?
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⋊3Q8⋊33C2, Dic7⋊4D4⋊25C2, 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
Subgroups: 1100 in 236 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×13], C22, C22 [×12], C7, C2×C4, C2×C4 [×4], C2×C4 [×10], D4 [×10], Q8 [×4], C23 [×2], C23 [×2], D7 [×2], C14, C14 [×2], C14 [×2], C42 [×5], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×5], C2×Q8 [×2], Dic7 [×4], Dic7 [×4], C28 [×5], D14 [×6], C2×C14, C2×C14 [×6], C4×D4 [×4], C4×Q8 [×2], C22.D4, C22.D4 [×3], C4.4D4 [×4], C4⋊1D4, Dic14 [×4], C4×D7 [×2], D28, C2×Dic7 [×6], C2×Dic7, C7⋊D4 [×8], C2×C28, C2×C28 [×4], C2×C28, C7×D4, C22×D7 [×2], C22×C14 [×2], C22.53C24, C4×Dic7, C4×Dic7 [×4], Dic7⋊C4 [×4], D14⋊C4 [×6], C23.D7, C23.D7 [×2], C7×C22⋊C4, C7×C22⋊C4 [×2], C7×C4⋊C4 [×2], C2×Dic14 [×2], C2×C4×D7 [×2], C2×D28, C22×Dic7, C2×C7⋊D4 [×4], C22×C28, D4×C14, Dic7⋊4D4 [×2], Dic7.D4 [×4], Dic7⋊3Q8 [×2], D14.5D4 [×2], C4×C7⋊D4 [×2], C23.18D14, C28⋊D4, C7×C22.D4, C14.672+ (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], C22.53C24, C23×D7, D4⋊6D14, D7×C4○D4 [×2], C14.672+ (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, 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)
(1 185 27 177)(2 186 28 178)(3 187 15 179)(4 188 16 180)(5 189 17 181)(6 190 18 182)(7 191 19 169)(8 192 20 170)(9 193 21 171)(10 194 22 172)(11 195 23 173)(12 196 24 174)(13 183 25 175)(14 184 26 176)(29 215 51 201)(30 216 52 202)(31 217 53 203)(32 218 54 204)(33 219 55 205)(34 220 56 206)(35 221 43 207)(36 222 44 208)(37 223 45 209)(38 224 46 210)(39 211 47 197)(40 212 48 198)(41 213 49 199)(42 214 50 200)(57 130 80 122)(58 131 81 123)(59 132 82 124)(60 133 83 125)(61 134 84 126)(62 135 71 113)(63 136 72 114)(64 137 73 115)(65 138 74 116)(66 139 75 117)(67 140 76 118)(68 127 77 119)(69 128 78 120)(70 129 79 121)(85 168 112 152)(86 155 99 153)(87 156 100 154)(88 157 101 141)(89 158 102 142)(90 159 103 143)(91 160 104 144)(92 161 105 145)(93 162 106 146)(94 163 107 147)(95 164 108 148)(96 165 109 149)(97 166 110 150)(98 167 111 151)
(1 118 8 125)(2 119 9 126)(3 120 10 113)(4 121 11 114)(5 122 12 115)(6 123 13 116)(7 124 14 117)(15 128 22 135)(16 129 23 136)(17 130 24 137)(18 131 25 138)(19 132 26 139)(20 133 27 140)(21 134 28 127)(29 145 36 152)(30 146 37 153)(31 147 38 154)(32 148 39 141)(33 149 40 142)(34 150 41 143)(35 151 42 144)(43 167 50 160)(44 168 51 161)(45 155 52 162)(46 156 53 163)(47 157 54 164)(48 158 55 165)(49 159 56 166)(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)(85 201 92 208)(86 202 93 209)(87 203 94 210)(88 204 95 197)(89 205 96 198)(90 206 97 199)(91 207 98 200)(99 216 106 223)(100 217 107 224)(101 218 108 211)(102 219 109 212)(103 220 110 213)(104 221 111 214)(105 222 112 215)
(1 99 20 93)(2 112 21 92)(3 111 22 91)(4 110 23 90)(5 109 24 89)(6 108 25 88)(7 107 26 87)(8 106 27 86)(9 105 28 85)(10 104 15 98)(11 103 16 97)(12 102 17 96)(13 101 18 95)(14 100 19 94)(29 61 44 77)(30 60 45 76)(31 59 46 75)(32 58 47 74)(33 57 48 73)(34 70 49 72)(35 69 50 71)(36 68 51 84)(37 67 52 83)(38 66 53 82)(39 65 54 81)(40 64 55 80)(41 63 56 79)(42 62 43 78)(113 221 128 200)(114 220 129 199)(115 219 130 198)(116 218 131 197)(117 217 132 210)(118 216 133 209)(119 215 134 208)(120 214 135 207)(121 213 136 206)(122 212 137 205)(123 211 138 204)(124 224 139 203)(125 223 140 202)(126 222 127 201)(141 182 164 183)(142 181 165 196)(143 180 166 195)(144 179 167 194)(145 178 168 193)(146 177 155 192)(147 176 156 191)(148 175 157 190)(149 174 158 189)(150 173 159 188)(151 172 160 187)(152 171 161 186)(153 170 162 185)(154 169 163 184)
(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 54)(16 55)(17 56)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(57 90)(58 91)(59 92)(60 93)(61 94)(62 95)(63 96)(64 97)(65 98)(66 85)(67 86)(68 87)(69 88)(70 89)(71 108)(72 109)(73 110)(74 111)(75 112)(76 99)(77 100)(78 101)(79 102)(80 103)(81 104)(82 105)(83 106)(84 107)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(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)(169 201)(170 202)(171 203)(172 204)(173 205)(174 206)(175 207)(176 208)(177 209)(178 210)(179 197)(180 198)(181 199)(182 200)(183 221)(184 222)(185 223)(186 224)(187 211)(188 212)(189 213)(190 214)(191 215)(192 216)(193 217)(194 218)(195 219)(196 220)
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), (1,185,27,177)(2,186,28,178)(3,187,15,179)(4,188,16,180)(5,189,17,181)(6,190,18,182)(7,191,19,169)(8,192,20,170)(9,193,21,171)(10,194,22,172)(11,195,23,173)(12,196,24,174)(13,183,25,175)(14,184,26,176)(29,215,51,201)(30,216,52,202)(31,217,53,203)(32,218,54,204)(33,219,55,205)(34,220,56,206)(35,221,43,207)(36,222,44,208)(37,223,45,209)(38,224,46,210)(39,211,47,197)(40,212,48,198)(41,213,49,199)(42,214,50,200)(57,130,80,122)(58,131,81,123)(59,132,82,124)(60,133,83,125)(61,134,84,126)(62,135,71,113)(63,136,72,114)(64,137,73,115)(65,138,74,116)(66,139,75,117)(67,140,76,118)(68,127,77,119)(69,128,78,120)(70,129,79,121)(85,168,112,152)(86,155,99,153)(87,156,100,154)(88,157,101,141)(89,158,102,142)(90,159,103,143)(91,160,104,144)(92,161,105,145)(93,162,106,146)(94,163,107,147)(95,164,108,148)(96,165,109,149)(97,166,110,150)(98,167,111,151), (1,118,8,125)(2,119,9,126)(3,120,10,113)(4,121,11,114)(5,122,12,115)(6,123,13,116)(7,124,14,117)(15,128,22,135)(16,129,23,136)(17,130,24,137)(18,131,25,138)(19,132,26,139)(20,133,27,140)(21,134,28,127)(29,145,36,152)(30,146,37,153)(31,147,38,154)(32,148,39,141)(33,149,40,142)(34,150,41,143)(35,151,42,144)(43,167,50,160)(44,168,51,161)(45,155,52,162)(46,156,53,163)(47,157,54,164)(48,158,55,165)(49,159,56,166)(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)(85,201,92,208)(86,202,93,209)(87,203,94,210)(88,204,95,197)(89,205,96,198)(90,206,97,199)(91,207,98,200)(99,216,106,223)(100,217,107,224)(101,218,108,211)(102,219,109,212)(103,220,110,213)(104,221,111,214)(105,222,112,215), (1,99,20,93)(2,112,21,92)(3,111,22,91)(4,110,23,90)(5,109,24,89)(6,108,25,88)(7,107,26,87)(8,106,27,86)(9,105,28,85)(10,104,15,98)(11,103,16,97)(12,102,17,96)(13,101,18,95)(14,100,19,94)(29,61,44,77)(30,60,45,76)(31,59,46,75)(32,58,47,74)(33,57,48,73)(34,70,49,72)(35,69,50,71)(36,68,51,84)(37,67,52,83)(38,66,53,82)(39,65,54,81)(40,64,55,80)(41,63,56,79)(42,62,43,78)(113,221,128,200)(114,220,129,199)(115,219,130,198)(116,218,131,197)(117,217,132,210)(118,216,133,209)(119,215,134,208)(120,214,135,207)(121,213,136,206)(122,212,137,205)(123,211,138,204)(124,224,139,203)(125,223,140,202)(126,222,127,201)(141,182,164,183)(142,181,165,196)(143,180,166,195)(144,179,167,194)(145,178,168,193)(146,177,155,192)(147,176,156,191)(148,175,157,190)(149,174,158,189)(150,173,159,188)(151,172,160,187)(152,171,161,186)(153,170,162,185)(154,169,163,184), (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)(15,54)(16,55)(17,56)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(57,90)(58,91)(59,92)(60,93)(61,94)(62,95)(63,96)(64,97)(65,98)(66,85)(67,86)(68,87)(69,88)(70,89)(71,108)(72,109)(73,110)(74,111)(75,112)(76,99)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(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)(169,201)(170,202)(171,203)(172,204)(173,205)(174,206)(175,207)(176,208)(177,209)(178,210)(179,197)(180,198)(181,199)(182,200)(183,221)(184,222)(185,223)(186,224)(187,211)(188,212)(189,213)(190,214)(191,215)(192,216)(193,217)(194,218)(195,219)(196,220)>;
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), (1,185,27,177)(2,186,28,178)(3,187,15,179)(4,188,16,180)(5,189,17,181)(6,190,18,182)(7,191,19,169)(8,192,20,170)(9,193,21,171)(10,194,22,172)(11,195,23,173)(12,196,24,174)(13,183,25,175)(14,184,26,176)(29,215,51,201)(30,216,52,202)(31,217,53,203)(32,218,54,204)(33,219,55,205)(34,220,56,206)(35,221,43,207)(36,222,44,208)(37,223,45,209)(38,224,46,210)(39,211,47,197)(40,212,48,198)(41,213,49,199)(42,214,50,200)(57,130,80,122)(58,131,81,123)(59,132,82,124)(60,133,83,125)(61,134,84,126)(62,135,71,113)(63,136,72,114)(64,137,73,115)(65,138,74,116)(66,139,75,117)(67,140,76,118)(68,127,77,119)(69,128,78,120)(70,129,79,121)(85,168,112,152)(86,155,99,153)(87,156,100,154)(88,157,101,141)(89,158,102,142)(90,159,103,143)(91,160,104,144)(92,161,105,145)(93,162,106,146)(94,163,107,147)(95,164,108,148)(96,165,109,149)(97,166,110,150)(98,167,111,151), (1,118,8,125)(2,119,9,126)(3,120,10,113)(4,121,11,114)(5,122,12,115)(6,123,13,116)(7,124,14,117)(15,128,22,135)(16,129,23,136)(17,130,24,137)(18,131,25,138)(19,132,26,139)(20,133,27,140)(21,134,28,127)(29,145,36,152)(30,146,37,153)(31,147,38,154)(32,148,39,141)(33,149,40,142)(34,150,41,143)(35,151,42,144)(43,167,50,160)(44,168,51,161)(45,155,52,162)(46,156,53,163)(47,157,54,164)(48,158,55,165)(49,159,56,166)(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)(85,201,92,208)(86,202,93,209)(87,203,94,210)(88,204,95,197)(89,205,96,198)(90,206,97,199)(91,207,98,200)(99,216,106,223)(100,217,107,224)(101,218,108,211)(102,219,109,212)(103,220,110,213)(104,221,111,214)(105,222,112,215), (1,99,20,93)(2,112,21,92)(3,111,22,91)(4,110,23,90)(5,109,24,89)(6,108,25,88)(7,107,26,87)(8,106,27,86)(9,105,28,85)(10,104,15,98)(11,103,16,97)(12,102,17,96)(13,101,18,95)(14,100,19,94)(29,61,44,77)(30,60,45,76)(31,59,46,75)(32,58,47,74)(33,57,48,73)(34,70,49,72)(35,69,50,71)(36,68,51,84)(37,67,52,83)(38,66,53,82)(39,65,54,81)(40,64,55,80)(41,63,56,79)(42,62,43,78)(113,221,128,200)(114,220,129,199)(115,219,130,198)(116,218,131,197)(117,217,132,210)(118,216,133,209)(119,215,134,208)(120,214,135,207)(121,213,136,206)(122,212,137,205)(123,211,138,204)(124,224,139,203)(125,223,140,202)(126,222,127,201)(141,182,164,183)(142,181,165,196)(143,180,166,195)(144,179,167,194)(145,178,168,193)(146,177,155,192)(147,176,156,191)(148,175,157,190)(149,174,158,189)(150,173,159,188)(151,172,160,187)(152,171,161,186)(153,170,162,185)(154,169,163,184), (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)(15,54)(16,55)(17,56)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(57,90)(58,91)(59,92)(60,93)(61,94)(62,95)(63,96)(64,97)(65,98)(66,85)(67,86)(68,87)(69,88)(70,89)(71,108)(72,109)(73,110)(74,111)(75,112)(76,99)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(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)(169,201)(170,202)(171,203)(172,204)(173,205)(174,206)(175,207)(176,208)(177,209)(178,210)(179,197)(180,198)(181,199)(182,200)(183,221)(184,222)(185,223)(186,224)(187,211)(188,212)(189,213)(190,214)(191,215)(192,216)(193,217)(194,218)(195,219)(196,220) );
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)], [(1,185,27,177),(2,186,28,178),(3,187,15,179),(4,188,16,180),(5,189,17,181),(6,190,18,182),(7,191,19,169),(8,192,20,170),(9,193,21,171),(10,194,22,172),(11,195,23,173),(12,196,24,174),(13,183,25,175),(14,184,26,176),(29,215,51,201),(30,216,52,202),(31,217,53,203),(32,218,54,204),(33,219,55,205),(34,220,56,206),(35,221,43,207),(36,222,44,208),(37,223,45,209),(38,224,46,210),(39,211,47,197),(40,212,48,198),(41,213,49,199),(42,214,50,200),(57,130,80,122),(58,131,81,123),(59,132,82,124),(60,133,83,125),(61,134,84,126),(62,135,71,113),(63,136,72,114),(64,137,73,115),(65,138,74,116),(66,139,75,117),(67,140,76,118),(68,127,77,119),(69,128,78,120),(70,129,79,121),(85,168,112,152),(86,155,99,153),(87,156,100,154),(88,157,101,141),(89,158,102,142),(90,159,103,143),(91,160,104,144),(92,161,105,145),(93,162,106,146),(94,163,107,147),(95,164,108,148),(96,165,109,149),(97,166,110,150),(98,167,111,151)], [(1,118,8,125),(2,119,9,126),(3,120,10,113),(4,121,11,114),(5,122,12,115),(6,123,13,116),(7,124,14,117),(15,128,22,135),(16,129,23,136),(17,130,24,137),(18,131,25,138),(19,132,26,139),(20,133,27,140),(21,134,28,127),(29,145,36,152),(30,146,37,153),(31,147,38,154),(32,148,39,141),(33,149,40,142),(34,150,41,143),(35,151,42,144),(43,167,50,160),(44,168,51,161),(45,155,52,162),(46,156,53,163),(47,157,54,164),(48,158,55,165),(49,159,56,166),(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),(85,201,92,208),(86,202,93,209),(87,203,94,210),(88,204,95,197),(89,205,96,198),(90,206,97,199),(91,207,98,200),(99,216,106,223),(100,217,107,224),(101,218,108,211),(102,219,109,212),(103,220,110,213),(104,221,111,214),(105,222,112,215)], [(1,99,20,93),(2,112,21,92),(3,111,22,91),(4,110,23,90),(5,109,24,89),(6,108,25,88),(7,107,26,87),(8,106,27,86),(9,105,28,85),(10,104,15,98),(11,103,16,97),(12,102,17,96),(13,101,18,95),(14,100,19,94),(29,61,44,77),(30,60,45,76),(31,59,46,75),(32,58,47,74),(33,57,48,73),(34,70,49,72),(35,69,50,71),(36,68,51,84),(37,67,52,83),(38,66,53,82),(39,65,54,81),(40,64,55,80),(41,63,56,79),(42,62,43,78),(113,221,128,200),(114,220,129,199),(115,219,130,198),(116,218,131,197),(117,217,132,210),(118,216,133,209),(119,215,134,208),(120,214,135,207),(121,213,136,206),(122,212,137,205),(123,211,138,204),(124,224,139,203),(125,223,140,202),(126,222,127,201),(141,182,164,183),(142,181,165,196),(143,180,166,195),(144,179,167,194),(145,178,168,193),(146,177,155,192),(147,176,156,191),(148,175,157,190),(149,174,158,189),(150,173,159,188),(151,172,160,187),(152,171,161,186),(153,170,162,185),(154,169,163,184)], [(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),(15,54),(16,55),(17,56),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(57,90),(58,91),(59,92),(60,93),(61,94),(62,95),(63,96),(64,97),(65,98),(66,85),(67,86),(68,87),(69,88),(70,89),(71,108),(72,109),(73,110),(74,111),(75,112),(76,99),(77,100),(78,101),(79,102),(80,103),(81,104),(82,105),(83,106),(84,107),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(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),(169,201),(170,202),(171,203),(172,204),(173,205),(174,206),(175,207),(176,208),(177,209),(178,210),(179,197),(180,198),(181,199),(182,200),(183,221),(184,222),(185,223),(186,224),(187,211),(188,212),(189,213),(190,214),(191,215),(192,216),(193,217),(194,218),(195,219),(196,220)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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