6,103,21,89)(7,104,22,90)(8,105,23,91)(9,106,24,92)(10,107,25,93)(11,108,26,94)(12,109,27,95)(13,110,28,96)(14,111,15,97)(29,84,43,60)(30,71,44,61)(31,72,45,62)(32,73,46,63)(33,74,47,64)(34,75,48,65)(35,76,49,66)(36,77,50,67)(37,78,51,68)(38,79,52,69)(39,80,53,70)(40,81,54,57)(41,82,55,58)(42,83,56,59)(113,220,138,206)(114,221,139,207)(115,222,140,208)(116,223,127,209)(117,224,128,210)(118,211,129,197)(119,212,130,198)(120,213,131,199)(121,214,132,200)(122,215,133,201)(123,216,134,202)(124,217,135,203)(125,218,136,204)(126,219,137,205)(141,189,157,174)(142,190,158,175)(143,191,159,176)(144,192,160,177)(145,193,161,178)(146,194,162,179)(147,195,163,180)(148,196,164,181)(149,183,165,182)(150,184,166,169)(151,185,167,170)(152,186,168,171)(153,187,155,172)(154,188,156,173), (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)(15,77,22,84)(16,78,23,71)(17,79,24,72)(18,80,25,73)(19,81,26,74)(20,82,27,75)(21,83,28,76)(29,97,36,90)(30,98,37,91)(31,85,38,92)(32,86,39,93)(33,87,40,94)(34,88,41,95)(35,89,42,96)(43,111,50,104)(44,112,51,105)(45,99,52,106)(46,100,53,107)(47,101,54,108)(48,102,55,109)(49,103,56,110)(113,171,120,178)(114,172,121,179)(115,173,122,180)(116,174,123,181)(117,175,124,182)(118,176,125,169)(119,177,126,170)(127,189,134,196)(128,190,135,183)(129,191,136,184)(130,192,137,185)(131,193,138,186)(132,194,139,187)(133,195,140,188)(141,202,148,209)(142,203,149,210)(143,204,150,197)(144,205,151,198)(145,206,152,199)(146,207,153,200)(147,208,154,201)(155,214,162,221)(156,215,163,222)(157,216,164,223)(158,217,165,224)(159,218,166,211)(160,219,167,212)(161,220,168,213), (1,126,23,130)(2,125,24,129)(3,124,25,128)(4,123,26,127)(5,122,27,140)(6,121,28,139)(7,120,15,138)(8,119,16,137)(9,118,17,136)(10,117,18,135)(11,116,19,134)(12,115,20,133)(13,114,21,132)(14,113,22,131)(29,152,50,161)(30,151,51,160)(31,150,52,159)(32,149,53,158)(33,148,54,157)(34,147,55,156)(35,146,56,155)(36,145,43,168)(37,144,44,167)(38,143,45,166)(39,142,46,165)(40,141,47,164)(41,154,48,163)(42,153,49,162)(57,174,74,196)(58,173,75,195)(59,172,76,194)(60,171,77,193)(61,170,78,192)(62,169,79,191)(63,182,80,190)(64,181,81,189)(65,180,82,188)(66,179,83,187)(67,178,84,186)(68,177,71,185)(69,176,72,184)(70,175,73,183)(85,204,106,211)(86,203,107,224)(87,202,108,223)(88,201,109,222)(89,200,110,221)(90,199,111,220)(91,198,112,219)(92,197,99,218)(93,210,100,217)(94,209,101,216)(95,208,102,215)(96,207,103,214)(97,206,104,213)(98,205,105,212), (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)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)(122,140)(123,127)(124,128)(125,129)(126,130)(141,164)(142,165)(143,166)(144,167)(145,168)(146,155)(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(154,163)(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)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224)>;
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), (1,112,16,98)(2,99,17,85)(3,100,18,86)(4,101,19,87)(5,102,20,88)(6,103,21,89)(7,104,22,90)(8,105,23,91)(9,106,24,92)(10,107,25,93)(11,108,26,94)(12,109,27,95)(13,110,28,96)(14,111,15,97)(29,84,43,60)(30,71,44,61)(31,72,45,62)(32,73,46,63)(33,74,47,64)(34,75,48,65)(35,76,49,66)(36,77,50,67)(37,78,51,68)(38,79,52,69)(39,80,53,70)(40,81,54,57)(41,82,55,58)(42,83,56,59)(113,220,138,206)(114,221,139,207)(115,222,140,208)(116,223,127,209)(117,224,128,210)(118,211,129,197)(119,212,130,198)(120,213,131,199)(121,214,132,200)(122,215,133,201)(123,216,134,202)(124,217,135,203)(125,218,136,204)(126,219,137,205)(141,189,157,174)(142,190,158,175)(143,191,159,176)(144,192,160,177)(145,193,161,178)(146,194,162,179)(147,195,163,180)(148,196,164,181)(149,183,165,182)(150,184,166,169)(151,185,167,170)(152,186,168,171)(153,187,155,172)(154,188,156,173), (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)(15,77,22,84)(16,78,23,71)(17,79,24,72)(18,80,25,73)(19,81,26,74)(20,82,27,75)(21,83,28,76)(29,97,36,90)(30,98,37,91)(31,85,38,92)(32,86,39,93)(33,87,40,94)(34,88,41,95)(35,89,42,96)(43,111,50,104)(44,112,51,105)(45,99,52,106)(46,100,53,107)(47,101,54,108)(48,102,55,109)(49,103,56,110)(113,171,120,178)(114,172,121,179)(115,173,122,180)(116,174,123,181)(117,175,124,182)(118,176,125,169)(119,177,126,170)(127,189,134,196)(128,190,135,183)(129,191,136,184)(130,192,137,185)(131,193,138,186)(132,194,139,187)(133,195,140,188)(141,202,148,209)(142,203,149,210)(143,204,150,197)(144,205,151,198)(145,206,152,199)(146,207,153,200)(147,208,154,201)(155,214,162,221)(156,215,163,222)(157,216,164,223)(158,217,165,224)(159,218,166,211)(160,219,167,212)(161,220,168,213), (1,126,23,130)(2,125,24,129)(3,124,25,128)(4,123,26,127)(5,122,27,140)(6,121,28,139)(7,120,15,138)(8,119,16,137)(9,118,17,136)(10,117,18,135)(11,116,19,134)(12,115,20,133)(13,114,21,132)(14,113,22,131)(29,152,50,161)(30,151,51,160)(31,150,52,159)(32,149,53,158)(33,148,54,157)(34,147,55,156)(35,146,56,155)(36,145,43,168)(37,144,44,167)(38,143,45,166)(39,142,46,165)(40,141,47,164)(41,154,48,163)(42,153,49,162)(57,174,74,196)(58,173,75,195)(59,172,76,194)(60,171,77,193)(61,170,78,192)(62,169,79,191)(63,182,80,190)(64,181,81,189)(65,180,82,188)(66,179,83,187)(67,178,84,186)(68,177,71,185)(69,176,72,184)(70,175,73,183)(85,204,106,211)(86,203,107,224)(87,202,108,223)(88,201,109,222)(89,200,110,221)(90,199,111,220)(91,198,112,219)(92,197,99,218)(93,210,100,217)(94,209,101,216)(95,208,102,215)(96,207,103,214)(97,206,104,213)(98,205,105,212), (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)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)(122,140)(123,127)(124,128)(125,129)(126,130)(141,164)(142,165)(143,166)(144,167)(145,168)(146,155)(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(154,163)(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)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224) );
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)], [(1,112,16,98),(2,99,17,85),(3,100,18,86),(4,101,19,87),(5,102,20,88),(6,103,21,89),(7,104,22,90),(8,105,23,91),(9,106,24,92),(10,107,25,93),(11,108,26,94),(12,109,27,95),(13,110,28,96),(14,111,15,97),(29,84,43,60),(30,71,44,61),(31,72,45,62),(32,73,46,63),(33,74,47,64),(34,75,48,65),(35,76,49,66),(36,77,50,67),(37,78,51,68),(38,79,52,69),(39,80,53,70),(40,81,54,57),(41,82,55,58),(42,83,56,59),(113,220,138,206),(114,221,139,207),(115,222,140,208),(116,223,127,209),(117,224,128,210),(118,211,129,197),(119,212,130,198),(120,213,131,199),(121,214,132,200),(122,215,133,201),(123,216,134,202),(124,217,135,203),(125,218,136,204),(126,219,137,205),(141,189,157,174),(142,190,158,175),(143,191,159,176),(144,192,160,177),(145,193,161,178),(146,194,162,179),(147,195,163,180),(148,196,164,181),(149,183,165,182),(150,184,166,169),(151,185,167,170),(152,186,168,171),(153,187,155,172),(154,188,156,173)], [(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),(15,77,22,84),(16,78,23,71),(17,79,24,72),(18,80,25,73),(19,81,26,74),(20,82,27,75),(21,83,28,76),(29,97,36,90),(30,98,37,91),(31,85,38,92),(32,86,39,93),(33,87,40,94),(34,88,41,95),(35,89,42,96),(43,111,50,104),(44,112,51,105),(45,99,52,106),(46,100,53,107),(47,101,54,108),(48,102,55,109),(49,103,56,110),(113,171,120,178),(114,172,121,179),(115,173,122,180),(116,174,123,181),(117,175,124,182),(118,176,125,169),(119,177,126,170),(127,189,134,196),(128,190,135,183),(129,191,136,184),(130,192,137,185),(131,193,138,186),(132,194,139,187),(133,195,140,188),(141,202,148,209),(142,203,149,210),(143,204,150,197),(144,205,151,198),(145,206,152,199),(146,207,153,200),(147,208,154,201),(155,214,162,221),(156,215,163,222),(157,216,164,223),(158,217,165,224),(159,218,166,211),(160,219,167,212),(161,220,168,213)], [(1,126,23,130),(2,125,24,129),(3,124,25,128),(4,123,26,127),(5,122,27,140),(6,121,28,139),(7,120,15,138),(8,119,16,137),(9,118,17,136),(10,117,18,135),(11,116,19,134),(12,115,20,133),(13,114,21,132),(14,113,22,131),(29,152,50,161),(30,151,51,160),(31,150,52,159),(32,149,53,158),(33,148,54,157),(34,147,55,156),(35,146,56,155),(36,145,43,168),(37,144,44,167),(38,143,45,166),(39,142,46,165),(40,141,47,164),(41,154,48,163),(42,153,49,162),(57,174,74,196),(58,173,75,195),(59,172,76,194),(60,171,77,193),(61,170,78,192),(62,169,79,191),(63,182,80,190),(64,181,81,189),(65,180,82,188),(66,179,83,187),(67,178,84,186),(68,177,71,185),(69,176,72,184),(70,175,73,183),(85,204,106,211),(86,203,107,224),(87,202,108,223),(88,201,109,222),(89,200,110,221),(90,199,111,220),(91,198,112,219),(92,197,99,218),(93,210,100,217),(94,209,101,216),(95,208,102,215),(96,207,103,214),(97,206,104,213),(98,205,105,212)], [(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),(113,131),(114,132),(115,133),(116,134),(117,135),(118,136),(119,137),(120,138),(121,139),(122,140),(123,127),(124,128),(125,129),(126,130),(141,164),(142,165),(143,166),(144,167),(145,168),(146,155),(147,156),(148,157),(149,158),(150,159),(151,160),(152,161),(153,162),(154,163),(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),(197,211),(198,212),(199,213),(200,214),(201,215),(202,216),(203,217),(204,218),(205,219),(206,220),(207,221),(208,222),(209,223),(210,224)])
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 | C14.692+ 1+4 | C22⋊Dic14 | D14.D4 | D14⋊D4 | Dic7.D4 | C22.D28 | C4.Dic14 | 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 C14.692+ 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] >;
C14.692+ 1+4 in GAP, Magma, Sage, TeX
C_{14}._{69}2_+^{1+4} % in TeX
G:=Group("C14.69ES+(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