?
G = C14.642+ (1+4) order 448 = 26·7
64th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.642+ (1+4), C28⋊2D4⋊31C2, C4⋊C4.198D14, D28⋊C4⋊34C2, D14⋊D4⋊33C2, (C2×D4).101D14, Dic7.Q8⋊28C2, C22⋊C4.68D14, (C22×C4).50D14, Dic7⋊D4⋊22C2, D14.31(C4○D4), Dic7⋊4D4⋊23C2, D14.D4⋊34C2, (C2×C28).600C23, (C2×C14).205C24, C22.D4⋊10D7, C2.66(D4⋊6D14), C23.29(C22×D7), D14⋊C4.132C22, Dic7.26(C4○D4), (C2×D28).157C22, (D4×C14).143C22, C23.D14⋊33C2, Dic7⋊C4.43C22, C4⋊Dic7.229C22, (C22×C14).37C23, C22.226(C23×D7), C23.11D14⋊14C2, (C22×C28).369C22, C7⋊9(C22.47C24), (C4×Dic7).212C22, (C2×Dic7).246C23, (C22×D7).211C23, C23.D7.127C22, (C22×Dic7).131C22, (D7×C4⋊C4)⋊34C2, (C4×C7⋊D4)⋊49C2, C2.67(D7×C4○D4), C4⋊C4⋊D7⋊29C2, C14.179(C2×C4○D4), (C2×C4×D7).114C22, (C2×C4).67(C22×D7), (C7×C4⋊C4).178C22, (C2×C7⋊D4).49C22, (C7×C22.D4)⋊13C2, (C7×C22⋊C4).53C22, SmallGroup(448,1114)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1100 in 238 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×5], C4 [×12], C22, C22 [×13], C7, C2×C4 [×5], C2×C4 [×14], D4 [×10], C23 [×2], C23 [×2], D7 [×3], C14 [×3], C14 [×2], C42 [×3], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×5], Dic7 [×2], Dic7 [×5], C28 [×5], D14 [×2], D14 [×5], C2×C14, C2×C14 [×6], C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4 [×4], C22.D4, C22.D4, C42.C2, C42⋊2C2 [×2], C4×D7 [×5], D28 [×2], C2×Dic7 [×6], C2×Dic7 [×2], C7⋊D4 [×7], C2×C28 [×5], C2×C28, C7×D4, C22×D7 [×2], C22×C14 [×2], C22.47C24, C4×Dic7 [×3], Dic7⋊C4 [×6], C4⋊Dic7 [×2], D14⋊C4 [×4], C23.D7 [×3], C7×C22⋊C4 [×3], C7×C4⋊C4 [×2], C2×C4×D7 [×4], C2×D28, C22×Dic7, C2×C7⋊D4 [×4], C22×C28, D4×C14, C23.11D14, C23.D14, Dic7⋊4D4, D14.D4, D14⋊D4 [×2], Dic7.Q8, D7×C4⋊C4, D28⋊C4, C4⋊C4⋊D7, C4×C7⋊D4 [×2], C28⋊2D4, Dic7⋊D4, C7×C22.D4, C14.642+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×4], C24, D14 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D7 [×7], C22.47C24, C23×D7, D4⋊6D14, D7×C4○D4 [×2], C14.642+ (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, dbd-1=ebe=a7b, cd=dc, ce=ec, 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)
(1 190 21 171)(2 191 22 172)(3 192 23 173)(4 193 24 174)(5 194 25 175)(6 195 26 176)(7 196 27 177)(8 183 28 178)(9 184 15 179)(10 185 16 180)(11 186 17 181)(12 187 18 182)(13 188 19 169)(14 189 20 170)(29 216 43 210)(30 217 44 197)(31 218 45 198)(32 219 46 199)(33 220 47 200)(34 221 48 201)(35 222 49 202)(36 223 50 203)(37 224 51 204)(38 211 52 205)(39 212 53 206)(40 213 54 207)(41 214 55 208)(42 215 56 209)(57 135 77 116)(58 136 78 117)(59 137 79 118)(60 138 80 119)(61 139 81 120)(62 140 82 121)(63 127 83 122)(64 128 84 123)(65 129 71 124)(66 130 72 125)(67 131 73 126)(68 132 74 113)(69 133 75 114)(70 134 76 115)(85 164 111 153)(86 165 112 154)(87 166 99 141)(88 167 100 142)(89 168 101 143)(90 155 102 144)(91 156 103 145)(92 157 104 146)(93 158 105 147)(94 159 106 148)(95 160 107 149)(96 161 108 150)(97 162 109 151)(98 163 110 152)
(1 121 8 114)(2 122 9 115)(3 123 10 116)(4 124 11 117)(5 125 12 118)(6 126 13 119)(7 113 14 120)(15 134 22 127)(16 135 23 128)(17 136 24 129)(18 137 25 130)(19 138 26 131)(20 139 27 132)(21 140 28 133)(29 154 36 147)(30 141 37 148)(31 142 38 149)(32 143 39 150)(33 144 40 151)(34 145 41 152)(35 146 42 153)(43 165 50 158)(44 166 51 159)(45 167 52 160)(46 168 53 161)(47 155 54 162)(48 156 55 163)(49 157 56 164)(57 173 64 180)(58 174 65 181)(59 175 66 182)(60 176 67 169)(61 177 68 170)(62 178 69 171)(63 179 70 172)(71 186 78 193)(72 187 79 194)(73 188 80 195)(74 189 81 196)(75 190 82 183)(76 191 83 184)(77 192 84 185)(85 202 92 209)(86 203 93 210)(87 204 94 197)(88 205 95 198)(89 206 96 199)(90 207 97 200)(91 208 98 201)(99 224 106 217)(100 211 107 218)(101 212 108 219)(102 213 109 220)(103 214 110 221)(104 215 111 222)(105 216 112 223)
(1 49 28 42)(2 48 15 41)(3 47 16 40)(4 46 17 39)(5 45 18 38)(6 44 19 37)(7 43 20 36)(8 56 21 35)(9 55 22 34)(10 54 23 33)(11 53 24 32)(12 52 25 31)(13 51 26 30)(14 50 27 29)(57 102 84 97)(58 101 71 96)(59 100 72 95)(60 99 73 94)(61 112 74 93)(62 111 75 92)(63 110 76 91)(64 109 77 90)(65 108 78 89)(66 107 79 88)(67 106 80 87)(68 105 81 86)(69 104 82 85)(70 103 83 98)(113 165 139 147)(114 164 140 146)(115 163 127 145)(116 162 128 144)(117 161 129 143)(118 160 130 142)(119 159 131 141)(120 158 132 154)(121 157 133 153)(122 156 134 152)(123 155 135 151)(124 168 136 150)(125 167 137 149)(126 166 138 148)(169 217 195 204)(170 216 196 203)(171 215 183 202)(172 214 184 201)(173 213 185 200)(174 212 186 199)(175 211 187 198)(176 224 188 197)(177 223 189 210)(178 222 190 209)(179 221 191 208)(180 220 192 207)(181 219 193 206)(182 218 194 205)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 94)(58 95)(59 96)(60 97)(61 98)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 91)(69 92)(70 93)(71 100)(72 101)(73 102)(74 103)(75 104)(76 105)(77 106)(78 107)(79 108)(80 109)(81 110)(82 111)(83 112)(84 99)(113 152)(114 153)(115 154)(116 141)(117 142)(118 143)(119 144)(120 145)(121 146)(122 147)(123 148)(124 149)(125 150)(126 151)(127 158)(128 159)(129 160)(130 161)(131 162)(132 163)(133 164)(134 165)(135 166)(136 167)(137 168)(138 155)(139 156)(140 157)(169 207)(170 208)(171 209)(172 210)(173 197)(174 198)(175 199)(176 200)(177 201)(178 202)(179 203)(180 204)(181 205)(182 206)(183 222)(184 223)(185 224)(186 211)(187 212)(188 213)(189 214)(190 215)(191 216)(192 217)(193 218)(194 219)(195 220)(196 221)
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), (1,190,21,171)(2,191,22,172)(3,192,23,173)(4,193,24,174)(5,194,25,175)(6,195,26,176)(7,196,27,177)(8,183,28,178)(9,184,15,179)(10,185,16,180)(11,186,17,181)(12,187,18,182)(13,188,19,169)(14,189,20,170)(29,216,43,210)(30,217,44,197)(31,218,45,198)(32,219,46,199)(33,220,47,200)(34,221,48,201)(35,222,49,202)(36,223,50,203)(37,224,51,204)(38,211,52,205)(39,212,53,206)(40,213,54,207)(41,214,55,208)(42,215,56,209)(57,135,77,116)(58,136,78,117)(59,137,79,118)(60,138,80,119)(61,139,81,120)(62,140,82,121)(63,127,83,122)(64,128,84,123)(65,129,71,124)(66,130,72,125)(67,131,73,126)(68,132,74,113)(69,133,75,114)(70,134,76,115)(85,164,111,153)(86,165,112,154)(87,166,99,141)(88,167,100,142)(89,168,101,143)(90,155,102,144)(91,156,103,145)(92,157,104,146)(93,158,105,147)(94,159,106,148)(95,160,107,149)(96,161,108,150)(97,162,109,151)(98,163,110,152), (1,121,8,114)(2,122,9,115)(3,123,10,116)(4,124,11,117)(5,125,12,118)(6,126,13,119)(7,113,14,120)(15,134,22,127)(16,135,23,128)(17,136,24,129)(18,137,25,130)(19,138,26,131)(20,139,27,132)(21,140,28,133)(29,154,36,147)(30,141,37,148)(31,142,38,149)(32,143,39,150)(33,144,40,151)(34,145,41,152)(35,146,42,153)(43,165,50,158)(44,166,51,159)(45,167,52,160)(46,168,53,161)(47,155,54,162)(48,156,55,163)(49,157,56,164)(57,173,64,180)(58,174,65,181)(59,175,66,182)(60,176,67,169)(61,177,68,170)(62,178,69,171)(63,179,70,172)(71,186,78,193)(72,187,79,194)(73,188,80,195)(74,189,81,196)(75,190,82,183)(76,191,83,184)(77,192,84,185)(85,202,92,209)(86,203,93,210)(87,204,94,197)(88,205,95,198)(89,206,96,199)(90,207,97,200)(91,208,98,201)(99,224,106,217)(100,211,107,218)(101,212,108,219)(102,213,109,220)(103,214,110,221)(104,215,111,222)(105,216,112,223), (1,49,28,42)(2,48,15,41)(3,47,16,40)(4,46,17,39)(5,45,18,38)(6,44,19,37)(7,43,20,36)(8,56,21,35)(9,55,22,34)(10,54,23,33)(11,53,24,32)(12,52,25,31)(13,51,26,30)(14,50,27,29)(57,102,84,97)(58,101,71,96)(59,100,72,95)(60,99,73,94)(61,112,74,93)(62,111,75,92)(63,110,76,91)(64,109,77,90)(65,108,78,89)(66,107,79,88)(67,106,80,87)(68,105,81,86)(69,104,82,85)(70,103,83,98)(113,165,139,147)(114,164,140,146)(115,163,127,145)(116,162,128,144)(117,161,129,143)(118,160,130,142)(119,159,131,141)(120,158,132,154)(121,157,133,153)(122,156,134,152)(123,155,135,151)(124,168,136,150)(125,167,137,149)(126,166,138,148)(169,217,195,204)(170,216,196,203)(171,215,183,202)(172,214,184,201)(173,213,185,200)(174,212,186,199)(175,211,187,198)(176,224,188,197)(177,223,189,210)(178,222,190,209)(179,221,191,208)(180,220,192,207)(181,219,193,206)(182,218,194,205), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,94)(58,95)(59,96)(60,97)(61,98)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,100)(72,101)(73,102)(74,103)(75,104)(76,105)(77,106)(78,107)(79,108)(80,109)(81,110)(82,111)(83,112)(84,99)(113,152)(114,153)(115,154)(116,141)(117,142)(118,143)(119,144)(120,145)(121,146)(122,147)(123,148)(124,149)(125,150)(126,151)(127,158)(128,159)(129,160)(130,161)(131,162)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,155)(139,156)(140,157)(169,207)(170,208)(171,209)(172,210)(173,197)(174,198)(175,199)(176,200)(177,201)(178,202)(179,203)(180,204)(181,205)(182,206)(183,222)(184,223)(185,224)(186,211)(187,212)(188,213)(189,214)(190,215)(191,216)(192,217)(193,218)(194,219)(195,220)(196,221)>;
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), (1,190,21,171)(2,191,22,172)(3,192,23,173)(4,193,24,174)(5,194,25,175)(6,195,26,176)(7,196,27,177)(8,183,28,178)(9,184,15,179)(10,185,16,180)(11,186,17,181)(12,187,18,182)(13,188,19,169)(14,189,20,170)(29,216,43,210)(30,217,44,197)(31,218,45,198)(32,219,46,199)(33,220,47,200)(34,221,48,201)(35,222,49,202)(36,223,50,203)(37,224,51,204)(38,211,52,205)(39,212,53,206)(40,213,54,207)(41,214,55,208)(42,215,56,209)(57,135,77,116)(58,136,78,117)(59,137,79,118)(60,138,80,119)(61,139,81,120)(62,140,82,121)(63,127,83,122)(64,128,84,123)(65,129,71,124)(66,130,72,125)(67,131,73,126)(68,132,74,113)(69,133,75,114)(70,134,76,115)(85,164,111,153)(86,165,112,154)(87,166,99,141)(88,167,100,142)(89,168,101,143)(90,155,102,144)(91,156,103,145)(92,157,104,146)(93,158,105,147)(94,159,106,148)(95,160,107,149)(96,161,108,150)(97,162,109,151)(98,163,110,152), (1,121,8,114)(2,122,9,115)(3,123,10,116)(4,124,11,117)(5,125,12,118)(6,126,13,119)(7,113,14,120)(15,134,22,127)(16,135,23,128)(17,136,24,129)(18,137,25,130)(19,138,26,131)(20,139,27,132)(21,140,28,133)(29,154,36,147)(30,141,37,148)(31,142,38,149)(32,143,39,150)(33,144,40,151)(34,145,41,152)(35,146,42,153)(43,165,50,158)(44,166,51,159)(45,167,52,160)(46,168,53,161)(47,155,54,162)(48,156,55,163)(49,157,56,164)(57,173,64,180)(58,174,65,181)(59,175,66,182)(60,176,67,169)(61,177,68,170)(62,178,69,171)(63,179,70,172)(71,186,78,193)(72,187,79,194)(73,188,80,195)(74,189,81,196)(75,190,82,183)(76,191,83,184)(77,192,84,185)(85,202,92,209)(86,203,93,210)(87,204,94,197)(88,205,95,198)(89,206,96,199)(90,207,97,200)(91,208,98,201)(99,224,106,217)(100,211,107,218)(101,212,108,219)(102,213,109,220)(103,214,110,221)(104,215,111,222)(105,216,112,223), (1,49,28,42)(2,48,15,41)(3,47,16,40)(4,46,17,39)(5,45,18,38)(6,44,19,37)(7,43,20,36)(8,56,21,35)(9,55,22,34)(10,54,23,33)(11,53,24,32)(12,52,25,31)(13,51,26,30)(14,50,27,29)(57,102,84,97)(58,101,71,96)(59,100,72,95)(60,99,73,94)(61,112,74,93)(62,111,75,92)(63,110,76,91)(64,109,77,90)(65,108,78,89)(66,107,79,88)(67,106,80,87)(68,105,81,86)(69,104,82,85)(70,103,83,98)(113,165,139,147)(114,164,140,146)(115,163,127,145)(116,162,128,144)(117,161,129,143)(118,160,130,142)(119,159,131,141)(120,158,132,154)(121,157,133,153)(122,156,134,152)(123,155,135,151)(124,168,136,150)(125,167,137,149)(126,166,138,148)(169,217,195,204)(170,216,196,203)(171,215,183,202)(172,214,184,201)(173,213,185,200)(174,212,186,199)(175,211,187,198)(176,224,188,197)(177,223,189,210)(178,222,190,209)(179,221,191,208)(180,220,192,207)(181,219,193,206)(182,218,194,205), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,94)(58,95)(59,96)(60,97)(61,98)(62,85)(63,86)(64,87)(65,88)(66,89)(67,90)(68,91)(69,92)(70,93)(71,100)(72,101)(73,102)(74,103)(75,104)(76,105)(77,106)(78,107)(79,108)(80,109)(81,110)(82,111)(83,112)(84,99)(113,152)(114,153)(115,154)(116,141)(117,142)(118,143)(119,144)(120,145)(121,146)(122,147)(123,148)(124,149)(125,150)(126,151)(127,158)(128,159)(129,160)(130,161)(131,162)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,155)(139,156)(140,157)(169,207)(170,208)(171,209)(172,210)(173,197)(174,198)(175,199)(176,200)(177,201)(178,202)(179,203)(180,204)(181,205)(182,206)(183,222)(184,223)(185,224)(186,211)(187,212)(188,213)(189,214)(190,215)(191,216)(192,217)(193,218)(194,219)(195,220)(196,221) );
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)], [(1,190,21,171),(2,191,22,172),(3,192,23,173),(4,193,24,174),(5,194,25,175),(6,195,26,176),(7,196,27,177),(8,183,28,178),(9,184,15,179),(10,185,16,180),(11,186,17,181),(12,187,18,182),(13,188,19,169),(14,189,20,170),(29,216,43,210),(30,217,44,197),(31,218,45,198),(32,219,46,199),(33,220,47,200),(34,221,48,201),(35,222,49,202),(36,223,50,203),(37,224,51,204),(38,211,52,205),(39,212,53,206),(40,213,54,207),(41,214,55,208),(42,215,56,209),(57,135,77,116),(58,136,78,117),(59,137,79,118),(60,138,80,119),(61,139,81,120),(62,140,82,121),(63,127,83,122),(64,128,84,123),(65,129,71,124),(66,130,72,125),(67,131,73,126),(68,132,74,113),(69,133,75,114),(70,134,76,115),(85,164,111,153),(86,165,112,154),(87,166,99,141),(88,167,100,142),(89,168,101,143),(90,155,102,144),(91,156,103,145),(92,157,104,146),(93,158,105,147),(94,159,106,148),(95,160,107,149),(96,161,108,150),(97,162,109,151),(98,163,110,152)], [(1,121,8,114),(2,122,9,115),(3,123,10,116),(4,124,11,117),(5,125,12,118),(6,126,13,119),(7,113,14,120),(15,134,22,127),(16,135,23,128),(17,136,24,129),(18,137,25,130),(19,138,26,131),(20,139,27,132),(21,140,28,133),(29,154,36,147),(30,141,37,148),(31,142,38,149),(32,143,39,150),(33,144,40,151),(34,145,41,152),(35,146,42,153),(43,165,50,158),(44,166,51,159),(45,167,52,160),(46,168,53,161),(47,155,54,162),(48,156,55,163),(49,157,56,164),(57,173,64,180),(58,174,65,181),(59,175,66,182),(60,176,67,169),(61,177,68,170),(62,178,69,171),(63,179,70,172),(71,186,78,193),(72,187,79,194),(73,188,80,195),(74,189,81,196),(75,190,82,183),(76,191,83,184),(77,192,84,185),(85,202,92,209),(86,203,93,210),(87,204,94,197),(88,205,95,198),(89,206,96,199),(90,207,97,200),(91,208,98,201),(99,224,106,217),(100,211,107,218),(101,212,108,219),(102,213,109,220),(103,214,110,221),(104,215,111,222),(105,216,112,223)], [(1,49,28,42),(2,48,15,41),(3,47,16,40),(4,46,17,39),(5,45,18,38),(6,44,19,37),(7,43,20,36),(8,56,21,35),(9,55,22,34),(10,54,23,33),(11,53,24,32),(12,52,25,31),(13,51,26,30),(14,50,27,29),(57,102,84,97),(58,101,71,96),(59,100,72,95),(60,99,73,94),(61,112,74,93),(62,111,75,92),(63,110,76,91),(64,109,77,90),(65,108,78,89),(66,107,79,88),(67,106,80,87),(68,105,81,86),(69,104,82,85),(70,103,83,98),(113,165,139,147),(114,164,140,146),(115,163,127,145),(116,162,128,144),(117,161,129,143),(118,160,130,142),(119,159,131,141),(120,158,132,154),(121,157,133,153),(122,156,134,152),(123,155,135,151),(124,168,136,150),(125,167,137,149),(126,166,138,148),(169,217,195,204),(170,216,196,203),(171,215,183,202),(172,214,184,201),(173,213,185,200),(174,212,186,199),(175,211,187,198),(176,224,188,197),(177,223,189,210),(178,222,190,209),(179,221,191,208),(180,220,192,207),(181,219,193,206),(182,218,194,205)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,94),(58,95),(59,96),(60,97),(61,98),(62,85),(63,86),(64,87),(65,88),(66,89),(67,90),(68,91),(69,92),(70,93),(71,100),(72,101),(73,102),(74,103),(75,104),(76,105),(77,106),(78,107),(79,108),(80,109),(81,110),(82,111),(83,112),(84,99),(113,152),(114,153),(115,154),(116,141),(117,142),(118,143),(119,144),(120,145),(121,146),(122,147),(123,148),(124,149),(125,150),(126,151),(127,158),(128,159),(129,160),(130,161),(131,162),(132,163),(133,164),(134,165),(135,166),(136,167),(137,168),(138,155),(139,156),(140,157),(169,207),(170,208),(171,209),(172,210),(173,197),(174,198),(175,199),(176,200),(177,201),(178,202),(179,203),(180,204),(181,205),(182,206),(183,222),(184,223),(185,224),(186,211),(187,212),(188,213),(189,214),(190,215),(191,216),(192,217),(193,218),(194,219),(195,220),(196,221)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 25 | 0 | 0 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 4 |
| 0 | 0 | 0 | 0 | 16 | 14 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 3 |
| 0 | 0 | 0 | 0 | 5 | 10 |
| 28 | 2 | 0 | 0 | 0 | 0 |
| 28 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 11 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 7 |
| 0 | 0 | 0 | 0 | 2 | 4 |
| 1 | 27 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 22 |
| 0 | 0 | 0 | 0 | 27 | 25 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,25,4,0,0,0,0,25,11,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,1,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,16,0,0,0,0,4,14],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,5,0,0,0,0,3,10],[28,28,0,0,0,0,2,1,0,0,0,0,0,0,28,11,0,0,0,0,0,1,0,0,0,0,0,0,25,2,0,0,0,0,7,4],[1,0,0,0,0,0,27,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,4,27,0,0,0,0,22,25] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 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 | 14 | 14 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4⋊6D14 | D7×C4○D4 |
| kernel | C14.642+ (1+4) | C23.11D14 | C23.D14 | Dic7⋊4D4 | D14.D4 | D14⋊D4 | Dic7.Q8 | D7×C4⋊C4 | D28⋊C4 | C4⋊C4⋊D7 | C4×C7⋊D4 | C28⋊2D4 | Dic7⋊D4 | C7×C22.D4 | C22.D4 | Dic7 | D14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 3 | 4 | 4 | 9 | 6 | 3 | 3 | 1 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_{14}._{64}2_+^{(1+4)} % in TeX
G:=Group("C14.64ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1114);
// by ID
G=gap.SmallGroup(448,1114);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,219,184,1571,297,18822]);
// 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,d*b*d^-1=e*b*e=a^7*b,c*d=d*c,c*e=e*c,e*d*e=a^7*b^2*d>;
// generators/relations
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