?
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
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)
Generators and relations
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 >
►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)
(1 107 16 90)(2 108 17 91)(3 109 18 92)(4 110 19 93)(5 111 20 94)(6 112 21 95)(7 99 22 96)(8 100 23 97)(9 101 24 98)(10 102 25 85)(11 103 26 86)(12 104 27 87)(13 105 28 88)(14 106 15 89)(29 79 54 65)(30 80 55 66)(31 81 56 67)(32 82 43 68)(33 83 44 69)(34 84 45 70)(35 71 46 57)(36 72 47 58)(37 73 48 59)(38 74 49 60)(39 75 50 61)(40 76 51 62)(41 77 52 63)(42 78 53 64)(113 213 127 199)(114 214 128 200)(115 215 129 201)(116 216 130 202)(117 217 131 203)(118 218 132 204)(119 219 133 205)(120 220 134 206)(121 221 135 207)(122 222 136 208)(123 223 137 209)(124 224 138 210)(125 211 139 197)(126 212 140 198)(141 193 158 180)(142 194 159 181)(143 195 160 182)(144 196 161 169)(145 183 162 170)(146 184 163 171)(147 185 164 172)(148 186 165 173)(149 187 166 174)(150 188 167 175)(151 189 168 176)(152 190 155 177)(153 191 156 178)(154 192 157 179)
(1 70 8 63)(2 57 9 64)(3 58 10 65)(4 59 11 66)(5 60 12 67)(6 61 13 68)(7 62 14 69)(15 83 22 76)(16 84 23 77)(17 71 24 78)(18 72 25 79)(19 73 26 80)(20 74 27 81)(21 75 28 82)(29 92 36 85)(30 93 37 86)(31 94 38 87)(32 95 39 88)(33 96 40 89)(34 97 41 90)(35 98 42 91)(43 112 50 105)(44 99 51 106)(45 100 52 107)(46 101 53 108)(47 102 54 109)(48 103 55 110)(49 104 56 111)(113 176 120 169)(114 177 121 170)(115 178 122 171)(116 179 123 172)(117 180 124 173)(118 181 125 174)(119 182 126 175)(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)(141 210 148 203)(142 197 149 204)(143 198 150 205)(144 199 151 206)(145 200 152 207)(146 201 153 208)(147 202 154 209)(155 221 162 214)(156 222 163 215)(157 223 164 216)(158 224 165 217)(159 211 166 218)(160 212 167 219)(161 213 168 220)
(1 125 23 132)(2 124 24 131)(3 123 25 130)(4 122 26 129)(5 121 27 128)(6 120 28 127)(7 119 15 140)(8 118 16 139)(9 117 17 138)(10 116 18 137)(11 115 19 136)(12 114 20 135)(13 113 21 134)(14 126 22 133)(29 147 47 157)(30 146 48 156)(31 145 49 155)(32 144 50 168)(33 143 51 167)(34 142 52 166)(35 141 53 165)(36 154 54 164)(37 153 55 163)(38 152 56 162)(39 151 43 161)(40 150 44 160)(41 149 45 159)(42 148 46 158)(57 180 78 186)(58 179 79 185)(59 178 80 184)(60 177 81 183)(61 176 82 196)(62 175 83 195)(63 174 84 194)(64 173 71 193)(65 172 72 192)(66 171 73 191)(67 170 74 190)(68 169 75 189)(69 182 76 188)(70 181 77 187)(85 202 109 223)(86 201 110 222)(87 200 111 221)(88 199 112 220)(89 198 99 219)(90 197 100 218)(91 210 101 217)(92 209 102 216)(93 208 103 215)(94 207 104 214)(95 206 105 213)(96 205 106 212)(97 204 107 211)(98 203 108 224)
(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)(113 134)(114 135)(115 136)(116 137)(117 138)(118 139)(119 140)(120 127)(121 128)(122 129)(123 130)(124 131)(125 132)(126 133)(141 165)(142 166)(143 167)(144 168)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)(151 161)(152 162)(153 163)(154 164)(169 196)(170 183)(171 184)(172 185)(173 186)(174 187)(175 188)(176 189)(177 190)(178 191)(179 192)(180 193)(181 194)(182 195)(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), (1,107,16,90)(2,108,17,91)(3,109,18,92)(4,110,19,93)(5,111,20,94)(6,112,21,95)(7,99,22,96)(8,100,23,97)(9,101,24,98)(10,102,25,85)(11,103,26,86)(12,104,27,87)(13,105,28,88)(14,106,15,89)(29,79,54,65)(30,80,55,66)(31,81,56,67)(32,82,43,68)(33,83,44,69)(34,84,45,70)(35,71,46,57)(36,72,47,58)(37,73,48,59)(38,74,49,60)(39,75,50,61)(40,76,51,62)(41,77,52,63)(42,78,53,64)(113,213,127,199)(114,214,128,200)(115,215,129,201)(116,216,130,202)(117,217,131,203)(118,218,132,204)(119,219,133,205)(120,220,134,206)(121,221,135,207)(122,222,136,208)(123,223,137,209)(124,224,138,210)(125,211,139,197)(126,212,140,198)(141,193,158,180)(142,194,159,181)(143,195,160,182)(144,196,161,169)(145,183,162,170)(146,184,163,171)(147,185,164,172)(148,186,165,173)(149,187,166,174)(150,188,167,175)(151,189,168,176)(152,190,155,177)(153,191,156,178)(154,192,157,179), (1,70,8,63)(2,57,9,64)(3,58,10,65)(4,59,11,66)(5,60,12,67)(6,61,13,68)(7,62,14,69)(15,83,22,76)(16,84,23,77)(17,71,24,78)(18,72,25,79)(19,73,26,80)(20,74,27,81)(21,75,28,82)(29,92,36,85)(30,93,37,86)(31,94,38,87)(32,95,39,88)(33,96,40,89)(34,97,41,90)(35,98,42,91)(43,112,50,105)(44,99,51,106)(45,100,52,107)(46,101,53,108)(47,102,54,109)(48,103,55,110)(49,104,56,111)(113,176,120,169)(114,177,121,170)(115,178,122,171)(116,179,123,172)(117,180,124,173)(118,181,125,174)(119,182,126,175)(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)(141,210,148,203)(142,197,149,204)(143,198,150,205)(144,199,151,206)(145,200,152,207)(146,201,153,208)(147,202,154,209)(155,221,162,214)(156,222,163,215)(157,223,164,216)(158,224,165,217)(159,211,166,218)(160,212,167,219)(161,213,168,220), (1,125,23,132)(2,124,24,131)(3,123,25,130)(4,122,26,129)(5,121,27,128)(6,120,28,127)(7,119,15,140)(8,118,16,139)(9,117,17,138)(10,116,18,137)(11,115,19,136)(12,114,20,135)(13,113,21,134)(14,126,22,133)(29,147,47,157)(30,146,48,156)(31,145,49,155)(32,144,50,168)(33,143,51,167)(34,142,52,166)(35,141,53,165)(36,154,54,164)(37,153,55,163)(38,152,56,162)(39,151,43,161)(40,150,44,160)(41,149,45,159)(42,148,46,158)(57,180,78,186)(58,179,79,185)(59,178,80,184)(60,177,81,183)(61,176,82,196)(62,175,83,195)(63,174,84,194)(64,173,71,193)(65,172,72,192)(66,171,73,191)(67,170,74,190)(68,169,75,189)(69,182,76,188)(70,181,77,187)(85,202,109,223)(86,201,110,222)(87,200,111,221)(88,199,112,220)(89,198,99,219)(90,197,100,218)(91,210,101,217)(92,209,102,216)(93,208,103,215)(94,207,104,214)(95,206,105,213)(96,205,106,212)(97,204,107,211)(98,203,108,224), (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)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(141,165)(142,166)(143,167)(144,168)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160)(151,161)(152,162)(153,163)(154,164)(169,196)(170,183)(171,184)(172,185)(173,186)(174,187)(175,188)(176,189)(177,190)(178,191)(179,192)(180,193)(181,194)(182,195)(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), (1,107,16,90)(2,108,17,91)(3,109,18,92)(4,110,19,93)(5,111,20,94)(6,112,21,95)(7,99,22,96)(8,100,23,97)(9,101,24,98)(10,102,25,85)(11,103,26,86)(12,104,27,87)(13,105,28,88)(14,106,15,89)(29,79,54,65)(30,80,55,66)(31,81,56,67)(32,82,43,68)(33,83,44,69)(34,84,45,70)(35,71,46,57)(36,72,47,58)(37,73,48,59)(38,74,49,60)(39,75,50,61)(40,76,51,62)(41,77,52,63)(42,78,53,64)(113,213,127,199)(114,214,128,200)(115,215,129,201)(116,216,130,202)(117,217,131,203)(118,218,132,204)(119,219,133,205)(120,220,134,206)(121,221,135,207)(122,222,136,208)(123,223,137,209)(124,224,138,210)(125,211,139,197)(126,212,140,198)(141,193,158,180)(142,194,159,181)(143,195,160,182)(144,196,161,169)(145,183,162,170)(146,184,163,171)(147,185,164,172)(148,186,165,173)(149,187,166,174)(150,188,167,175)(151,189,168,176)(152,190,155,177)(153,191,156,178)(154,192,157,179), (1,70,8,63)(2,57,9,64)(3,58,10,65)(4,59,11,66)(5,60,12,67)(6,61,13,68)(7,62,14,69)(15,83,22,76)(16,84,23,77)(17,71,24,78)(18,72,25,79)(19,73,26,80)(20,74,27,81)(21,75,28,82)(29,92,36,85)(30,93,37,86)(31,94,38,87)(32,95,39,88)(33,96,40,89)(34,97,41,90)(35,98,42,91)(43,112,50,105)(44,99,51,106)(45,100,52,107)(46,101,53,108)(47,102,54,109)(48,103,55,110)(49,104,56,111)(113,176,120,169)(114,177,121,170)(115,178,122,171)(116,179,123,172)(117,180,124,173)(118,181,125,174)(119,182,126,175)(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)(141,210,148,203)(142,197,149,204)(143,198,150,205)(144,199,151,206)(145,200,152,207)(146,201,153,208)(147,202,154,209)(155,221,162,214)(156,222,163,215)(157,223,164,216)(158,224,165,217)(159,211,166,218)(160,212,167,219)(161,213,168,220), (1,125,23,132)(2,124,24,131)(3,123,25,130)(4,122,26,129)(5,121,27,128)(6,120,28,127)(7,119,15,140)(8,118,16,139)(9,117,17,138)(10,116,18,137)(11,115,19,136)(12,114,20,135)(13,113,21,134)(14,126,22,133)(29,147,47,157)(30,146,48,156)(31,145,49,155)(32,144,50,168)(33,143,51,167)(34,142,52,166)(35,141,53,165)(36,154,54,164)(37,153,55,163)(38,152,56,162)(39,151,43,161)(40,150,44,160)(41,149,45,159)(42,148,46,158)(57,180,78,186)(58,179,79,185)(59,178,80,184)(60,177,81,183)(61,176,82,196)(62,175,83,195)(63,174,84,194)(64,173,71,193)(65,172,72,192)(66,171,73,191)(67,170,74,190)(68,169,75,189)(69,182,76,188)(70,181,77,187)(85,202,109,223)(86,201,110,222)(87,200,111,221)(88,199,112,220)(89,198,99,219)(90,197,100,218)(91,210,101,217)(92,209,102,216)(93,208,103,215)(94,207,104,214)(95,206,105,213)(96,205,106,212)(97,204,107,211)(98,203,108,224), (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)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(141,165)(142,166)(143,167)(144,168)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160)(151,161)(152,162)(153,163)(154,164)(169,196)(170,183)(171,184)(172,185)(173,186)(174,187)(175,188)(176,189)(177,190)(178,191)(179,192)(180,193)(181,194)(182,195)(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)], [(1,107,16,90),(2,108,17,91),(3,109,18,92),(4,110,19,93),(5,111,20,94),(6,112,21,95),(7,99,22,96),(8,100,23,97),(9,101,24,98),(10,102,25,85),(11,103,26,86),(12,104,27,87),(13,105,28,88),(14,106,15,89),(29,79,54,65),(30,80,55,66),(31,81,56,67),(32,82,43,68),(33,83,44,69),(34,84,45,70),(35,71,46,57),(36,72,47,58),(37,73,48,59),(38,74,49,60),(39,75,50,61),(40,76,51,62),(41,77,52,63),(42,78,53,64),(113,213,127,199),(114,214,128,200),(115,215,129,201),(116,216,130,202),(117,217,131,203),(118,218,132,204),(119,219,133,205),(120,220,134,206),(121,221,135,207),(122,222,136,208),(123,223,137,209),(124,224,138,210),(125,211,139,197),(126,212,140,198),(141,193,158,180),(142,194,159,181),(143,195,160,182),(144,196,161,169),(145,183,162,170),(146,184,163,171),(147,185,164,172),(148,186,165,173),(149,187,166,174),(150,188,167,175),(151,189,168,176),(152,190,155,177),(153,191,156,178),(154,192,157,179)], [(1,70,8,63),(2,57,9,64),(3,58,10,65),(4,59,11,66),(5,60,12,67),(6,61,13,68),(7,62,14,69),(15,83,22,76),(16,84,23,77),(17,71,24,78),(18,72,25,79),(19,73,26,80),(20,74,27,81),(21,75,28,82),(29,92,36,85),(30,93,37,86),(31,94,38,87),(32,95,39,88),(33,96,40,89),(34,97,41,90),(35,98,42,91),(43,112,50,105),(44,99,51,106),(45,100,52,107),(46,101,53,108),(47,102,54,109),(48,103,55,110),(49,104,56,111),(113,176,120,169),(114,177,121,170),(115,178,122,171),(116,179,123,172),(117,180,124,173),(118,181,125,174),(119,182,126,175),(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),(141,210,148,203),(142,197,149,204),(143,198,150,205),(144,199,151,206),(145,200,152,207),(146,201,153,208),(147,202,154,209),(155,221,162,214),(156,222,163,215),(157,223,164,216),(158,224,165,217),(159,211,166,218),(160,212,167,219),(161,213,168,220)], [(1,125,23,132),(2,124,24,131),(3,123,25,130),(4,122,26,129),(5,121,27,128),(6,120,28,127),(7,119,15,140),(8,118,16,139),(9,117,17,138),(10,116,18,137),(11,115,19,136),(12,114,20,135),(13,113,21,134),(14,126,22,133),(29,147,47,157),(30,146,48,156),(31,145,49,155),(32,144,50,168),(33,143,51,167),(34,142,52,166),(35,141,53,165),(36,154,54,164),(37,153,55,163),(38,152,56,162),(39,151,43,161),(40,150,44,160),(41,149,45,159),(42,148,46,158),(57,180,78,186),(58,179,79,185),(59,178,80,184),(60,177,81,183),(61,176,82,196),(62,175,83,195),(63,174,84,194),(64,173,71,193),(65,172,72,192),(66,171,73,191),(67,170,74,190),(68,169,75,189),(69,182,76,188),(70,181,77,187),(85,202,109,223),(86,201,110,222),(87,200,111,221),(88,199,112,220),(89,198,99,219),(90,197,100,218),(91,210,101,217),(92,209,102,216),(93,208,103,215),(94,207,104,214),(95,206,105,213),(96,205,106,212),(97,204,107,211),(98,203,108,224)], [(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),(113,134),(114,135),(115,136),(116,137),(117,138),(118,139),(119,140),(120,127),(121,128),(122,129),(123,130),(124,131),(125,132),(126,133),(141,165),(142,166),(143,167),(144,168),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160),(151,161),(152,162),(153,163),(154,164),(169,196),(170,183),(171,184),(172,185),(173,186),(174,187),(175,188),(176,189),(177,190),(178,191),(179,192),(180,193),(181,194),(182,195),(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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