G = C14.672+ 1+4 order 448 = 26·7
67th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.672+ 1+4, C4⋊C4.201D14, C28⋊D4.8C2, (C2×D4).104D14, (C2×C28).75C23, C22⋊C4.31D14, (C22×C4).51D14, Dic7⋊4D4⋊25C2, Dic7⋊3Q8⋊33C2, D14.5D4⋊31C2, (C2×C14).208C24, C22.D4⋊13D7, C2.69(D4⋊6D14), C23.32(C22×D7), D14⋊C4.133C22, Dic7.38(C4○D4), Dic7.D4⋊35C2, (C2×D28).159C22, (D4×C14).146C22, (C22×C14).40C23, (C22×D7).89C23, C22.229(C23×D7), C23.D7.46C22, C23.18D14⋊16C2, Dic7⋊C4.140C22, (C22×C28).370C22, C7⋊3(C22.53C24), (C4×Dic7).127C22, (C2×Dic7).248C23, (C2×Dic14).170C22, (C22×Dic7).134C22, (C4×C7⋊D4)⋊50C2, C2.70(D7×C4○D4), C14.182(C2×C4○D4), (C2×C4×D7).213C22, (C2×C4).70(C22×D7), (C7×C4⋊C4).181C22, (C2×C7⋊D4).52C22, (C7×C22.D4)⋊16C2, (C7×C22⋊C4).56C22, SmallGroup(448,1117)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.672+ 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, cd=dc, ce=ec, ede=a7b2d >
Subgroups: 1100 in 236 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic7, Dic7, C28, D14, C2×C14, C2×C14, C4×D4, C4×Q8, C22.D4, C22.D4, C4.4D4, C4⋊1D4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C22.53C24, C4×Dic7, C4×Dic7, Dic7⋊C4, D14⋊C4, C23.D7, C23.D7, C7×C22⋊C4, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, Dic7⋊4D4, Dic7.D4, Dic7⋊3Q8, D14.5D4, C4×C7⋊D4, C23.18D14, C28⋊D4, C7×C22.D4, C14.672+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.53C24, C23×D7, D4⋊6D14, D7×C4○D4, C14.672+ 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)
(1 190 15 177)(2 191 16 178)(3 192 17 179)(4 193 18 180)(5 194 19 181)(6 195 20 182)(7 196 21 169)(8 183 22 170)(9 184 23 171)(10 185 24 172)(11 186 25 173)(12 187 26 174)(13 188 27 175)(14 189 28 176)(29 221 51 203)(30 222 52 204)(31 223 53 205)(32 224 54 206)(33 211 55 207)(34 212 56 208)(35 213 43 209)(36 214 44 210)(37 215 45 197)(38 216 46 198)(39 217 47 199)(40 218 48 200)(41 219 49 201)(42 220 50 202)(57 128 71 114)(58 129 72 115)(59 130 73 116)(60 131 74 117)(61 132 75 118)(62 133 76 119)(63 134 77 120)(64 135 78 121)(65 136 79 122)(66 137 80 123)(67 138 81 124)(68 139 82 125)(69 140 83 126)(70 127 84 113)(85 156 103 150)(86 157 104 151)(87 158 105 152)(88 159 106 153)(89 160 107 154)(90 161 108 141)(91 162 109 142)(92 163 110 143)(93 164 111 144)(94 165 112 145)(95 166 99 146)(96 167 100 147)(97 168 101 148)(98 155 102 149)
(1 124 8 117)(2 125 9 118)(3 126 10 119)(4 113 11 120)(5 114 12 121)(6 115 13 122)(7 116 14 123)(15 138 22 131)(16 139 23 132)(17 140 24 133)(18 127 25 134)(19 128 26 135)(20 129 27 136)(21 130 28 137)(29 142 36 149)(30 143 37 150)(31 144 38 151)(32 145 39 152)(33 146 40 153)(34 147 41 154)(35 148 42 141)(43 168 50 161)(44 155 51 162)(45 156 52 163)(46 157 53 164)(47 158 54 165)(48 159 55 166)(49 160 56 167)(57 174 64 181)(58 175 65 182)(59 176 66 169)(60 177 67 170)(61 178 68 171)(62 179 69 172)(63 180 70 173)(71 187 78 194)(72 188 79 195)(73 189 80 196)(74 190 81 183)(75 191 82 184)(76 192 83 185)(77 193 84 186)(85 204 92 197)(86 205 93 198)(87 206 94 199)(88 207 95 200)(89 208 96 201)(90 209 97 202)(91 210 98 203)(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)
(1 103 22 92)(2 102 23 91)(3 101 24 90)(4 100 25 89)(5 99 26 88)(6 112 27 87)(7 111 28 86)(8 110 15 85)(9 109 16 98)(10 108 17 97)(11 107 18 96)(12 106 19 95)(13 105 20 94)(14 104 21 93)(29 61 44 82)(30 60 45 81)(31 59 46 80)(32 58 47 79)(33 57 48 78)(34 70 49 77)(35 69 50 76)(36 68 51 75)(37 67 52 74)(38 66 53 73)(39 65 54 72)(40 64 55 71)(41 63 56 84)(42 62 43 83)(113 219 134 208)(114 218 135 207)(115 217 136 206)(116 216 137 205)(117 215 138 204)(118 214 139 203)(119 213 140 202)(120 212 127 201)(121 211 128 200)(122 224 129 199)(123 223 130 198)(124 222 131 197)(125 221 132 210)(126 220 133 209)(141 179 168 185)(142 178 155 184)(143 177 156 183)(144 176 157 196)(145 175 158 195)(146 174 159 194)(147 173 160 193)(148 172 161 192)(149 171 162 191)(150 170 163 190)(151 169 164 189)(152 182 165 188)(153 181 166 187)(154 180 167 186)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 29)(15 52)(16 53)(17 54)(18 55)(19 56)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 85)(68 86)(69 87)(70 88)(71 107)(72 108)(73 109)(74 110)(75 111)(76 112)(77 99)(78 100)(79 101)(80 102)(81 103)(82 104)(83 105)(84 106)(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)(127 166)(128 167)(129 168)(130 155)(131 156)(132 157)(133 158)(134 159)(135 160)(136 161)(137 162)(138 163)(139 164)(140 165)(169 203)(170 204)(171 205)(172 206)(173 207)(174 208)(175 209)(176 210)(177 197)(178 198)(179 199)(180 200)(181 201)(182 202)(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), (1,190,15,177)(2,191,16,178)(3,192,17,179)(4,193,18,180)(5,194,19,181)(6,195,20,182)(7,196,21,169)(8,183,22,170)(9,184,23,171)(10,185,24,172)(11,186,25,173)(12,187,26,174)(13,188,27,175)(14,189,28,176)(29,221,51,203)(30,222,52,204)(31,223,53,205)(32,224,54,206)(33,211,55,207)(34,212,56,208)(35,213,43,209)(36,214,44,210)(37,215,45,197)(38,216,46,198)(39,217,47,199)(40,218,48,200)(41,219,49,201)(42,220,50,202)(57,128,71,114)(58,129,72,115)(59,130,73,116)(60,131,74,117)(61,132,75,118)(62,133,76,119)(63,134,77,120)(64,135,78,121)(65,136,79,122)(66,137,80,123)(67,138,81,124)(68,139,82,125)(69,140,83,126)(70,127,84,113)(85,156,103,150)(86,157,104,151)(87,158,105,152)(88,159,106,153)(89,160,107,154)(90,161,108,141)(91,162,109,142)(92,163,110,143)(93,164,111,144)(94,165,112,145)(95,166,99,146)(96,167,100,147)(97,168,101,148)(98,155,102,149), (1,124,8,117)(2,125,9,118)(3,126,10,119)(4,113,11,120)(5,114,12,121)(6,115,13,122)(7,116,14,123)(15,138,22,131)(16,139,23,132)(17,140,24,133)(18,127,25,134)(19,128,26,135)(20,129,27,136)(21,130,28,137)(29,142,36,149)(30,143,37,150)(31,144,38,151)(32,145,39,152)(33,146,40,153)(34,147,41,154)(35,148,42,141)(43,168,50,161)(44,155,51,162)(45,156,52,163)(46,157,53,164)(47,158,54,165)(48,159,55,166)(49,160,56,167)(57,174,64,181)(58,175,65,182)(59,176,66,169)(60,177,67,170)(61,178,68,171)(62,179,69,172)(63,180,70,173)(71,187,78,194)(72,188,79,195)(73,189,80,196)(74,190,81,183)(75,191,82,184)(76,192,83,185)(77,193,84,186)(85,204,92,197)(86,205,93,198)(87,206,94,199)(88,207,95,200)(89,208,96,201)(90,209,97,202)(91,210,98,203)(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), (1,103,22,92)(2,102,23,91)(3,101,24,90)(4,100,25,89)(5,99,26,88)(6,112,27,87)(7,111,28,86)(8,110,15,85)(9,109,16,98)(10,108,17,97)(11,107,18,96)(12,106,19,95)(13,105,20,94)(14,104,21,93)(29,61,44,82)(30,60,45,81)(31,59,46,80)(32,58,47,79)(33,57,48,78)(34,70,49,77)(35,69,50,76)(36,68,51,75)(37,67,52,74)(38,66,53,73)(39,65,54,72)(40,64,55,71)(41,63,56,84)(42,62,43,83)(113,219,134,208)(114,218,135,207)(115,217,136,206)(116,216,137,205)(117,215,138,204)(118,214,139,203)(119,213,140,202)(120,212,127,201)(121,211,128,200)(122,224,129,199)(123,223,130,198)(124,222,131,197)(125,221,132,210)(126,220,133,209)(141,179,168,185)(142,178,155,184)(143,177,156,183)(144,176,157,196)(145,175,158,195)(146,174,159,194)(147,173,160,193)(148,172,161,192)(149,171,162,191)(150,170,163,190)(151,169,164,189)(152,182,165,188)(153,181,166,187)(154,180,167,186), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,29)(15,52)(16,53)(17,54)(18,55)(19,56)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,85)(68,86)(69,87)(70,88)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,99)(78,100)(79,101)(80,102)(81,103)(82,104)(83,105)(84,106)(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)(127,166)(128,167)(129,168)(130,155)(131,156)(132,157)(133,158)(134,159)(135,160)(136,161)(137,162)(138,163)(139,164)(140,165)(169,203)(170,204)(171,205)(172,206)(173,207)(174,208)(175,209)(176,210)(177,197)(178,198)(179,199)(180,200)(181,201)(182,202)(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), (1,190,15,177)(2,191,16,178)(3,192,17,179)(4,193,18,180)(5,194,19,181)(6,195,20,182)(7,196,21,169)(8,183,22,170)(9,184,23,171)(10,185,24,172)(11,186,25,173)(12,187,26,174)(13,188,27,175)(14,189,28,176)(29,221,51,203)(30,222,52,204)(31,223,53,205)(32,224,54,206)(33,211,55,207)(34,212,56,208)(35,213,43,209)(36,214,44,210)(37,215,45,197)(38,216,46,198)(39,217,47,199)(40,218,48,200)(41,219,49,201)(42,220,50,202)(57,128,71,114)(58,129,72,115)(59,130,73,116)(60,131,74,117)(61,132,75,118)(62,133,76,119)(63,134,77,120)(64,135,78,121)(65,136,79,122)(66,137,80,123)(67,138,81,124)(68,139,82,125)(69,140,83,126)(70,127,84,113)(85,156,103,150)(86,157,104,151)(87,158,105,152)(88,159,106,153)(89,160,107,154)(90,161,108,141)(91,162,109,142)(92,163,110,143)(93,164,111,144)(94,165,112,145)(95,166,99,146)(96,167,100,147)(97,168,101,148)(98,155,102,149), (1,124,8,117)(2,125,9,118)(3,126,10,119)(4,113,11,120)(5,114,12,121)(6,115,13,122)(7,116,14,123)(15,138,22,131)(16,139,23,132)(17,140,24,133)(18,127,25,134)(19,128,26,135)(20,129,27,136)(21,130,28,137)(29,142,36,149)(30,143,37,150)(31,144,38,151)(32,145,39,152)(33,146,40,153)(34,147,41,154)(35,148,42,141)(43,168,50,161)(44,155,51,162)(45,156,52,163)(46,157,53,164)(47,158,54,165)(48,159,55,166)(49,160,56,167)(57,174,64,181)(58,175,65,182)(59,176,66,169)(60,177,67,170)(61,178,68,171)(62,179,69,172)(63,180,70,173)(71,187,78,194)(72,188,79,195)(73,189,80,196)(74,190,81,183)(75,191,82,184)(76,192,83,185)(77,193,84,186)(85,204,92,197)(86,205,93,198)(87,206,94,199)(88,207,95,200)(89,208,96,201)(90,209,97,202)(91,210,98,203)(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), (1,103,22,92)(2,102,23,91)(3,101,24,90)(4,100,25,89)(5,99,26,88)(6,112,27,87)(7,111,28,86)(8,110,15,85)(9,109,16,98)(10,108,17,97)(11,107,18,96)(12,106,19,95)(13,105,20,94)(14,104,21,93)(29,61,44,82)(30,60,45,81)(31,59,46,80)(32,58,47,79)(33,57,48,78)(34,70,49,77)(35,69,50,76)(36,68,51,75)(37,67,52,74)(38,66,53,73)(39,65,54,72)(40,64,55,71)(41,63,56,84)(42,62,43,83)(113,219,134,208)(114,218,135,207)(115,217,136,206)(116,216,137,205)(117,215,138,204)(118,214,139,203)(119,213,140,202)(120,212,127,201)(121,211,128,200)(122,224,129,199)(123,223,130,198)(124,222,131,197)(125,221,132,210)(126,220,133,209)(141,179,168,185)(142,178,155,184)(143,177,156,183)(144,176,157,196)(145,175,158,195)(146,174,159,194)(147,173,160,193)(148,172,161,192)(149,171,162,191)(150,170,163,190)(151,169,164,189)(152,182,165,188)(153,181,166,187)(154,180,167,186), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,29)(15,52)(16,53)(17,54)(18,55)(19,56)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,85)(68,86)(69,87)(70,88)(71,107)(72,108)(73,109)(74,110)(75,111)(76,112)(77,99)(78,100)(79,101)(80,102)(81,103)(82,104)(83,105)(84,106)(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)(127,166)(128,167)(129,168)(130,155)(131,156)(132,157)(133,158)(134,159)(135,160)(136,161)(137,162)(138,163)(139,164)(140,165)(169,203)(170,204)(171,205)(172,206)(173,207)(174,208)(175,209)(176,210)(177,197)(178,198)(179,199)(180,200)(181,201)(182,202)(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)], [(1,190,15,177),(2,191,16,178),(3,192,17,179),(4,193,18,180),(5,194,19,181),(6,195,20,182),(7,196,21,169),(8,183,22,170),(9,184,23,171),(10,185,24,172),(11,186,25,173),(12,187,26,174),(13,188,27,175),(14,189,28,176),(29,221,51,203),(30,222,52,204),(31,223,53,205),(32,224,54,206),(33,211,55,207),(34,212,56,208),(35,213,43,209),(36,214,44,210),(37,215,45,197),(38,216,46,198),(39,217,47,199),(40,218,48,200),(41,219,49,201),(42,220,50,202),(57,128,71,114),(58,129,72,115),(59,130,73,116),(60,131,74,117),(61,132,75,118),(62,133,76,119),(63,134,77,120),(64,135,78,121),(65,136,79,122),(66,137,80,123),(67,138,81,124),(68,139,82,125),(69,140,83,126),(70,127,84,113),(85,156,103,150),(86,157,104,151),(87,158,105,152),(88,159,106,153),(89,160,107,154),(90,161,108,141),(91,162,109,142),(92,163,110,143),(93,164,111,144),(94,165,112,145),(95,166,99,146),(96,167,100,147),(97,168,101,148),(98,155,102,149)], [(1,124,8,117),(2,125,9,118),(3,126,10,119),(4,113,11,120),(5,114,12,121),(6,115,13,122),(7,116,14,123),(15,138,22,131),(16,139,23,132),(17,140,24,133),(18,127,25,134),(19,128,26,135),(20,129,27,136),(21,130,28,137),(29,142,36,149),(30,143,37,150),(31,144,38,151),(32,145,39,152),(33,146,40,153),(34,147,41,154),(35,148,42,141),(43,168,50,161),(44,155,51,162),(45,156,52,163),(46,157,53,164),(47,158,54,165),(48,159,55,166),(49,160,56,167),(57,174,64,181),(58,175,65,182),(59,176,66,169),(60,177,67,170),(61,178,68,171),(62,179,69,172),(63,180,70,173),(71,187,78,194),(72,188,79,195),(73,189,80,196),(74,190,81,183),(75,191,82,184),(76,192,83,185),(77,193,84,186),(85,204,92,197),(86,205,93,198),(87,206,94,199),(88,207,95,200),(89,208,96,201),(90,209,97,202),(91,210,98,203),(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)], [(1,103,22,92),(2,102,23,91),(3,101,24,90),(4,100,25,89),(5,99,26,88),(6,112,27,87),(7,111,28,86),(8,110,15,85),(9,109,16,98),(10,108,17,97),(11,107,18,96),(12,106,19,95),(13,105,20,94),(14,104,21,93),(29,61,44,82),(30,60,45,81),(31,59,46,80),(32,58,47,79),(33,57,48,78),(34,70,49,77),(35,69,50,76),(36,68,51,75),(37,67,52,74),(38,66,53,73),(39,65,54,72),(40,64,55,71),(41,63,56,84),(42,62,43,83),(113,219,134,208),(114,218,135,207),(115,217,136,206),(116,216,137,205),(117,215,138,204),(118,214,139,203),(119,213,140,202),(120,212,127,201),(121,211,128,200),(122,224,129,199),(123,223,130,198),(124,222,131,197),(125,221,132,210),(126,220,133,209),(141,179,168,185),(142,178,155,184),(143,177,156,183),(144,176,157,196),(145,175,158,195),(146,174,159,194),(147,173,160,193),(148,172,161,192),(149,171,162,191),(150,170,163,190),(151,169,164,189),(152,182,165,188),(153,181,166,187),(154,180,167,186)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,29),(15,52),(16,53),(17,54),(18,55),(19,56),(20,43),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49),(27,50),(28,51),(57,89),(58,90),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,85),(68,86),(69,87),(70,88),(71,107),(72,108),(73,109),(74,110),(75,111),(76,112),(77,99),(78,100),(79,101),(80,102),(81,103),(82,104),(83,105),(84,106),(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),(127,166),(128,167),(129,168),(130,155),(131,156),(132,157),(133,158),(134,159),(135,160),(136,161),(137,162),(138,163),(139,164),(140,165),(169,203),(170,204),(171,205),(172,206),(173,207),(174,208),(175,209),(176,210),(177,197),(178,198),(179,199),(180,200),(181,201),(182,202),(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)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4O | 4P | 4Q | 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 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | ··· | 14 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | D4⋊6D14 | D7×C4○D4 |
| kernel | C14.672+ 1+4 | Dic7⋊4D4 | Dic7.D4 | Dic7⋊3Q8 | D14.5D4 | C4×C7⋊D4 | C23.18D14 | C28⋊D4 | C7×C22.D4 | C22.D4 | Dic7 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C2 | C2 |
| # reps | 1 | 2 | 4 | 2 | 2 | 2 | 1 | 1 | 1 | 3 | 8 | 9 | 6 | 3 | 3 | 1 | 6 | 12 |
Matrix representation of C14.672+ 1+4 ►in GL6(𝔽29)
| 19 | 21 | 0 | 0 | 0 | 0 |
| 17 | 28 | 0 | 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 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 28 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 5 |
| 0 | 0 | 0 | 0 | 5 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 26 | 0 | 0 |
| 0 | 0 | 2 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 26 | 0 | 0 | 0 | 0 |
| 19 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 27 |
| 0 | 0 | 0 | 0 | 27 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 7 | 0 | 0 |
| 0 | 0 | 5 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 16 | 28 |
G:=sub<GL(6,GF(29))| [19,17,0,0,0,0,21,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,28,0,0,0,0,0,12,0,0,0,0,0,0,18,5,0,0,0,0,5,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,2,0,0,0,0,26,18,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,19,0,0,0,0,26,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,27,0,0,0,0,27,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,5,0,0,0,0,7,16,0,0,0,0,0,0,1,16,0,0,0,0,0,28] >;
C14.672+ 1+4 in GAP, Magma, Sage, TeX
C_{14}._{67}2_+^{1+4} % in TeX
G:=Group("C14.67ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1117);
// by ID
G=gap.SmallGroup(448,1117);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,184,1571,297,18822]);
// 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,b*d=d*b,e*b*e=a^7*b,c*d=d*c,c*e=e*c,e*d*e=a^7*b^2*d>;
// generators/relations