G = C14.1082- 1+4 order 448 = 26·7
63rd non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.1082- 1+4, C14.1472+ 1+4, (C2×C28)⋊18D4, C28⋊7D4⋊49C2, C28⋊2D4⋊43C2, C28.431(C2×D4), D14⋊3Q8⋊45C2, (C2×D4).238D14, (C2×Q8).195D14, Dic7⋊D4⋊45C2, C28.48D4⋊49C2, (C2×C28).654C23, (C2×C14).319C24, D14⋊C4.79C22, C14.169(C22×D4), (C22×C4).289D14, C2.71(D4⋊8D14), (D4×C14).315C22, (C2×D28).231C22, Dic7⋊C4.93C22, C4⋊Dic7.393C22, (Q8×C14).245C22, C22.328(C23×D7), C23.140(C22×D7), C23.D7.77C22, (C22×C28).321C22, (C22×C14).245C23, C7⋊5(C22.31C24), (C2×Dic7).165C23, (C22×D7).140C23, C2.71(D4.10D14), (C2×Dic14).259C22, (C22×Dic7).168C22, (C2×C4○D4)⋊11D7, (C2×C4)⋊8(C7⋊D4), (C14×C4○D4)⋊11C2, (C2×C4○D28)⋊33C2, (C2×C4⋊Dic7)⋊48C2, (C2×C14).84(C2×D4), C4.101(C2×C7⋊D4), C22.2(C2×C7⋊D4), (C2×C4×D7).170C22, C2.42(C22×C7⋊D4), (C2×C4).641(C22×D7), (C2×C7⋊D4).82C22, SmallGroup(448,1286)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.1082- 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, bd=db, ebe-1=a7b, dcd-1=a7c, ce=ec, ede-1=b2d >
Subgroups: 1300 in 294 conjugacy classes, 111 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C2×C4○D4, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×C14, C22×C14, C22.31C24, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C22×Dic7, C2×C7⋊D4, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C28.48D4, C2×C4⋊Dic7, C28⋊7D4, C28⋊2D4, Dic7⋊D4, D14⋊3Q8, C2×C4○D28, C14×C4○D4, C14.1082- 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, 2- 1+4, C7⋊D4, C22×D7, C22.31C24, C2×C7⋊D4, C23×D7, D4⋊8D14, D4.10D14, C22×C7⋊D4, C14.1082- 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)
(1 195 40 21)(2 194 41 20)(3 193 42 19)(4 192 29 18)(5 191 30 17)(6 190 31 16)(7 189 32 15)(8 188 33 28)(9 187 34 27)(10 186 35 26)(11 185 36 25)(12 184 37 24)(13 183 38 23)(14 196 39 22)(43 134 170 95)(44 133 171 94)(45 132 172 93)(46 131 173 92)(47 130 174 91)(48 129 175 90)(49 128 176 89)(50 127 177 88)(51 140 178 87)(52 139 179 86)(53 138 180 85)(54 137 181 98)(55 136 182 97)(56 135 169 96)(57 162 221 113)(58 161 222 126)(59 160 223 125)(60 159 224 124)(61 158 211 123)(62 157 212 122)(63 156 213 121)(64 155 214 120)(65 168 215 119)(66 167 216 118)(67 166 217 117)(68 165 218 116)(69 164 219 115)(70 163 220 114)(71 202 106 145)(72 201 107 144)(73 200 108 143)(74 199 109 142)(75 198 110 141)(76 197 111 154)(77 210 112 153)(78 209 99 152)(79 208 100 151)(80 207 101 150)(81 206 102 149)(82 205 103 148)(83 204 104 147)(84 203 105 146)
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 121)(10 122)(11 123)(12 124)(13 125)(14 126)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 165)(30 166)(31 167)(32 168)(33 155)(34 156)(35 157)(36 158)(37 159)(38 160)(39 161)(40 162)(41 163)(42 164)(43 199)(44 200)(45 201)(46 202)(47 203)(48 204)(49 205)(50 206)(51 207)(52 208)(53 209)(54 210)(55 197)(56 198)(71 131)(72 132)(73 133)(74 134)(75 135)(76 136)(77 137)(78 138)(79 139)(80 140)(81 127)(82 128)(83 129)(84 130)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(141 169)(142 170)(143 171)(144 172)(145 173)(146 174)(147 175)(148 176)(149 177)(150 178)(151 179)(152 180)(153 181)(154 182)(183 223)(184 224)(185 211)(186 212)(187 213)(188 214)(189 215)(190 216)(191 217)(192 218)(193 219)(194 220)(195 221)(196 222)
(1 21 40 195)(2 20 41 194)(3 19 42 193)(4 18 29 192)(5 17 30 191)(6 16 31 190)(7 15 32 189)(8 28 33 188)(9 27 34 187)(10 26 35 186)(11 25 36 185)(12 24 37 184)(13 23 38 183)(14 22 39 196)(43 127 170 88)(44 140 171 87)(45 139 172 86)(46 138 173 85)(47 137 174 98)(48 136 175 97)(49 135 176 96)(50 134 177 95)(51 133 178 94)(52 132 179 93)(53 131 180 92)(54 130 181 91)(55 129 182 90)(56 128 169 89)(57 155 221 120)(58 168 222 119)(59 167 223 118)(60 166 224 117)(61 165 211 116)(62 164 212 115)(63 163 213 114)(64 162 214 113)(65 161 215 126)(66 160 216 125)(67 159 217 124)(68 158 218 123)(69 157 219 122)(70 156 220 121)(71 145 106 202)(72 144 107 201)(73 143 108 200)(74 142 109 199)(75 141 110 198)(76 154 111 197)(77 153 112 210)(78 152 99 209)(79 151 100 208)(80 150 101 207)(81 149 102 206)(82 148 103 205)(83 147 104 204)(84 146 105 203)
(1 171 33 51)(2 172 34 52)(3 173 35 53)(4 174 36 54)(5 175 37 55)(6 176 38 56)(7 177 39 43)(8 178 40 44)(9 179 41 45)(10 180 42 46)(11 181 29 47)(12 182 30 48)(13 169 31 49)(14 170 32 50)(15 134 196 88)(16 135 183 89)(17 136 184 90)(18 137 185 91)(19 138 186 92)(20 139 187 93)(21 140 188 94)(22 127 189 95)(23 128 190 96)(24 129 191 97)(25 130 192 98)(26 131 193 85)(27 132 194 86)(28 133 195 87)(57 80 214 108)(58 81 215 109)(59 82 216 110)(60 83 217 111)(61 84 218 112)(62 71 219 99)(63 72 220 100)(64 73 221 101)(65 74 222 102)(66 75 223 103)(67 76 224 104)(68 77 211 105)(69 78 212 106)(70 79 213 107)(113 143 155 207)(114 144 156 208)(115 145 157 209)(116 146 158 210)(117 147 159 197)(118 148 160 198)(119 149 161 199)(120 150 162 200)(121 151 163 201)(122 152 164 202)(123 153 165 203)(124 154 166 204)(125 141 167 205)(126 142 168 206)
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), (1,195,40,21)(2,194,41,20)(3,193,42,19)(4,192,29,18)(5,191,30,17)(6,190,31,16)(7,189,32,15)(8,188,33,28)(9,187,34,27)(10,186,35,26)(11,185,36,25)(12,184,37,24)(13,183,38,23)(14,196,39,22)(43,134,170,95)(44,133,171,94)(45,132,172,93)(46,131,173,92)(47,130,174,91)(48,129,175,90)(49,128,176,89)(50,127,177,88)(51,140,178,87)(52,139,179,86)(53,138,180,85)(54,137,181,98)(55,136,182,97)(56,135,169,96)(57,162,221,113)(58,161,222,126)(59,160,223,125)(60,159,224,124)(61,158,211,123)(62,157,212,122)(63,156,213,121)(64,155,214,120)(65,168,215,119)(66,167,216,118)(67,166,217,117)(68,165,218,116)(69,164,219,115)(70,163,220,114)(71,202,106,145)(72,201,107,144)(73,200,108,143)(74,199,109,142)(75,198,110,141)(76,197,111,154)(77,210,112,153)(78,209,99,152)(79,208,100,151)(80,207,101,150)(81,206,102,149)(82,205,103,148)(83,204,104,147)(84,203,105,146), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,165)(30,166)(31,167)(32,168)(33,155)(34,156)(35,157)(36,158)(37,159)(38,160)(39,161)(40,162)(41,163)(42,164)(43,199)(44,200)(45,201)(46,202)(47,203)(48,204)(49,205)(50,206)(51,207)(52,208)(53,209)(54,210)(55,197)(56,198)(71,131)(72,132)(73,133)(74,134)(75,135)(76,136)(77,137)(78,138)(79,139)(80,140)(81,127)(82,128)(83,129)(84,130)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(141,169)(142,170)(143,171)(144,172)(145,173)(146,174)(147,175)(148,176)(149,177)(150,178)(151,179)(152,180)(153,181)(154,182)(183,223)(184,224)(185,211)(186,212)(187,213)(188,214)(189,215)(190,216)(191,217)(192,218)(193,219)(194,220)(195,221)(196,222), (1,21,40,195)(2,20,41,194)(3,19,42,193)(4,18,29,192)(5,17,30,191)(6,16,31,190)(7,15,32,189)(8,28,33,188)(9,27,34,187)(10,26,35,186)(11,25,36,185)(12,24,37,184)(13,23,38,183)(14,22,39,196)(43,127,170,88)(44,140,171,87)(45,139,172,86)(46,138,173,85)(47,137,174,98)(48,136,175,97)(49,135,176,96)(50,134,177,95)(51,133,178,94)(52,132,179,93)(53,131,180,92)(54,130,181,91)(55,129,182,90)(56,128,169,89)(57,155,221,120)(58,168,222,119)(59,167,223,118)(60,166,224,117)(61,165,211,116)(62,164,212,115)(63,163,213,114)(64,162,214,113)(65,161,215,126)(66,160,216,125)(67,159,217,124)(68,158,218,123)(69,157,219,122)(70,156,220,121)(71,145,106,202)(72,144,107,201)(73,143,108,200)(74,142,109,199)(75,141,110,198)(76,154,111,197)(77,153,112,210)(78,152,99,209)(79,151,100,208)(80,150,101,207)(81,149,102,206)(82,148,103,205)(83,147,104,204)(84,146,105,203), (1,171,33,51)(2,172,34,52)(3,173,35,53)(4,174,36,54)(5,175,37,55)(6,176,38,56)(7,177,39,43)(8,178,40,44)(9,179,41,45)(10,180,42,46)(11,181,29,47)(12,182,30,48)(13,169,31,49)(14,170,32,50)(15,134,196,88)(16,135,183,89)(17,136,184,90)(18,137,185,91)(19,138,186,92)(20,139,187,93)(21,140,188,94)(22,127,189,95)(23,128,190,96)(24,129,191,97)(25,130,192,98)(26,131,193,85)(27,132,194,86)(28,133,195,87)(57,80,214,108)(58,81,215,109)(59,82,216,110)(60,83,217,111)(61,84,218,112)(62,71,219,99)(63,72,220,100)(64,73,221,101)(65,74,222,102)(66,75,223,103)(67,76,224,104)(68,77,211,105)(69,78,212,106)(70,79,213,107)(113,143,155,207)(114,144,156,208)(115,145,157,209)(116,146,158,210)(117,147,159,197)(118,148,160,198)(119,149,161,199)(120,150,162,200)(121,151,163,201)(122,152,164,202)(123,153,165,203)(124,154,166,204)(125,141,167,205)(126,142,168,206)>;
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), (1,195,40,21)(2,194,41,20)(3,193,42,19)(4,192,29,18)(5,191,30,17)(6,190,31,16)(7,189,32,15)(8,188,33,28)(9,187,34,27)(10,186,35,26)(11,185,36,25)(12,184,37,24)(13,183,38,23)(14,196,39,22)(43,134,170,95)(44,133,171,94)(45,132,172,93)(46,131,173,92)(47,130,174,91)(48,129,175,90)(49,128,176,89)(50,127,177,88)(51,140,178,87)(52,139,179,86)(53,138,180,85)(54,137,181,98)(55,136,182,97)(56,135,169,96)(57,162,221,113)(58,161,222,126)(59,160,223,125)(60,159,224,124)(61,158,211,123)(62,157,212,122)(63,156,213,121)(64,155,214,120)(65,168,215,119)(66,167,216,118)(67,166,217,117)(68,165,218,116)(69,164,219,115)(70,163,220,114)(71,202,106,145)(72,201,107,144)(73,200,108,143)(74,199,109,142)(75,198,110,141)(76,197,111,154)(77,210,112,153)(78,209,99,152)(79,208,100,151)(80,207,101,150)(81,206,102,149)(82,205,103,148)(83,204,104,147)(84,203,105,146), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,65)(16,66)(17,67)(18,68)(19,69)(20,70)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,165)(30,166)(31,167)(32,168)(33,155)(34,156)(35,157)(36,158)(37,159)(38,160)(39,161)(40,162)(41,163)(42,164)(43,199)(44,200)(45,201)(46,202)(47,203)(48,204)(49,205)(50,206)(51,207)(52,208)(53,209)(54,210)(55,197)(56,198)(71,131)(72,132)(73,133)(74,134)(75,135)(76,136)(77,137)(78,138)(79,139)(80,140)(81,127)(82,128)(83,129)(84,130)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(141,169)(142,170)(143,171)(144,172)(145,173)(146,174)(147,175)(148,176)(149,177)(150,178)(151,179)(152,180)(153,181)(154,182)(183,223)(184,224)(185,211)(186,212)(187,213)(188,214)(189,215)(190,216)(191,217)(192,218)(193,219)(194,220)(195,221)(196,222), (1,21,40,195)(2,20,41,194)(3,19,42,193)(4,18,29,192)(5,17,30,191)(6,16,31,190)(7,15,32,189)(8,28,33,188)(9,27,34,187)(10,26,35,186)(11,25,36,185)(12,24,37,184)(13,23,38,183)(14,22,39,196)(43,127,170,88)(44,140,171,87)(45,139,172,86)(46,138,173,85)(47,137,174,98)(48,136,175,97)(49,135,176,96)(50,134,177,95)(51,133,178,94)(52,132,179,93)(53,131,180,92)(54,130,181,91)(55,129,182,90)(56,128,169,89)(57,155,221,120)(58,168,222,119)(59,167,223,118)(60,166,224,117)(61,165,211,116)(62,164,212,115)(63,163,213,114)(64,162,214,113)(65,161,215,126)(66,160,216,125)(67,159,217,124)(68,158,218,123)(69,157,219,122)(70,156,220,121)(71,145,106,202)(72,144,107,201)(73,143,108,200)(74,142,109,199)(75,141,110,198)(76,154,111,197)(77,153,112,210)(78,152,99,209)(79,151,100,208)(80,150,101,207)(81,149,102,206)(82,148,103,205)(83,147,104,204)(84,146,105,203), (1,171,33,51)(2,172,34,52)(3,173,35,53)(4,174,36,54)(5,175,37,55)(6,176,38,56)(7,177,39,43)(8,178,40,44)(9,179,41,45)(10,180,42,46)(11,181,29,47)(12,182,30,48)(13,169,31,49)(14,170,32,50)(15,134,196,88)(16,135,183,89)(17,136,184,90)(18,137,185,91)(19,138,186,92)(20,139,187,93)(21,140,188,94)(22,127,189,95)(23,128,190,96)(24,129,191,97)(25,130,192,98)(26,131,193,85)(27,132,194,86)(28,133,195,87)(57,80,214,108)(58,81,215,109)(59,82,216,110)(60,83,217,111)(61,84,218,112)(62,71,219,99)(63,72,220,100)(64,73,221,101)(65,74,222,102)(66,75,223,103)(67,76,224,104)(68,77,211,105)(69,78,212,106)(70,79,213,107)(113,143,155,207)(114,144,156,208)(115,145,157,209)(116,146,158,210)(117,147,159,197)(118,148,160,198)(119,149,161,199)(120,150,162,200)(121,151,163,201)(122,152,164,202)(123,153,165,203)(124,154,166,204)(125,141,167,205)(126,142,168,206) );
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)], [(1,195,40,21),(2,194,41,20),(3,193,42,19),(4,192,29,18),(5,191,30,17),(6,190,31,16),(7,189,32,15),(8,188,33,28),(9,187,34,27),(10,186,35,26),(11,185,36,25),(12,184,37,24),(13,183,38,23),(14,196,39,22),(43,134,170,95),(44,133,171,94),(45,132,172,93),(46,131,173,92),(47,130,174,91),(48,129,175,90),(49,128,176,89),(50,127,177,88),(51,140,178,87),(52,139,179,86),(53,138,180,85),(54,137,181,98),(55,136,182,97),(56,135,169,96),(57,162,221,113),(58,161,222,126),(59,160,223,125),(60,159,224,124),(61,158,211,123),(62,157,212,122),(63,156,213,121),(64,155,214,120),(65,168,215,119),(66,167,216,118),(67,166,217,117),(68,165,218,116),(69,164,219,115),(70,163,220,114),(71,202,106,145),(72,201,107,144),(73,200,108,143),(74,199,109,142),(75,198,110,141),(76,197,111,154),(77,210,112,153),(78,209,99,152),(79,208,100,151),(80,207,101,150),(81,206,102,149),(82,205,103,148),(83,204,104,147),(84,203,105,146)], [(1,113),(2,114),(3,115),(4,116),(5,117),(6,118),(7,119),(8,120),(9,121),(10,122),(11,123),(12,124),(13,125),(14,126),(15,65),(16,66),(17,67),(18,68),(19,69),(20,70),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,165),(30,166),(31,167),(32,168),(33,155),(34,156),(35,157),(36,158),(37,159),(38,160),(39,161),(40,162),(41,163),(42,164),(43,199),(44,200),(45,201),(46,202),(47,203),(48,204),(49,205),(50,206),(51,207),(52,208),(53,209),(54,210),(55,197),(56,198),(71,131),(72,132),(73,133),(74,134),(75,135),(76,136),(77,137),(78,138),(79,139),(80,140),(81,127),(82,128),(83,129),(84,130),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(141,169),(142,170),(143,171),(144,172),(145,173),(146,174),(147,175),(148,176),(149,177),(150,178),(151,179),(152,180),(153,181),(154,182),(183,223),(184,224),(185,211),(186,212),(187,213),(188,214),(189,215),(190,216),(191,217),(192,218),(193,219),(194,220),(195,221),(196,222)], [(1,21,40,195),(2,20,41,194),(3,19,42,193),(4,18,29,192),(5,17,30,191),(6,16,31,190),(7,15,32,189),(8,28,33,188),(9,27,34,187),(10,26,35,186),(11,25,36,185),(12,24,37,184),(13,23,38,183),(14,22,39,196),(43,127,170,88),(44,140,171,87),(45,139,172,86),(46,138,173,85),(47,137,174,98),(48,136,175,97),(49,135,176,96),(50,134,177,95),(51,133,178,94),(52,132,179,93),(53,131,180,92),(54,130,181,91),(55,129,182,90),(56,128,169,89),(57,155,221,120),(58,168,222,119),(59,167,223,118),(60,166,224,117),(61,165,211,116),(62,164,212,115),(63,163,213,114),(64,162,214,113),(65,161,215,126),(66,160,216,125),(67,159,217,124),(68,158,218,123),(69,157,219,122),(70,156,220,121),(71,145,106,202),(72,144,107,201),(73,143,108,200),(74,142,109,199),(75,141,110,198),(76,154,111,197),(77,153,112,210),(78,152,99,209),(79,151,100,208),(80,150,101,207),(81,149,102,206),(82,148,103,205),(83,147,104,204),(84,146,105,203)], [(1,171,33,51),(2,172,34,52),(3,173,35,53),(4,174,36,54),(5,175,37,55),(6,176,38,56),(7,177,39,43),(8,178,40,44),(9,179,41,45),(10,180,42,46),(11,181,29,47),(12,182,30,48),(13,169,31,49),(14,170,32,50),(15,134,196,88),(16,135,183,89),(17,136,184,90),(18,137,185,91),(19,138,186,92),(20,139,187,93),(21,140,188,94),(22,127,189,95),(23,128,190,96),(24,129,191,97),(25,130,192,98),(26,131,193,85),(27,132,194,86),(28,133,195,87),(57,80,214,108),(58,81,215,109),(59,82,216,110),(60,83,217,111),(61,84,218,112),(62,71,219,99),(63,72,220,100),(64,73,221,101),(65,74,222,102),(66,75,223,103),(67,76,224,104),(68,77,211,105),(69,78,212,106),(70,79,213,107),(113,143,155,207),(114,144,156,208),(115,145,157,209),(116,146,158,210),(117,147,159,197),(118,148,160,198),(119,149,161,199),(120,150,162,200),(121,151,163,201),(122,152,164,202),(123,153,165,203),(124,154,166,204),(125,141,167,205),(126,142,168,206)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | 2+ 1+4 | 2- 1+4 | D4⋊8D14 | D4.10D14 |
| kernel | C14.1082- 1+4 | C28.48D4 | C2×C4⋊Dic7 | C28⋊7D4 | C28⋊2D4 | Dic7⋊D4 | D14⋊3Q8 | C2×C4○D28 | C14×C4○D4 | C2×C28 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 4 | 3 | 9 | 9 | 3 | 24 | 1 | 1 | 6 | 6 |
Matrix representation of C14.1082- 1+4 ►in GL8(𝔽29)
| 25 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 7 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 21 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 | 0 | 1 |
| 0 | 0 | 0 | 0 | 22 | 0 | 11 | 28 |
| 0 | 0 | 0 | 0 | 0 | 22 | 28 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 0 | 27 | 28 | 0 |
| 8 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 24 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 28 | 28 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 | 0 | 1 |
| 0 | 0 | 0 | 0 | 5 | 7 | 18 | 28 |
| 0 | 0 | 0 | 0 | 7 | 24 | 1 | 18 |
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 | 0 | 13 |
| 0 | 0 | 0 | 0 | 0 | 16 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 22 | 13 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 0 | 16 |
G:=sub<GL(8,GF(29))| [25,1,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[21,5,0,0,0,0,0,0,28,8,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,21,25,0,0,0,0,0,0,0,0,18,1,22,0,0,0,0,0,1,11,0,22,0,0,0,0,1,0,11,28,0,0,0,0,0,1,28,18],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,14,1,0,0,0,0,0,0,0,0,0,1,27,0,0,0,0,0,1,0,0,27,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0],[8,24,0,0,0,0,0,0,1,21,0,0,0,0,0,0,0,0,4,13,0,0,0,0,0,0,21,25,0,0,0,0,0,0,0,0,11,1,5,7,0,0,0,0,28,11,7,24,0,0,0,0,28,0,18,1,0,0,0,0,0,1,28,18],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,15,28,0,0,0,0,0,0,0,0,13,0,0,7,0,0,0,0,0,16,22,0,0,0,0,0,0,16,13,0,0,0,0,0,13,0,0,16] >;
C14.1082- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{108}2_-^{1+4} % in TeX
G:=Group("C14.108ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1286);
// by ID
G=gap.SmallGroup(448,1286);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,675,570,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=b^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=b^-1,b*d=d*b,e*b*e^-1=a^7*b,d*c*d^-1=a^7*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations