?
G = C14.632+ (1+4) order 448 = 26·7
63rd non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.632+ (1+4), C14.832- (1+4), C4⋊C4.107D14, D14⋊Q8⋊31C2, D14.7(C4○D4), (C2×D4).100D14, C28⋊2D4.11C2, Dic7.Q8⋊27C2, C22⋊C4.28D14, C4.Dic14⋊29C2, C22.D4⋊9D7, Dic7⋊4D4⋊22C2, D14.D4⋊33C2, C28.48D4⋊22C2, (C2×C14).204C24, (C2×C28).180C23, (C22×C4).259D14, C2.65(D4⋊6D14), C23.28(C22×D7), D14⋊C4.108C22, C22⋊Dic14⋊33C2, (D4×C14).142C22, C23.D14⋊32C2, Dic7⋊C4.42C22, C4⋊Dic7.228C22, (C22×C14).36C23, C22.225(C23×D7), C23.D7.44C22, C23.18D14⋊15C2, (C22×C28).115C22, C7⋊6(C22.33C24), (C2×Dic14).37C22, (C2×Dic7).106C23, (C4×Dic7).124C22, (C22×D7).210C23, C2.44(D4.10D14), (C22×Dic7).130C22, (D7×C4⋊C4)⋊33C2, (C4×C7⋊D4)⋊7C2, C2.66(D7×C4○D4), C14.178(C2×C4○D4), (C2×C4×D7).113C22, (C2×C4).66(C22×D7), (C7×C4⋊C4).177C22, (C2×C7⋊D4).48C22, (C7×C22.D4)⋊12C2, (C7×C22⋊C4).52C22, SmallGroup(448,1113)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 940 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×10], C7, C2×C4 [×5], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23, D7 [×2], C14 [×3], C14 [×2], C42 [×2], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×12], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, Dic7 [×7], C28 [×5], D14 [×2], D14 [×2], C2×C14, C2×C14 [×6], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8 [×3], C22.D4, C22.D4 [×3], C42.C2 [×2], C42⋊2C2 [×2], Dic14, C4×D7 [×4], C2×Dic7 [×7], C2×Dic7, C7⋊D4 [×4], C2×C28 [×5], C2×C28, C7×D4, C22×D7, C22×C14 [×2], C22.33C24, C4×Dic7 [×2], Dic7⋊C4 [×8], C4⋊Dic7 [×4], D14⋊C4 [×2], C23.D7 [×5], C7×C22⋊C4 [×3], C7×C4⋊C4 [×2], C2×Dic14, C2×C4×D7 [×3], C22×Dic7, C2×C7⋊D4 [×2], C22×C28, D4×C14, C22⋊Dic14, C23.D14 [×2], Dic7⋊4D4, D14.D4 [×2], Dic7.Q8, C4.Dic14, D7×C4⋊C4, D14⋊Q8, C28.48D4, C4×C7⋊D4, C23.18D14, C28⋊2D4, C7×C22.D4, C14.632+ (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, C23×D7, D4⋊6D14, D7×C4○D4, D4.10D14, C14.632+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, dbd-1=ebe=a7b, cd=dc, ce=ec, ede=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)
(1 187 27 170)(2 188 28 171)(3 189 15 172)(4 190 16 173)(5 191 17 174)(6 192 18 175)(7 193 19 176)(8 194 20 177)(9 195 21 178)(10 196 22 179)(11 183 23 180)(12 184 24 181)(13 185 25 182)(14 186 26 169)(29 217 49 205)(30 218 50 206)(31 219 51 207)(32 220 52 208)(33 221 53 209)(34 222 54 210)(35 223 55 197)(36 224 56 198)(37 211 43 199)(38 212 44 200)(39 213 45 201)(40 214 46 202)(41 215 47 203)(42 216 48 204)(57 134 71 125)(58 135 72 126)(59 136 73 113)(60 137 74 114)(61 138 75 115)(62 139 76 116)(63 140 77 117)(64 127 78 118)(65 128 79 119)(66 129 80 120)(67 130 81 121)(68 131 82 122)(69 132 83 123)(70 133 84 124)(85 156 99 144)(86 157 100 145)(87 158 101 146)(88 159 102 147)(89 160 103 148)(90 161 104 149)(91 162 105 150)(92 163 106 151)(93 164 107 152)(94 165 108 153)(95 166 109 154)(96 167 110 141)(97 168 111 142)(98 155 112 143)
(1 115 8 122)(2 116 9 123)(3 117 10 124)(4 118 11 125)(5 119 12 126)(6 120 13 113)(7 121 14 114)(15 140 22 133)(16 127 23 134)(17 128 24 135)(18 129 25 136)(19 130 26 137)(20 131 27 138)(21 132 28 139)(29 150 36 143)(30 151 37 144)(31 152 38 145)(32 153 39 146)(33 154 40 147)(34 141 41 148)(35 142 42 149)(43 156 50 163)(44 157 51 164)(45 158 52 165)(46 159 53 166)(47 160 54 167)(48 161 55 168)(49 162 56 155)(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)(71 190 78 183)(72 191 79 184)(73 192 80 185)(74 193 81 186)(75 194 82 187)(76 195 83 188)(77 196 84 189)(85 206 92 199)(86 207 93 200)(87 208 94 201)(88 209 95 202)(89 210 96 203)(90 197 97 204)(91 198 98 205)(99 218 106 211)(100 219 107 212)(101 220 108 213)(102 221 109 214)(103 222 110 215)(104 223 111 216)(105 224 112 217)
(1 47 27 41)(2 46 28 40)(3 45 15 39)(4 44 16 38)(5 43 17 37)(6 56 18 36)(7 55 19 35)(8 54 20 34)(9 53 21 33)(10 52 22 32)(11 51 23 31)(12 50 24 30)(13 49 25 29)(14 48 26 42)(57 100 71 86)(58 99 72 85)(59 112 73 98)(60 111 74 97)(61 110 75 96)(62 109 76 95)(63 108 77 94)(64 107 78 93)(65 106 79 92)(66 105 80 91)(67 104 81 90)(68 103 82 89)(69 102 83 88)(70 101 84 87)(113 162 136 150)(114 161 137 149)(115 160 138 148)(116 159 139 147)(117 158 140 146)(118 157 127 145)(119 156 128 144)(120 155 129 143)(121 168 130 142)(122 167 131 141)(123 166 132 154)(124 165 133 153)(125 164 134 152)(126 163 135 151)(169 223 186 197)(170 222 187 210)(171 221 188 209)(172 220 189 208)(173 219 190 207)(174 218 191 206)(175 217 192 205)(176 216 193 204)(177 215 194 203)(178 214 195 202)(179 213 196 201)(180 212 183 200)(181 211 184 199)(182 224 185 198)
(1 41)(2 42)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(113 146)(114 147)(115 148)(116 149)(117 150)(118 151)(119 152)(120 153)(121 154)(122 141)(123 142)(124 143)(125 144)(126 145)(127 163)(128 164)(129 165)(130 166)(131 167)(132 168)(133 155)(134 156)(135 157)(136 158)(137 159)(138 160)(139 161)(140 162)(169 209)(170 210)(171 197)(172 198)(173 199)(174 200)(175 201)(176 202)(177 203)(178 204)(179 205)(180 206)(181 207)(182 208)(183 218)(184 219)(185 220)(186 221)(187 222)(188 223)(189 224)(190 211)(191 212)(192 213)(193 214)(194 215)(195 216)(196 217)
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), (1,187,27,170)(2,188,28,171)(3,189,15,172)(4,190,16,173)(5,191,17,174)(6,192,18,175)(7,193,19,176)(8,194,20,177)(9,195,21,178)(10,196,22,179)(11,183,23,180)(12,184,24,181)(13,185,25,182)(14,186,26,169)(29,217,49,205)(30,218,50,206)(31,219,51,207)(32,220,52,208)(33,221,53,209)(34,222,54,210)(35,223,55,197)(36,224,56,198)(37,211,43,199)(38,212,44,200)(39,213,45,201)(40,214,46,202)(41,215,47,203)(42,216,48,204)(57,134,71,125)(58,135,72,126)(59,136,73,113)(60,137,74,114)(61,138,75,115)(62,139,76,116)(63,140,77,117)(64,127,78,118)(65,128,79,119)(66,129,80,120)(67,130,81,121)(68,131,82,122)(69,132,83,123)(70,133,84,124)(85,156,99,144)(86,157,100,145)(87,158,101,146)(88,159,102,147)(89,160,103,148)(90,161,104,149)(91,162,105,150)(92,163,106,151)(93,164,107,152)(94,165,108,153)(95,166,109,154)(96,167,110,141)(97,168,111,142)(98,155,112,143), (1,115,8,122)(2,116,9,123)(3,117,10,124)(4,118,11,125)(5,119,12,126)(6,120,13,113)(7,121,14,114)(15,140,22,133)(16,127,23,134)(17,128,24,135)(18,129,25,136)(19,130,26,137)(20,131,27,138)(21,132,28,139)(29,150,36,143)(30,151,37,144)(31,152,38,145)(32,153,39,146)(33,154,40,147)(34,141,41,148)(35,142,42,149)(43,156,50,163)(44,157,51,164)(45,158,52,165)(46,159,53,166)(47,160,54,167)(48,161,55,168)(49,162,56,155)(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)(71,190,78,183)(72,191,79,184)(73,192,80,185)(74,193,81,186)(75,194,82,187)(76,195,83,188)(77,196,84,189)(85,206,92,199)(86,207,93,200)(87,208,94,201)(88,209,95,202)(89,210,96,203)(90,197,97,204)(91,198,98,205)(99,218,106,211)(100,219,107,212)(101,220,108,213)(102,221,109,214)(103,222,110,215)(104,223,111,216)(105,224,112,217), (1,47,27,41)(2,46,28,40)(3,45,15,39)(4,44,16,38)(5,43,17,37)(6,56,18,36)(7,55,19,35)(8,54,20,34)(9,53,21,33)(10,52,22,32)(11,51,23,31)(12,50,24,30)(13,49,25,29)(14,48,26,42)(57,100,71,86)(58,99,72,85)(59,112,73,98)(60,111,74,97)(61,110,75,96)(62,109,76,95)(63,108,77,94)(64,107,78,93)(65,106,79,92)(66,105,80,91)(67,104,81,90)(68,103,82,89)(69,102,83,88)(70,101,84,87)(113,162,136,150)(114,161,137,149)(115,160,138,148)(116,159,139,147)(117,158,140,146)(118,157,127,145)(119,156,128,144)(120,155,129,143)(121,168,130,142)(122,167,131,141)(123,166,132,154)(124,165,133,153)(125,164,134,152)(126,163,135,151)(169,223,186,197)(170,222,187,210)(171,221,188,209)(172,220,189,208)(173,219,190,207)(174,218,191,206)(175,217,192,205)(176,216,193,204)(177,215,194,203)(178,214,195,202)(179,213,196,201)(180,212,183,200)(181,211,184,199)(182,224,185,198), (1,41)(2,42)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(113,146)(114,147)(115,148)(116,149)(117,150)(118,151)(119,152)(120,153)(121,154)(122,141)(123,142)(124,143)(125,144)(126,145)(127,163)(128,164)(129,165)(130,166)(131,167)(132,168)(133,155)(134,156)(135,157)(136,158)(137,159)(138,160)(139,161)(140,162)(169,209)(170,210)(171,197)(172,198)(173,199)(174,200)(175,201)(176,202)(177,203)(178,204)(179,205)(180,206)(181,207)(182,208)(183,218)(184,219)(185,220)(186,221)(187,222)(188,223)(189,224)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217)>;
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), (1,187,27,170)(2,188,28,171)(3,189,15,172)(4,190,16,173)(5,191,17,174)(6,192,18,175)(7,193,19,176)(8,194,20,177)(9,195,21,178)(10,196,22,179)(11,183,23,180)(12,184,24,181)(13,185,25,182)(14,186,26,169)(29,217,49,205)(30,218,50,206)(31,219,51,207)(32,220,52,208)(33,221,53,209)(34,222,54,210)(35,223,55,197)(36,224,56,198)(37,211,43,199)(38,212,44,200)(39,213,45,201)(40,214,46,202)(41,215,47,203)(42,216,48,204)(57,134,71,125)(58,135,72,126)(59,136,73,113)(60,137,74,114)(61,138,75,115)(62,139,76,116)(63,140,77,117)(64,127,78,118)(65,128,79,119)(66,129,80,120)(67,130,81,121)(68,131,82,122)(69,132,83,123)(70,133,84,124)(85,156,99,144)(86,157,100,145)(87,158,101,146)(88,159,102,147)(89,160,103,148)(90,161,104,149)(91,162,105,150)(92,163,106,151)(93,164,107,152)(94,165,108,153)(95,166,109,154)(96,167,110,141)(97,168,111,142)(98,155,112,143), (1,115,8,122)(2,116,9,123)(3,117,10,124)(4,118,11,125)(5,119,12,126)(6,120,13,113)(7,121,14,114)(15,140,22,133)(16,127,23,134)(17,128,24,135)(18,129,25,136)(19,130,26,137)(20,131,27,138)(21,132,28,139)(29,150,36,143)(30,151,37,144)(31,152,38,145)(32,153,39,146)(33,154,40,147)(34,141,41,148)(35,142,42,149)(43,156,50,163)(44,157,51,164)(45,158,52,165)(46,159,53,166)(47,160,54,167)(48,161,55,168)(49,162,56,155)(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)(71,190,78,183)(72,191,79,184)(73,192,80,185)(74,193,81,186)(75,194,82,187)(76,195,83,188)(77,196,84,189)(85,206,92,199)(86,207,93,200)(87,208,94,201)(88,209,95,202)(89,210,96,203)(90,197,97,204)(91,198,98,205)(99,218,106,211)(100,219,107,212)(101,220,108,213)(102,221,109,214)(103,222,110,215)(104,223,111,216)(105,224,112,217), (1,47,27,41)(2,46,28,40)(3,45,15,39)(4,44,16,38)(5,43,17,37)(6,56,18,36)(7,55,19,35)(8,54,20,34)(9,53,21,33)(10,52,22,32)(11,51,23,31)(12,50,24,30)(13,49,25,29)(14,48,26,42)(57,100,71,86)(58,99,72,85)(59,112,73,98)(60,111,74,97)(61,110,75,96)(62,109,76,95)(63,108,77,94)(64,107,78,93)(65,106,79,92)(66,105,80,91)(67,104,81,90)(68,103,82,89)(69,102,83,88)(70,101,84,87)(113,162,136,150)(114,161,137,149)(115,160,138,148)(116,159,139,147)(117,158,140,146)(118,157,127,145)(119,156,128,144)(120,155,129,143)(121,168,130,142)(122,167,131,141)(123,166,132,154)(124,165,133,153)(125,164,134,152)(126,163,135,151)(169,223,186,197)(170,222,187,210)(171,221,188,209)(172,220,189,208)(173,219,190,207)(174,218,191,206)(175,217,192,205)(176,216,193,204)(177,215,194,203)(178,214,195,202)(179,213,196,201)(180,212,183,200)(181,211,184,199)(182,224,185,198), (1,41)(2,42)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,106)(72,107)(73,108)(74,109)(75,110)(76,111)(77,112)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(113,146)(114,147)(115,148)(116,149)(117,150)(118,151)(119,152)(120,153)(121,154)(122,141)(123,142)(124,143)(125,144)(126,145)(127,163)(128,164)(129,165)(130,166)(131,167)(132,168)(133,155)(134,156)(135,157)(136,158)(137,159)(138,160)(139,161)(140,162)(169,209)(170,210)(171,197)(172,198)(173,199)(174,200)(175,201)(176,202)(177,203)(178,204)(179,205)(180,206)(181,207)(182,208)(183,218)(184,219)(185,220)(186,221)(187,222)(188,223)(189,224)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217) );
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)], [(1,187,27,170),(2,188,28,171),(3,189,15,172),(4,190,16,173),(5,191,17,174),(6,192,18,175),(7,193,19,176),(8,194,20,177),(9,195,21,178),(10,196,22,179),(11,183,23,180),(12,184,24,181),(13,185,25,182),(14,186,26,169),(29,217,49,205),(30,218,50,206),(31,219,51,207),(32,220,52,208),(33,221,53,209),(34,222,54,210),(35,223,55,197),(36,224,56,198),(37,211,43,199),(38,212,44,200),(39,213,45,201),(40,214,46,202),(41,215,47,203),(42,216,48,204),(57,134,71,125),(58,135,72,126),(59,136,73,113),(60,137,74,114),(61,138,75,115),(62,139,76,116),(63,140,77,117),(64,127,78,118),(65,128,79,119),(66,129,80,120),(67,130,81,121),(68,131,82,122),(69,132,83,123),(70,133,84,124),(85,156,99,144),(86,157,100,145),(87,158,101,146),(88,159,102,147),(89,160,103,148),(90,161,104,149),(91,162,105,150),(92,163,106,151),(93,164,107,152),(94,165,108,153),(95,166,109,154),(96,167,110,141),(97,168,111,142),(98,155,112,143)], [(1,115,8,122),(2,116,9,123),(3,117,10,124),(4,118,11,125),(5,119,12,126),(6,120,13,113),(7,121,14,114),(15,140,22,133),(16,127,23,134),(17,128,24,135),(18,129,25,136),(19,130,26,137),(20,131,27,138),(21,132,28,139),(29,150,36,143),(30,151,37,144),(31,152,38,145),(32,153,39,146),(33,154,40,147),(34,141,41,148),(35,142,42,149),(43,156,50,163),(44,157,51,164),(45,158,52,165),(46,159,53,166),(47,160,54,167),(48,161,55,168),(49,162,56,155),(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),(71,190,78,183),(72,191,79,184),(73,192,80,185),(74,193,81,186),(75,194,82,187),(76,195,83,188),(77,196,84,189),(85,206,92,199),(86,207,93,200),(87,208,94,201),(88,209,95,202),(89,210,96,203),(90,197,97,204),(91,198,98,205),(99,218,106,211),(100,219,107,212),(101,220,108,213),(102,221,109,214),(103,222,110,215),(104,223,111,216),(105,224,112,217)], [(1,47,27,41),(2,46,28,40),(3,45,15,39),(4,44,16,38),(5,43,17,37),(6,56,18,36),(7,55,19,35),(8,54,20,34),(9,53,21,33),(10,52,22,32),(11,51,23,31),(12,50,24,30),(13,49,25,29),(14,48,26,42),(57,100,71,86),(58,99,72,85),(59,112,73,98),(60,111,74,97),(61,110,75,96),(62,109,76,95),(63,108,77,94),(64,107,78,93),(65,106,79,92),(66,105,80,91),(67,104,81,90),(68,103,82,89),(69,102,83,88),(70,101,84,87),(113,162,136,150),(114,161,137,149),(115,160,138,148),(116,159,139,147),(117,158,140,146),(118,157,127,145),(119,156,128,144),(120,155,129,143),(121,168,130,142),(122,167,131,141),(123,166,132,154),(124,165,133,153),(125,164,134,152),(126,163,135,151),(169,223,186,197),(170,222,187,210),(171,221,188,209),(172,220,189,208),(173,219,190,207),(174,218,191,206),(175,217,192,205),(176,216,193,204),(177,215,194,203),(178,214,195,202),(179,213,196,201),(180,212,183,200),(181,211,184,199),(182,224,185,198)], [(1,41),(2,42),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,49),(16,50),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(57,92),(58,93),(59,94),(60,95),(61,96),(62,97),(63,98),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,106),(72,107),(73,108),(74,109),(75,110),(76,111),(77,112),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105),(113,146),(114,147),(115,148),(116,149),(117,150),(118,151),(119,152),(120,153),(121,154),(122,141),(123,142),(124,143),(125,144),(126,145),(127,163),(128,164),(129,165),(130,166),(131,167),(132,168),(133,155),(134,156),(135,157),(136,158),(137,159),(138,160),(139,161),(140,162),(169,209),(170,210),(171,197),(172,198),(173,199),(174,200),(175,201),(176,202),(177,203),(178,204),(179,205),(180,206),(181,207),(182,208),(183,218),(184,219),(185,220),(186,221),(187,222),(188,223),(189,224),(190,211),(191,212),(192,213),(193,214),(194,215),(195,216),(196,217)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 21 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 26 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 1 | 7 | 8 | 17 |
| 0 | 0 | 0 | 0 | 20 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 | 27 | 22 |
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 17 | 23 | 14 | 11 |
| 0 | 0 | 0 | 0 | 23 | 0 | 22 | 0 |
| 0 | 0 | 0 | 0 | 14 | 23 | 23 | 6 |
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 26 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 12 | 17 | 1 |
| 0 | 0 | 0 | 0 | 26 | 10 | 0 | 12 |
| 28 | 0 | 0 | 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 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 12 | 17 | 1 |
| 0 | 0 | 0 | 0 | 21 | 10 | 2 | 12 |
G:=sub<GL(8,GF(29))| [21,8,0,0,0,0,0,0,21,26,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,23,1,20,11,0,0,0,0,0,7,0,9,0,0,0,0,17,8,6,27,0,0,0,0,0,17,0,22],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,7,17,23,14,0,0,0,0,0,23,0,23,0,0,0,0,18,14,22,23,0,0,0,0,0,11,0,6],[28,26,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,21,7,20,26,0,0,0,0,28,8,12,10,0,0,0,0,0,0,17,0,0,0,0,0,0,0,1,12],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,21,5,20,21,0,0,0,0,28,8,12,10,0,0,0,0,0,0,17,2,0,0,0,0,0,0,1,12] >;
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 | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| 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 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | D4⋊6D14 | D7×C4○D4 | D4.10D14 |
| kernel | C14.632+ (1+4) | C22⋊Dic14 | C23.D14 | Dic7⋊4D4 | D14.D4 | Dic7.Q8 | C4.Dic14 | D7×C4⋊C4 | D14⋊Q8 | C28.48D4 | C4×C7⋊D4 | C23.18D14 | C28⋊2D4 | C7×C22.D4 | C22.D4 | D14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 9 | 6 | 3 | 3 | 1 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{63}2_+^{(1+4)} % in TeX
G:=Group("C14.63ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1113);
// by ID
G=gap.SmallGroup(448,1113);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,219,184,675,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=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=b^2*d>;
// generators/relations
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ℤ
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