?
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×C14).146C24, (C2×C28).625C23, (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, (C2×Dic7).67C23, (C4×Dic7).93C22, 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
Subgroups: 844 in 218 conjugacy classes, 95 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×2], C22 [×8], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×14], D4 [×5], Q8, C23, C23 [×2], C14 [×3], C14 [×4], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×13], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, Dic7 [×8], C28 [×4], C2×C14, C2×C14 [×2], C2×C14 [×8], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8 [×3], C22.D4 [×4], C42.C2 [×2], C42⋊2C2 [×2], Dic14, C2×Dic7 [×8], C2×Dic7 [×5], C2×C28 [×2], C2×C28 [×2], C2×C28, C7×D4 [×5], C22×C14, C22×C14 [×2], C22.33C24, C4×Dic7 [×2], Dic7⋊C4 [×10], C4⋊Dic7, C4⋊Dic7 [×2], C23.D7 [×8], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14, C22×Dic7 [×2], C22×Dic7 [×2], C22×C28, D4×C14, D4×C14 [×2], C22⋊Dic14 [×2], C23.D14 [×2], Dic7.Q8 [×2], C2×Dic7⋊C4, C28.48D4, D4×Dic7 [×2], C23.18D14 [×4], C7×C4⋊D4, C14.352+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×D7 [×7], C22.33C24, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, D4⋊6D14, D4.10D14, C14.352+ (1+4)
Generators and relations
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 >
►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 63 27 77)(2 64 28 78)(3 65 15 79)(4 66 16 80)(5 67 17 81)(6 68 18 82)(7 69 19 83)(8 70 20 84)(9 57 21 71)(10 58 22 72)(11 59 23 73)(12 60 24 74)(13 61 25 75)(14 62 26 76)(29 95 43 110)(30 96 44 111)(31 97 45 112)(32 98 46 99)(33 85 47 100)(34 86 48 101)(35 87 49 102)(36 88 50 103)(37 89 51 104)(38 90 52 105)(39 91 53 106)(40 92 54 107)(41 93 55 108)(42 94 56 109)(113 181 127 196)(114 182 128 183)(115 169 129 184)(116 170 130 185)(117 171 131 186)(118 172 132 187)(119 173 133 188)(120 174 134 189)(121 175 135 190)(122 176 136 191)(123 177 137 192)(124 178 138 193)(125 179 139 194)(126 180 140 195)(141 197 166 221)(142 198 167 222)(143 199 168 223)(144 200 155 224)(145 201 156 211)(146 202 157 212)(147 203 158 213)(148 204 159 214)(149 205 160 215)(150 206 161 216)(151 207 162 217)(152 208 163 218)(153 209 164 219)(154 210 165 220)
(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)(91 106)(92 107)(93 108)(94 109)(95 110)(96 111)(97 112)(98 99)(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)(197 221)(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)
(1 154 27 165)(2 153 28 164)(3 152 15 163)(4 151 16 162)(5 150 17 161)(6 149 18 160)(7 148 19 159)(8 147 20 158)(9 146 21 157)(10 145 22 156)(11 144 23 155)(12 143 24 168)(13 142 25 167)(14 141 26 166)(29 122 43 136)(30 121 44 135)(31 120 45 134)(32 119 46 133)(33 118 47 132)(34 117 48 131)(35 116 49 130)(36 115 50 129)(37 114 51 128)(38 113 52 127)(39 126 53 140)(40 125 54 139)(41 124 55 138)(42 123 56 137)(57 209 71 219)(58 208 72 218)(59 207 73 217)(60 206 74 216)(61 205 75 215)(62 204 76 214)(63 203 77 213)(64 202 78 212)(65 201 79 211)(66 200 80 224)(67 199 81 223)(68 198 82 222)(69 197 83 221)(70 210 84 220)(85 179 100 194)(86 178 101 193)(87 177 102 192)(88 176 103 191)(89 175 104 190)(90 174 105 189)(91 173 106 188)(92 172 107 187)(93 171 108 186)(94 170 109 185)(95 169 110 184)(96 182 111 183)(97 181 112 196)(98 180 99 195)
(1 40 8 33)(2 41 9 34)(3 42 10 35)(4 29 11 36)(5 30 12 37)(6 31 13 38)(7 32 14 39)(15 56 22 49)(16 43 23 50)(17 44 24 51)(18 45 25 52)(19 46 26 53)(20 47 27 54)(21 48 28 55)(57 93 64 86)(58 94 65 87)(59 95 66 88)(60 96 67 89)(61 97 68 90)(62 98 69 91)(63 85 70 92)(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)(113 160 120 167)(114 161 121 168)(115 162 122 155)(116 163 123 156)(117 164 124 157)(118 165 125 158)(119 166 126 159)(127 149 134 142)(128 150 135 143)(129 151 136 144)(130 152 137 145)(131 153 138 146)(132 154 139 147)(133 141 140 148)(169 224 176 217)(170 211 177 218)(171 212 178 219)(172 213 179 220)(173 214 180 221)(174 215 181 222)(175 216 182 223)(183 199 190 206)(184 200 191 207)(185 201 192 208)(186 202 193 209)(187 203 194 210)(188 204 195 197)(189 205 196 198)
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), (1,63,27,77)(2,64,28,78)(3,65,15,79)(4,66,16,80)(5,67,17,81)(6,68,18,82)(7,69,19,83)(8,70,20,84)(9,57,21,71)(10,58,22,72)(11,59,23,73)(12,60,24,74)(13,61,25,75)(14,62,26,76)(29,95,43,110)(30,96,44,111)(31,97,45,112)(32,98,46,99)(33,85,47,100)(34,86,48,101)(35,87,49,102)(36,88,50,103)(37,89,51,104)(38,90,52,105)(39,91,53,106)(40,92,54,107)(41,93,55,108)(42,94,56,109)(113,181,127,196)(114,182,128,183)(115,169,129,184)(116,170,130,185)(117,171,131,186)(118,172,132,187)(119,173,133,188)(120,174,134,189)(121,175,135,190)(122,176,136,191)(123,177,137,192)(124,178,138,193)(125,179,139,194)(126,180,140,195)(141,197,166,221)(142,198,167,222)(143,199,168,223)(144,200,155,224)(145,201,156,211)(146,202,157,212)(147,203,158,213)(148,204,159,214)(149,205,160,215)(150,206,161,216)(151,207,162,217)(152,208,163,218)(153,209,164,219)(154,210,165,220), (57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,99)(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)(197,221)(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), (1,154,27,165)(2,153,28,164)(3,152,15,163)(4,151,16,162)(5,150,17,161)(6,149,18,160)(7,148,19,159)(8,147,20,158)(9,146,21,157)(10,145,22,156)(11,144,23,155)(12,143,24,168)(13,142,25,167)(14,141,26,166)(29,122,43,136)(30,121,44,135)(31,120,45,134)(32,119,46,133)(33,118,47,132)(34,117,48,131)(35,116,49,130)(36,115,50,129)(37,114,51,128)(38,113,52,127)(39,126,53,140)(40,125,54,139)(41,124,55,138)(42,123,56,137)(57,209,71,219)(58,208,72,218)(59,207,73,217)(60,206,74,216)(61,205,75,215)(62,204,76,214)(63,203,77,213)(64,202,78,212)(65,201,79,211)(66,200,80,224)(67,199,81,223)(68,198,82,222)(69,197,83,221)(70,210,84,220)(85,179,100,194)(86,178,101,193)(87,177,102,192)(88,176,103,191)(89,175,104,190)(90,174,105,189)(91,173,106,188)(92,172,107,187)(93,171,108,186)(94,170,109,185)(95,169,110,184)(96,182,111,183)(97,181,112,196)(98,180,99,195), (1,40,8,33)(2,41,9,34)(3,42,10,35)(4,29,11,36)(5,30,12,37)(6,31,13,38)(7,32,14,39)(15,56,22,49)(16,43,23,50)(17,44,24,51)(18,45,25,52)(19,46,26,53)(20,47,27,54)(21,48,28,55)(57,93,64,86)(58,94,65,87)(59,95,66,88)(60,96,67,89)(61,97,68,90)(62,98,69,91)(63,85,70,92)(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)(113,160,120,167)(114,161,121,168)(115,162,122,155)(116,163,123,156)(117,164,124,157)(118,165,125,158)(119,166,126,159)(127,149,134,142)(128,150,135,143)(129,151,136,144)(130,152,137,145)(131,153,138,146)(132,154,139,147)(133,141,140,148)(169,224,176,217)(170,211,177,218)(171,212,178,219)(172,213,179,220)(173,214,180,221)(174,215,181,222)(175,216,182,223)(183,199,190,206)(184,200,191,207)(185,201,192,208)(186,202,193,209)(187,203,194,210)(188,204,195,197)(189,205,196,198)>;
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), (1,63,27,77)(2,64,28,78)(3,65,15,79)(4,66,16,80)(5,67,17,81)(6,68,18,82)(7,69,19,83)(8,70,20,84)(9,57,21,71)(10,58,22,72)(11,59,23,73)(12,60,24,74)(13,61,25,75)(14,62,26,76)(29,95,43,110)(30,96,44,111)(31,97,45,112)(32,98,46,99)(33,85,47,100)(34,86,48,101)(35,87,49,102)(36,88,50,103)(37,89,51,104)(38,90,52,105)(39,91,53,106)(40,92,54,107)(41,93,55,108)(42,94,56,109)(113,181,127,196)(114,182,128,183)(115,169,129,184)(116,170,130,185)(117,171,131,186)(118,172,132,187)(119,173,133,188)(120,174,134,189)(121,175,135,190)(122,176,136,191)(123,177,137,192)(124,178,138,193)(125,179,139,194)(126,180,140,195)(141,197,166,221)(142,198,167,222)(143,199,168,223)(144,200,155,224)(145,201,156,211)(146,202,157,212)(147,203,158,213)(148,204,159,214)(149,205,160,215)(150,206,161,216)(151,207,162,217)(152,208,163,218)(153,209,164,219)(154,210,165,220), (57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105)(91,106)(92,107)(93,108)(94,109)(95,110)(96,111)(97,112)(98,99)(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)(197,221)(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), (1,154,27,165)(2,153,28,164)(3,152,15,163)(4,151,16,162)(5,150,17,161)(6,149,18,160)(7,148,19,159)(8,147,20,158)(9,146,21,157)(10,145,22,156)(11,144,23,155)(12,143,24,168)(13,142,25,167)(14,141,26,166)(29,122,43,136)(30,121,44,135)(31,120,45,134)(32,119,46,133)(33,118,47,132)(34,117,48,131)(35,116,49,130)(36,115,50,129)(37,114,51,128)(38,113,52,127)(39,126,53,140)(40,125,54,139)(41,124,55,138)(42,123,56,137)(57,209,71,219)(58,208,72,218)(59,207,73,217)(60,206,74,216)(61,205,75,215)(62,204,76,214)(63,203,77,213)(64,202,78,212)(65,201,79,211)(66,200,80,224)(67,199,81,223)(68,198,82,222)(69,197,83,221)(70,210,84,220)(85,179,100,194)(86,178,101,193)(87,177,102,192)(88,176,103,191)(89,175,104,190)(90,174,105,189)(91,173,106,188)(92,172,107,187)(93,171,108,186)(94,170,109,185)(95,169,110,184)(96,182,111,183)(97,181,112,196)(98,180,99,195), (1,40,8,33)(2,41,9,34)(3,42,10,35)(4,29,11,36)(5,30,12,37)(6,31,13,38)(7,32,14,39)(15,56,22,49)(16,43,23,50)(17,44,24,51)(18,45,25,52)(19,46,26,53)(20,47,27,54)(21,48,28,55)(57,93,64,86)(58,94,65,87)(59,95,66,88)(60,96,67,89)(61,97,68,90)(62,98,69,91)(63,85,70,92)(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)(113,160,120,167)(114,161,121,168)(115,162,122,155)(116,163,123,156)(117,164,124,157)(118,165,125,158)(119,166,126,159)(127,149,134,142)(128,150,135,143)(129,151,136,144)(130,152,137,145)(131,153,138,146)(132,154,139,147)(133,141,140,148)(169,224,176,217)(170,211,177,218)(171,212,178,219)(172,213,179,220)(173,214,180,221)(174,215,181,222)(175,216,182,223)(183,199,190,206)(184,200,191,207)(185,201,192,208)(186,202,193,209)(187,203,194,210)(188,204,195,197)(189,205,196,198) );
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)], [(1,63,27,77),(2,64,28,78),(3,65,15,79),(4,66,16,80),(5,67,17,81),(6,68,18,82),(7,69,19,83),(8,70,20,84),(9,57,21,71),(10,58,22,72),(11,59,23,73),(12,60,24,74),(13,61,25,75),(14,62,26,76),(29,95,43,110),(30,96,44,111),(31,97,45,112),(32,98,46,99),(33,85,47,100),(34,86,48,101),(35,87,49,102),(36,88,50,103),(37,89,51,104),(38,90,52,105),(39,91,53,106),(40,92,54,107),(41,93,55,108),(42,94,56,109),(113,181,127,196),(114,182,128,183),(115,169,129,184),(116,170,130,185),(117,171,131,186),(118,172,132,187),(119,173,133,188),(120,174,134,189),(121,175,135,190),(122,176,136,191),(123,177,137,192),(124,178,138,193),(125,179,139,194),(126,180,140,195),(141,197,166,221),(142,198,167,222),(143,199,168,223),(144,200,155,224),(145,201,156,211),(146,202,157,212),(147,203,158,213),(148,204,159,214),(149,205,160,215),(150,206,161,216),(151,207,162,217),(152,208,163,218),(153,209,164,219),(154,210,165,220)], [(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105),(91,106),(92,107),(93,108),(94,109),(95,110),(96,111),(97,112),(98,99),(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),(197,221),(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)], [(1,154,27,165),(2,153,28,164),(3,152,15,163),(4,151,16,162),(5,150,17,161),(6,149,18,160),(7,148,19,159),(8,147,20,158),(9,146,21,157),(10,145,22,156),(11,144,23,155),(12,143,24,168),(13,142,25,167),(14,141,26,166),(29,122,43,136),(30,121,44,135),(31,120,45,134),(32,119,46,133),(33,118,47,132),(34,117,48,131),(35,116,49,130),(36,115,50,129),(37,114,51,128),(38,113,52,127),(39,126,53,140),(40,125,54,139),(41,124,55,138),(42,123,56,137),(57,209,71,219),(58,208,72,218),(59,207,73,217),(60,206,74,216),(61,205,75,215),(62,204,76,214),(63,203,77,213),(64,202,78,212),(65,201,79,211),(66,200,80,224),(67,199,81,223),(68,198,82,222),(69,197,83,221),(70,210,84,220),(85,179,100,194),(86,178,101,193),(87,177,102,192),(88,176,103,191),(89,175,104,190),(90,174,105,189),(91,173,106,188),(92,172,107,187),(93,171,108,186),(94,170,109,185),(95,169,110,184),(96,182,111,183),(97,181,112,196),(98,180,99,195)], [(1,40,8,33),(2,41,9,34),(3,42,10,35),(4,29,11,36),(5,30,12,37),(6,31,13,38),(7,32,14,39),(15,56,22,49),(16,43,23,50),(17,44,24,51),(18,45,25,52),(19,46,26,53),(20,47,27,54),(21,48,28,55),(57,93,64,86),(58,94,65,87),(59,95,66,88),(60,96,67,89),(61,97,68,90),(62,98,69,91),(63,85,70,92),(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),(113,160,120,167),(114,161,121,168),(115,162,122,155),(116,163,123,156),(117,164,124,157),(118,165,125,158),(119,166,126,159),(127,149,134,142),(128,150,135,143),(129,151,136,144),(130,152,137,145),(131,153,138,146),(132,154,139,147),(133,141,140,148),(169,224,176,217),(170,211,177,218),(171,212,178,219),(172,213,179,220),(173,214,180,221),(174,215,181,222),(175,216,182,223),(183,199,190,206),(184,200,191,207),(185,201,192,208),(186,202,193,209),(187,203,194,210),(188,204,195,197),(189,205,196,198)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀