?
G = C14.802- (1+4) order 448 = 26·7
35th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.802- (1+4), C4⋊C4.104D14, (C2×D4).160D14, Dic7.Q8⋊25C2, (C2×C28).68C23, C22⋊C4.67D14, C4.Dic14⋊28C2, (D4×Dic7).13C2, Dic7⋊3Q8⋊31C2, (C2×C14).194C24, (C22×C4).255D14, C22.D4.2D7, Dic7.25(C4○D4), C22⋊Dic14⋊29C2, (D4×C14).132C22, C23.D14⋊28C2, Dic7⋊C4.39C22, C4⋊Dic7.225C22, (C22×C14).30C23, (C2×Dic7).99C23, C22.215(C23×D7), C23.199(C22×D7), C23.D7.40C22, C23.21D14⋊11C2, C23.11D14⋊12C2, C22.18(D4⋊2D7), (C22×C28).112C22, C7⋊8(C22.46C24), (C4×Dic7).121C22, C23.18D14.2C2, C2.41(D4.10D14), (C2×Dic14).167C22, (C22×Dic7).127C22, C2.58(D7×C4○D4), (C2×Dic7⋊C4)⋊26C2, C14.170(C2×C4○D4), C2.52(C2×D4⋊2D7), (C2×C14).46(C4○D4), (C7×C4⋊C4).174C22, (C2×C4).296(C22×D7), (C7×C22⋊C4).49C22, (C7×C22.D4).2C2, SmallGroup(448,1103)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 780 in 214 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×14], C22, C22 [×2], C22 [×5], C7, C2×C4 [×5], C2×C4 [×16], D4 [×2], Q8 [×2], C23 [×2], C14 [×3], C14 [×3], C42 [×5], C22⋊C4 [×3], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4, C22×C4 [×3], C2×D4, C2×Q8, Dic7 [×2], Dic7 [×7], C28 [×5], C2×C14, C2×C14 [×2], C2×C14 [×5], C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4, C22.D4, C42.C2 [×3], C42⋊2C2 [×2], Dic14 [×2], C2×Dic7 [×8], C2×Dic7 [×6], C2×C28 [×5], C2×C28 [×2], C7×D4 [×2], C22×C14 [×2], C22.46C24, C4×Dic7 [×5], Dic7⋊C4 [×10], C4⋊Dic7 [×4], C23.D7 [×5], C7×C22⋊C4 [×3], C7×C4⋊C4 [×2], C2×Dic14, C22×Dic7 [×3], C22×C28, D4×C14, C23.11D14 [×2], C22⋊Dic14 [×2], C23.D14 [×2], Dic7⋊3Q8, Dic7.Q8 [×2], C4.Dic14, C2×Dic7⋊C4, C23.21D14, D4×Dic7, C23.18D14, C7×C22.D4, C14.802- (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], C22.46C24, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, D7×C4○D4, D4.10D14, C14.802- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=e2=a7b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=a7b-1, bd=db, be=eb, cd=dc, ece-1=a7c, 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)
(1 143 119 106)(2 142 120 105)(3 141 121 104)(4 154 122 103)(5 153 123 102)(6 152 124 101)(7 151 125 100)(8 150 126 99)(9 149 113 112)(10 148 114 111)(11 147 115 110)(12 146 116 109)(13 145 117 108)(14 144 118 107)(15 67 215 181)(16 66 216 180)(17 65 217 179)(18 64 218 178)(19 63 219 177)(20 62 220 176)(21 61 221 175)(22 60 222 174)(23 59 223 173)(24 58 224 172)(25 57 211 171)(26 70 212 170)(27 69 213 169)(28 68 214 182)(29 88 71 45)(30 87 72 44)(31 86 73 43)(32 85 74 56)(33 98 75 55)(34 97 76 54)(35 96 77 53)(36 95 78 52)(37 94 79 51)(38 93 80 50)(39 92 81 49)(40 91 82 48)(41 90 83 47)(42 89 84 46)(127 198 189 159)(128 197 190 158)(129 210 191 157)(130 209 192 156)(131 208 193 155)(132 207 194 168)(133 206 195 167)(134 205 196 166)(135 204 183 165)(136 203 184 164)(137 202 185 163)(138 201 186 162)(139 200 187 161)(140 199 188 160)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 43)(14 44)(15 139)(16 140)(17 127)(18 128)(19 129)(20 130)(21 131)(22 132)(23 133)(24 134)(25 135)(26 136)(27 137)(28 138)(29 99)(30 100)(31 101)(32 102)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 111)(42 112)(57 158)(58 159)(59 160)(60 161)(61 162)(62 163)(63 164)(64 165)(65 166)(66 167)(67 168)(68 155)(69 156)(70 157)(71 150)(72 151)(73 152)(74 153)(75 154)(76 141)(77 142)(78 143)(79 144)(80 145)(81 146)(82 147)(83 148)(84 149)(85 116)(86 117)(87 118)(88 119)(89 120)(90 121)(91 122)(92 123)(93 124)(94 125)(95 126)(96 113)(97 114)(98 115)(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)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 219)(192 220)(193 221)(194 222)(195 223)(196 224)
(1 134 126 189)(2 133 113 188)(3 132 114 187)(4 131 115 186)(5 130 116 185)(6 129 117 184)(7 128 118 183)(8 127 119 196)(9 140 120 195)(10 139 121 194)(11 138 122 193)(12 137 123 192)(13 136 124 191)(14 135 125 190)(15 90 222 54)(16 89 223 53)(17 88 224 52)(18 87 211 51)(19 86 212 50)(20 85 213 49)(21 98 214 48)(22 97 215 47)(23 96 216 46)(24 95 217 45)(25 94 218 44)(26 93 219 43)(27 92 220 56)(28 91 221 55)(29 58 78 179)(30 57 79 178)(31 70 80 177)(32 69 81 176)(33 68 82 175)(34 67 83 174)(35 66 84 173)(36 65 71 172)(37 64 72 171)(38 63 73 170)(39 62 74 169)(40 61 75 182)(41 60 76 181)(42 59 77 180)(99 159 143 205)(100 158 144 204)(101 157 145 203)(102 156 146 202)(103 155 147 201)(104 168 148 200)(105 167 149 199)(106 166 150 198)(107 165 151 197)(108 164 152 210)(109 163 153 209)(110 162 154 208)(111 161 141 207)(112 160 142 206)
(1 159 126 205)(2 160 113 206)(3 161 114 207)(4 162 115 208)(5 163 116 209)(6 164 117 210)(7 165 118 197)(8 166 119 198)(9 167 120 199)(10 168 121 200)(11 155 122 201)(12 156 123 202)(13 157 124 203)(14 158 125 204)(15 34 222 83)(16 35 223 84)(17 36 224 71)(18 37 211 72)(19 38 212 73)(20 39 213 74)(21 40 214 75)(22 41 215 76)(23 42 216 77)(24 29 217 78)(25 30 218 79)(26 31 219 80)(27 32 220 81)(28 33 221 82)(43 63 93 170)(44 64 94 171)(45 65 95 172)(46 66 96 173)(47 67 97 174)(48 68 98 175)(49 69 85 176)(50 70 86 177)(51 57 87 178)(52 58 88 179)(53 59 89 180)(54 60 90 181)(55 61 91 182)(56 62 92 169)(99 196 143 127)(100 183 144 128)(101 184 145 129)(102 185 146 130)(103 186 147 131)(104 187 148 132)(105 188 149 133)(106 189 150 134)(107 190 151 135)(108 191 152 136)(109 192 153 137)(110 193 154 138)(111 194 141 139)(112 195 142 140)
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), (1,143,119,106)(2,142,120,105)(3,141,121,104)(4,154,122,103)(5,153,123,102)(6,152,124,101)(7,151,125,100)(8,150,126,99)(9,149,113,112)(10,148,114,111)(11,147,115,110)(12,146,116,109)(13,145,117,108)(14,144,118,107)(15,67,215,181)(16,66,216,180)(17,65,217,179)(18,64,218,178)(19,63,219,177)(20,62,220,176)(21,61,221,175)(22,60,222,174)(23,59,223,173)(24,58,224,172)(25,57,211,171)(26,70,212,170)(27,69,213,169)(28,68,214,182)(29,88,71,45)(30,87,72,44)(31,86,73,43)(32,85,74,56)(33,98,75,55)(34,97,76,54)(35,96,77,53)(36,95,78,52)(37,94,79,51)(38,93,80,50)(39,92,81,49)(40,91,82,48)(41,90,83,47)(42,89,84,46)(127,198,189,159)(128,197,190,158)(129,210,191,157)(130,209,192,156)(131,208,193,155)(132,207,194,168)(133,206,195,167)(134,205,196,166)(135,204,183,165)(136,203,184,164)(137,202,185,163)(138,201,186,162)(139,200,187,161)(140,199,188,160), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,43)(14,44)(15,139)(16,140)(17,127)(18,128)(19,129)(20,130)(21,131)(22,132)(23,133)(24,134)(25,135)(26,136)(27,137)(28,138)(29,99)(30,100)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,111)(42,112)(57,158)(58,159)(59,160)(60,161)(61,162)(62,163)(63,164)(64,165)(65,166)(66,167)(67,168)(68,155)(69,156)(70,157)(71,150)(72,151)(73,152)(74,153)(75,154)(76,141)(77,142)(78,143)(79,144)(80,145)(81,146)(82,147)(83,148)(84,149)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,113)(97,114)(98,115)(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)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (1,134,126,189)(2,133,113,188)(3,132,114,187)(4,131,115,186)(5,130,116,185)(6,129,117,184)(7,128,118,183)(8,127,119,196)(9,140,120,195)(10,139,121,194)(11,138,122,193)(12,137,123,192)(13,136,124,191)(14,135,125,190)(15,90,222,54)(16,89,223,53)(17,88,224,52)(18,87,211,51)(19,86,212,50)(20,85,213,49)(21,98,214,48)(22,97,215,47)(23,96,216,46)(24,95,217,45)(25,94,218,44)(26,93,219,43)(27,92,220,56)(28,91,221,55)(29,58,78,179)(30,57,79,178)(31,70,80,177)(32,69,81,176)(33,68,82,175)(34,67,83,174)(35,66,84,173)(36,65,71,172)(37,64,72,171)(38,63,73,170)(39,62,74,169)(40,61,75,182)(41,60,76,181)(42,59,77,180)(99,159,143,205)(100,158,144,204)(101,157,145,203)(102,156,146,202)(103,155,147,201)(104,168,148,200)(105,167,149,199)(106,166,150,198)(107,165,151,197)(108,164,152,210)(109,163,153,209)(110,162,154,208)(111,161,141,207)(112,160,142,206), (1,159,126,205)(2,160,113,206)(3,161,114,207)(4,162,115,208)(5,163,116,209)(6,164,117,210)(7,165,118,197)(8,166,119,198)(9,167,120,199)(10,168,121,200)(11,155,122,201)(12,156,123,202)(13,157,124,203)(14,158,125,204)(15,34,222,83)(16,35,223,84)(17,36,224,71)(18,37,211,72)(19,38,212,73)(20,39,213,74)(21,40,214,75)(22,41,215,76)(23,42,216,77)(24,29,217,78)(25,30,218,79)(26,31,219,80)(27,32,220,81)(28,33,221,82)(43,63,93,170)(44,64,94,171)(45,65,95,172)(46,66,96,173)(47,67,97,174)(48,68,98,175)(49,69,85,176)(50,70,86,177)(51,57,87,178)(52,58,88,179)(53,59,89,180)(54,60,90,181)(55,61,91,182)(56,62,92,169)(99,196,143,127)(100,183,144,128)(101,184,145,129)(102,185,146,130)(103,186,147,131)(104,187,148,132)(105,188,149,133)(106,189,150,134)(107,190,151,135)(108,191,152,136)(109,192,153,137)(110,193,154,138)(111,194,141,139)(112,195,142,140)>;
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), (1,143,119,106)(2,142,120,105)(3,141,121,104)(4,154,122,103)(5,153,123,102)(6,152,124,101)(7,151,125,100)(8,150,126,99)(9,149,113,112)(10,148,114,111)(11,147,115,110)(12,146,116,109)(13,145,117,108)(14,144,118,107)(15,67,215,181)(16,66,216,180)(17,65,217,179)(18,64,218,178)(19,63,219,177)(20,62,220,176)(21,61,221,175)(22,60,222,174)(23,59,223,173)(24,58,224,172)(25,57,211,171)(26,70,212,170)(27,69,213,169)(28,68,214,182)(29,88,71,45)(30,87,72,44)(31,86,73,43)(32,85,74,56)(33,98,75,55)(34,97,76,54)(35,96,77,53)(36,95,78,52)(37,94,79,51)(38,93,80,50)(39,92,81,49)(40,91,82,48)(41,90,83,47)(42,89,84,46)(127,198,189,159)(128,197,190,158)(129,210,191,157)(130,209,192,156)(131,208,193,155)(132,207,194,168)(133,206,195,167)(134,205,196,166)(135,204,183,165)(136,203,184,164)(137,202,185,163)(138,201,186,162)(139,200,187,161)(140,199,188,160), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,43)(14,44)(15,139)(16,140)(17,127)(18,128)(19,129)(20,130)(21,131)(22,132)(23,133)(24,134)(25,135)(26,136)(27,137)(28,138)(29,99)(30,100)(31,101)(32,102)(33,103)(34,104)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,111)(42,112)(57,158)(58,159)(59,160)(60,161)(61,162)(62,163)(63,164)(64,165)(65,166)(66,167)(67,168)(68,155)(69,156)(70,157)(71,150)(72,151)(73,152)(74,153)(75,154)(76,141)(77,142)(78,143)(79,144)(80,145)(81,146)(82,147)(83,148)(84,149)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,113)(97,114)(98,115)(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)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (1,134,126,189)(2,133,113,188)(3,132,114,187)(4,131,115,186)(5,130,116,185)(6,129,117,184)(7,128,118,183)(8,127,119,196)(9,140,120,195)(10,139,121,194)(11,138,122,193)(12,137,123,192)(13,136,124,191)(14,135,125,190)(15,90,222,54)(16,89,223,53)(17,88,224,52)(18,87,211,51)(19,86,212,50)(20,85,213,49)(21,98,214,48)(22,97,215,47)(23,96,216,46)(24,95,217,45)(25,94,218,44)(26,93,219,43)(27,92,220,56)(28,91,221,55)(29,58,78,179)(30,57,79,178)(31,70,80,177)(32,69,81,176)(33,68,82,175)(34,67,83,174)(35,66,84,173)(36,65,71,172)(37,64,72,171)(38,63,73,170)(39,62,74,169)(40,61,75,182)(41,60,76,181)(42,59,77,180)(99,159,143,205)(100,158,144,204)(101,157,145,203)(102,156,146,202)(103,155,147,201)(104,168,148,200)(105,167,149,199)(106,166,150,198)(107,165,151,197)(108,164,152,210)(109,163,153,209)(110,162,154,208)(111,161,141,207)(112,160,142,206), (1,159,126,205)(2,160,113,206)(3,161,114,207)(4,162,115,208)(5,163,116,209)(6,164,117,210)(7,165,118,197)(8,166,119,198)(9,167,120,199)(10,168,121,200)(11,155,122,201)(12,156,123,202)(13,157,124,203)(14,158,125,204)(15,34,222,83)(16,35,223,84)(17,36,224,71)(18,37,211,72)(19,38,212,73)(20,39,213,74)(21,40,214,75)(22,41,215,76)(23,42,216,77)(24,29,217,78)(25,30,218,79)(26,31,219,80)(27,32,220,81)(28,33,221,82)(43,63,93,170)(44,64,94,171)(45,65,95,172)(46,66,96,173)(47,67,97,174)(48,68,98,175)(49,69,85,176)(50,70,86,177)(51,57,87,178)(52,58,88,179)(53,59,89,180)(54,60,90,181)(55,61,91,182)(56,62,92,169)(99,196,143,127)(100,183,144,128)(101,184,145,129)(102,185,146,130)(103,186,147,131)(104,187,148,132)(105,188,149,133)(106,189,150,134)(107,190,151,135)(108,191,152,136)(109,192,153,137)(110,193,154,138)(111,194,141,139)(112,195,142,140) );
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)], [(1,143,119,106),(2,142,120,105),(3,141,121,104),(4,154,122,103),(5,153,123,102),(6,152,124,101),(7,151,125,100),(8,150,126,99),(9,149,113,112),(10,148,114,111),(11,147,115,110),(12,146,116,109),(13,145,117,108),(14,144,118,107),(15,67,215,181),(16,66,216,180),(17,65,217,179),(18,64,218,178),(19,63,219,177),(20,62,220,176),(21,61,221,175),(22,60,222,174),(23,59,223,173),(24,58,224,172),(25,57,211,171),(26,70,212,170),(27,69,213,169),(28,68,214,182),(29,88,71,45),(30,87,72,44),(31,86,73,43),(32,85,74,56),(33,98,75,55),(34,97,76,54),(35,96,77,53),(36,95,78,52),(37,94,79,51),(38,93,80,50),(39,92,81,49),(40,91,82,48),(41,90,83,47),(42,89,84,46),(127,198,189,159),(128,197,190,158),(129,210,191,157),(130,209,192,156),(131,208,193,155),(132,207,194,168),(133,206,195,167),(134,205,196,166),(135,204,183,165),(136,203,184,164),(137,202,185,163),(138,201,186,162),(139,200,187,161),(140,199,188,160)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,43),(14,44),(15,139),(16,140),(17,127),(18,128),(19,129),(20,130),(21,131),(22,132),(23,133),(24,134),(25,135),(26,136),(27,137),(28,138),(29,99),(30,100),(31,101),(32,102),(33,103),(34,104),(35,105),(36,106),(37,107),(38,108),(39,109),(40,110),(41,111),(42,112),(57,158),(58,159),(59,160),(60,161),(61,162),(62,163),(63,164),(64,165),(65,166),(66,167),(67,168),(68,155),(69,156),(70,157),(71,150),(72,151),(73,152),(74,153),(75,154),(76,141),(77,142),(78,143),(79,144),(80,145),(81,146),(82,147),(83,148),(84,149),(85,116),(86,117),(87,118),(88,119),(89,120),(90,121),(91,122),(92,123),(93,124),(94,125),(95,126),(96,113),(97,114),(98,115),(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),(183,211),(184,212),(185,213),(186,214),(187,215),(188,216),(189,217),(190,218),(191,219),(192,220),(193,221),(194,222),(195,223),(196,224)], [(1,134,126,189),(2,133,113,188),(3,132,114,187),(4,131,115,186),(5,130,116,185),(6,129,117,184),(7,128,118,183),(8,127,119,196),(9,140,120,195),(10,139,121,194),(11,138,122,193),(12,137,123,192),(13,136,124,191),(14,135,125,190),(15,90,222,54),(16,89,223,53),(17,88,224,52),(18,87,211,51),(19,86,212,50),(20,85,213,49),(21,98,214,48),(22,97,215,47),(23,96,216,46),(24,95,217,45),(25,94,218,44),(26,93,219,43),(27,92,220,56),(28,91,221,55),(29,58,78,179),(30,57,79,178),(31,70,80,177),(32,69,81,176),(33,68,82,175),(34,67,83,174),(35,66,84,173),(36,65,71,172),(37,64,72,171),(38,63,73,170),(39,62,74,169),(40,61,75,182),(41,60,76,181),(42,59,77,180),(99,159,143,205),(100,158,144,204),(101,157,145,203),(102,156,146,202),(103,155,147,201),(104,168,148,200),(105,167,149,199),(106,166,150,198),(107,165,151,197),(108,164,152,210),(109,163,153,209),(110,162,154,208),(111,161,141,207),(112,160,142,206)], [(1,159,126,205),(2,160,113,206),(3,161,114,207),(4,162,115,208),(5,163,116,209),(6,164,117,210),(7,165,118,197),(8,166,119,198),(9,167,120,199),(10,168,121,200),(11,155,122,201),(12,156,123,202),(13,157,124,203),(14,158,125,204),(15,34,222,83),(16,35,223,84),(17,36,224,71),(18,37,211,72),(19,38,212,73),(20,39,213,74),(21,40,214,75),(22,41,215,76),(23,42,216,77),(24,29,217,78),(25,30,218,79),(26,31,219,80),(27,32,220,81),(28,33,221,82),(43,63,93,170),(44,64,94,171),(45,65,95,172),(46,66,96,173),(47,67,97,174),(48,68,98,175),(49,69,85,176),(50,70,86,177),(51,57,87,178),(52,58,88,179),(53,59,89,180),(54,60,90,181),(55,61,91,182),(56,62,92,169),(99,196,143,127),(100,183,144,128),(101,184,145,129),(102,185,146,130),(103,186,147,131),(104,187,148,132),(105,188,149,133),(106,189,150,134),(107,190,151,135),(108,191,152,136),(109,192,153,137),(110,193,154,138),(111,194,141,139),(112,195,142,140)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 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 | 0 | 25 |
| 0 | 0 | 0 | 0 | 22 | 7 |
| 28 | 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 0 | 24 | 25 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 14 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 27 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 27 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 0 | 24 | 25 |
| 12 | 10 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 17 | 0 | 0 |
| 0 | 0 | 24 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,22,0,0,0,0,25,7],[28,0,0,0,0,0,4,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,4,24,0,0,0,0,3,25],[28,14,0,0,0,0,0,1,0,0,0,0,0,0,28,27,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,28,27,0,0,0,0,0,1,0,0,0,0,0,0,4,24,0,0,0,0,3,25],[12,0,0,0,0,0,10,17,0,0,0,0,0,0,12,24,0,0,0,0,17,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 4Q | 4R | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| order | 1 | 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 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | 2- (1+4) | D4⋊2D7 | D7×C4○D4 | D4.10D14 |
| kernel | C14.802- (1+4) | C23.11D14 | C22⋊Dic14 | C23.D14 | Dic7⋊3Q8 | Dic7.Q8 | C4.Dic14 | C2×Dic7⋊C4 | C23.21D14 | D4×Dic7 | C23.18D14 | C7×C22.D4 | C22.D4 | Dic7 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 9 | 6 | 3 | 3 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{80}2_-^{(1+4)} % in TeX
G:=Group("C14.80ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1103);
// by ID
G=gap.SmallGroup(448,1103);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,100,346,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=e^2=a^7*b^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,a*e=e*a,c*b*c=a^7*b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^7*c,e*d*e^-1=b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀