?
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, C23.140(C22×D7), C22.328(C23×D7), C23.D7.77C22, (C22×C14).245C23, (C22×C28).321C22, 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
Subgroups: 1300 in 294 conjugacy classes, 111 normal (29 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×14], C7, C2×C4 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×16], Q8 [×4], C23, C23 [×2], C23 [×2], D7 [×2], C14 [×3], C14 [×4], C22⋊C4 [×8], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×7], C2×Q8, C2×Q8, C4○D4 [×8], Dic7 [×6], C28 [×4], C28 [×2], D14 [×6], C2×C14, C2×C14 [×2], C2×C14 [×8], C2×C4⋊C4, C4⋊D4 [×8], C22⋊Q8 [×4], C2×C4○D4, C2×C4○D4, Dic14 [×2], C4×D7 [×4], D28 [×2], C2×Dic7 [×6], C2×Dic7 [×2], C7⋊D4 [×8], C2×C28 [×2], C2×C28 [×6], C2×C28 [×4], C7×D4 [×6], C7×Q8 [×2], C22×D7 [×2], C22×C14, C22×C14 [×2], C22.31C24, Dic7⋊C4 [×4], C4⋊Dic7 [×4], D14⋊C4 [×4], C23.D7 [×4], C2×Dic14, C2×C4×D7 [×2], C2×D28, C4○D28 [×4], C22×Dic7 [×2], C2×C7⋊D4 [×6], C22×C28, C22×C28 [×2], D4×C14, D4×C14 [×2], Q8×C14, C7×C4○D4 [×4], C28.48D4 [×2], C2×C4⋊Dic7, C28⋊7D4 [×2], C28⋊2D4 [×2], Dic7⋊D4 [×4], D14⋊3Q8 [×2], C2×C4○D28, C14×C4○D4, C14.1082- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, 2+ (1+4), 2- (1+4), C7⋊D4 [×4], C22×D7 [×7], C22.31C24, C2×C7⋊D4 [×6], C23×D7, D4⋊8D14, D4.10D14, C22×C7⋊D4, C14.1082- (1+4)
Generators and relations
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 >
►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)
(1 162 25 204)(2 161 26 203)(3 160 27 202)(4 159 28 201)(5 158 15 200)(6 157 16 199)(7 156 17 198)(8 155 18 197)(9 168 19 210)(10 167 20 209)(11 166 21 208)(12 165 22 207)(13 164 23 206)(14 163 24 205)(29 43 93 80)(30 56 94 79)(31 55 95 78)(32 54 96 77)(33 53 97 76)(34 52 98 75)(35 51 85 74)(36 50 86 73)(37 49 87 72)(38 48 88 71)(39 47 89 84)(40 46 90 83)(41 45 91 82)(42 44 92 81)(57 145 195 171)(58 144 196 170)(59 143 183 169)(60 142 184 182)(61 141 185 181)(62 154 186 180)(63 153 187 179)(64 152 188 178)(65 151 189 177)(66 150 190 176)(67 149 191 175)(68 148 192 174)(69 147 193 173)(70 146 194 172)(99 131 213 126)(100 130 214 125)(101 129 215 124)(102 128 216 123)(103 127 217 122)(104 140 218 121)(105 139 219 120)(106 138 220 119)(107 137 221 118)(108 136 222 117)(109 135 223 116)(110 134 224 115)(111 133 211 114)(112 132 212 113)
(1 108)(2 109)(3 110)(4 111)(5 112)(6 99)(7 100)(8 101)(9 102)(10 103)(11 104)(12 105)(13 106)(14 107)(15 212)(16 213)(17 214)(18 215)(19 216)(20 217)(21 218)(22 219)(23 220)(24 221)(25 222)(26 223)(27 224)(28 211)(29 177)(30 178)(31 179)(32 180)(33 181)(34 182)(35 169)(36 170)(37 171)(38 172)(39 173)(40 174)(41 175)(42 176)(43 189)(44 190)(45 191)(46 192)(47 193)(48 194)(49 195)(50 196)(51 183)(52 184)(53 185)(54 186)(55 187)(56 188)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 78)(64 79)(65 80)(66 81)(67 82)(68 83)(69 84)(70 71)(85 143)(86 144)(87 145)(88 146)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 153)(96 154)(97 141)(98 142)(113 158)(114 159)(115 160)(116 161)(117 162)(118 163)(119 164)(120 165)(121 166)(122 167)(123 168)(124 155)(125 156)(126 157)(127 209)(128 210)(129 197)(130 198)(131 199)(132 200)(133 201)(134 202)(135 203)(136 204)(137 205)(138 206)(139 207)(140 208)
(1 204 25 162)(2 203 26 161)(3 202 27 160)(4 201 28 159)(5 200 15 158)(6 199 16 157)(7 198 17 156)(8 197 18 155)(9 210 19 168)(10 209 20 167)(11 208 21 166)(12 207 22 165)(13 206 23 164)(14 205 24 163)(29 80 93 43)(30 79 94 56)(31 78 95 55)(32 77 96 54)(33 76 97 53)(34 75 98 52)(35 74 85 51)(36 73 86 50)(37 72 87 49)(38 71 88 48)(39 84 89 47)(40 83 90 46)(41 82 91 45)(42 81 92 44)(57 152 195 178)(58 151 196 177)(59 150 183 176)(60 149 184 175)(61 148 185 174)(62 147 186 173)(63 146 187 172)(64 145 188 171)(65 144 189 170)(66 143 190 169)(67 142 191 182)(68 141 192 181)(69 154 193 180)(70 153 194 179)(99 138 213 119)(100 137 214 118)(101 136 215 117)(102 135 216 116)(103 134 217 115)(104 133 218 114)(105 132 219 113)(106 131 220 126)(107 130 221 125)(108 129 222 124)(109 128 223 123)(110 127 224 122)(111 140 211 121)(112 139 212 120)
(1 67 18 184)(2 68 19 185)(3 69 20 186)(4 70 21 187)(5 57 22 188)(6 58 23 189)(7 59 24 190)(8 60 25 191)(9 61 26 192)(10 62 27 193)(11 63 28 194)(12 64 15 195)(13 65 16 196)(14 66 17 183)(29 119 86 131)(30 120 87 132)(31 121 88 133)(32 122 89 134)(33 123 90 135)(34 124 91 136)(35 125 92 137)(36 126 93 138)(37 113 94 139)(38 114 95 140)(39 115 96 127)(40 116 97 128)(41 117 98 129)(42 118 85 130)(43 99 73 220)(44 100 74 221)(45 101 75 222)(46 102 76 223)(47 103 77 224)(48 104 78 211)(49 105 79 212)(50 106 80 213)(51 107 81 214)(52 108 82 215)(53 109 83 216)(54 110 84 217)(55 111 71 218)(56 112 72 219)(141 210 174 161)(142 197 175 162)(143 198 176 163)(144 199 177 164)(145 200 178 165)(146 201 179 166)(147 202 180 167)(148 203 181 168)(149 204 182 155)(150 205 169 156)(151 206 170 157)(152 207 171 158)(153 208 172 159)(154 209 173 160)
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), (1,162,25,204)(2,161,26,203)(3,160,27,202)(4,159,28,201)(5,158,15,200)(6,157,16,199)(7,156,17,198)(8,155,18,197)(9,168,19,210)(10,167,20,209)(11,166,21,208)(12,165,22,207)(13,164,23,206)(14,163,24,205)(29,43,93,80)(30,56,94,79)(31,55,95,78)(32,54,96,77)(33,53,97,76)(34,52,98,75)(35,51,85,74)(36,50,86,73)(37,49,87,72)(38,48,88,71)(39,47,89,84)(40,46,90,83)(41,45,91,82)(42,44,92,81)(57,145,195,171)(58,144,196,170)(59,143,183,169)(60,142,184,182)(61,141,185,181)(62,154,186,180)(63,153,187,179)(64,152,188,178)(65,151,189,177)(66,150,190,176)(67,149,191,175)(68,148,192,174)(69,147,193,173)(70,146,194,172)(99,131,213,126)(100,130,214,125)(101,129,215,124)(102,128,216,123)(103,127,217,122)(104,140,218,121)(105,139,219,120)(106,138,220,119)(107,137,221,118)(108,136,222,117)(109,135,223,116)(110,134,224,115)(111,133,211,114)(112,132,212,113), (1,108)(2,109)(3,110)(4,111)(5,112)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,212)(16,213)(17,214)(18,215)(19,216)(20,217)(21,218)(22,219)(23,220)(24,221)(25,222)(26,223)(27,224)(28,211)(29,177)(30,178)(31,179)(32,180)(33,181)(34,182)(35,169)(36,170)(37,171)(38,172)(39,173)(40,174)(41,175)(42,176)(43,189)(44,190)(45,191)(46,192)(47,193)(48,194)(49,195)(50,196)(51,183)(52,184)(53,185)(54,186)(55,187)(56,188)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,71)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,141)(98,142)(113,158)(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)(123,168)(124,155)(125,156)(126,157)(127,209)(128,210)(129,197)(130,198)(131,199)(132,200)(133,201)(134,202)(135,203)(136,204)(137,205)(138,206)(139,207)(140,208), (1,204,25,162)(2,203,26,161)(3,202,27,160)(4,201,28,159)(5,200,15,158)(6,199,16,157)(7,198,17,156)(8,197,18,155)(9,210,19,168)(10,209,20,167)(11,208,21,166)(12,207,22,165)(13,206,23,164)(14,205,24,163)(29,80,93,43)(30,79,94,56)(31,78,95,55)(32,77,96,54)(33,76,97,53)(34,75,98,52)(35,74,85,51)(36,73,86,50)(37,72,87,49)(38,71,88,48)(39,84,89,47)(40,83,90,46)(41,82,91,45)(42,81,92,44)(57,152,195,178)(58,151,196,177)(59,150,183,176)(60,149,184,175)(61,148,185,174)(62,147,186,173)(63,146,187,172)(64,145,188,171)(65,144,189,170)(66,143,190,169)(67,142,191,182)(68,141,192,181)(69,154,193,180)(70,153,194,179)(99,138,213,119)(100,137,214,118)(101,136,215,117)(102,135,216,116)(103,134,217,115)(104,133,218,114)(105,132,219,113)(106,131,220,126)(107,130,221,125)(108,129,222,124)(109,128,223,123)(110,127,224,122)(111,140,211,121)(112,139,212,120), (1,67,18,184)(2,68,19,185)(3,69,20,186)(4,70,21,187)(5,57,22,188)(6,58,23,189)(7,59,24,190)(8,60,25,191)(9,61,26,192)(10,62,27,193)(11,63,28,194)(12,64,15,195)(13,65,16,196)(14,66,17,183)(29,119,86,131)(30,120,87,132)(31,121,88,133)(32,122,89,134)(33,123,90,135)(34,124,91,136)(35,125,92,137)(36,126,93,138)(37,113,94,139)(38,114,95,140)(39,115,96,127)(40,116,97,128)(41,117,98,129)(42,118,85,130)(43,99,73,220)(44,100,74,221)(45,101,75,222)(46,102,76,223)(47,103,77,224)(48,104,78,211)(49,105,79,212)(50,106,80,213)(51,107,81,214)(52,108,82,215)(53,109,83,216)(54,110,84,217)(55,111,71,218)(56,112,72,219)(141,210,174,161)(142,197,175,162)(143,198,176,163)(144,199,177,164)(145,200,178,165)(146,201,179,166)(147,202,180,167)(148,203,181,168)(149,204,182,155)(150,205,169,156)(151,206,170,157)(152,207,171,158)(153,208,172,159)(154,209,173,160)>;
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), (1,162,25,204)(2,161,26,203)(3,160,27,202)(4,159,28,201)(5,158,15,200)(6,157,16,199)(7,156,17,198)(8,155,18,197)(9,168,19,210)(10,167,20,209)(11,166,21,208)(12,165,22,207)(13,164,23,206)(14,163,24,205)(29,43,93,80)(30,56,94,79)(31,55,95,78)(32,54,96,77)(33,53,97,76)(34,52,98,75)(35,51,85,74)(36,50,86,73)(37,49,87,72)(38,48,88,71)(39,47,89,84)(40,46,90,83)(41,45,91,82)(42,44,92,81)(57,145,195,171)(58,144,196,170)(59,143,183,169)(60,142,184,182)(61,141,185,181)(62,154,186,180)(63,153,187,179)(64,152,188,178)(65,151,189,177)(66,150,190,176)(67,149,191,175)(68,148,192,174)(69,147,193,173)(70,146,194,172)(99,131,213,126)(100,130,214,125)(101,129,215,124)(102,128,216,123)(103,127,217,122)(104,140,218,121)(105,139,219,120)(106,138,220,119)(107,137,221,118)(108,136,222,117)(109,135,223,116)(110,134,224,115)(111,133,211,114)(112,132,212,113), (1,108)(2,109)(3,110)(4,111)(5,112)(6,99)(7,100)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,212)(16,213)(17,214)(18,215)(19,216)(20,217)(21,218)(22,219)(23,220)(24,221)(25,222)(26,223)(27,224)(28,211)(29,177)(30,178)(31,179)(32,180)(33,181)(34,182)(35,169)(36,170)(37,171)(38,172)(39,173)(40,174)(41,175)(42,176)(43,189)(44,190)(45,191)(46,192)(47,193)(48,194)(49,195)(50,196)(51,183)(52,184)(53,185)(54,186)(55,187)(56,188)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,78)(64,79)(65,80)(66,81)(67,82)(68,83)(69,84)(70,71)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,141)(98,142)(113,158)(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)(123,168)(124,155)(125,156)(126,157)(127,209)(128,210)(129,197)(130,198)(131,199)(132,200)(133,201)(134,202)(135,203)(136,204)(137,205)(138,206)(139,207)(140,208), (1,204,25,162)(2,203,26,161)(3,202,27,160)(4,201,28,159)(5,200,15,158)(6,199,16,157)(7,198,17,156)(8,197,18,155)(9,210,19,168)(10,209,20,167)(11,208,21,166)(12,207,22,165)(13,206,23,164)(14,205,24,163)(29,80,93,43)(30,79,94,56)(31,78,95,55)(32,77,96,54)(33,76,97,53)(34,75,98,52)(35,74,85,51)(36,73,86,50)(37,72,87,49)(38,71,88,48)(39,84,89,47)(40,83,90,46)(41,82,91,45)(42,81,92,44)(57,152,195,178)(58,151,196,177)(59,150,183,176)(60,149,184,175)(61,148,185,174)(62,147,186,173)(63,146,187,172)(64,145,188,171)(65,144,189,170)(66,143,190,169)(67,142,191,182)(68,141,192,181)(69,154,193,180)(70,153,194,179)(99,138,213,119)(100,137,214,118)(101,136,215,117)(102,135,216,116)(103,134,217,115)(104,133,218,114)(105,132,219,113)(106,131,220,126)(107,130,221,125)(108,129,222,124)(109,128,223,123)(110,127,224,122)(111,140,211,121)(112,139,212,120), (1,67,18,184)(2,68,19,185)(3,69,20,186)(4,70,21,187)(5,57,22,188)(6,58,23,189)(7,59,24,190)(8,60,25,191)(9,61,26,192)(10,62,27,193)(11,63,28,194)(12,64,15,195)(13,65,16,196)(14,66,17,183)(29,119,86,131)(30,120,87,132)(31,121,88,133)(32,122,89,134)(33,123,90,135)(34,124,91,136)(35,125,92,137)(36,126,93,138)(37,113,94,139)(38,114,95,140)(39,115,96,127)(40,116,97,128)(41,117,98,129)(42,118,85,130)(43,99,73,220)(44,100,74,221)(45,101,75,222)(46,102,76,223)(47,103,77,224)(48,104,78,211)(49,105,79,212)(50,106,80,213)(51,107,81,214)(52,108,82,215)(53,109,83,216)(54,110,84,217)(55,111,71,218)(56,112,72,219)(141,210,174,161)(142,197,175,162)(143,198,176,163)(144,199,177,164)(145,200,178,165)(146,201,179,166)(147,202,180,167)(148,203,181,168)(149,204,182,155)(150,205,169,156)(151,206,170,157)(152,207,171,158)(153,208,172,159)(154,209,173,160) );
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)], [(1,162,25,204),(2,161,26,203),(3,160,27,202),(4,159,28,201),(5,158,15,200),(6,157,16,199),(7,156,17,198),(8,155,18,197),(9,168,19,210),(10,167,20,209),(11,166,21,208),(12,165,22,207),(13,164,23,206),(14,163,24,205),(29,43,93,80),(30,56,94,79),(31,55,95,78),(32,54,96,77),(33,53,97,76),(34,52,98,75),(35,51,85,74),(36,50,86,73),(37,49,87,72),(38,48,88,71),(39,47,89,84),(40,46,90,83),(41,45,91,82),(42,44,92,81),(57,145,195,171),(58,144,196,170),(59,143,183,169),(60,142,184,182),(61,141,185,181),(62,154,186,180),(63,153,187,179),(64,152,188,178),(65,151,189,177),(66,150,190,176),(67,149,191,175),(68,148,192,174),(69,147,193,173),(70,146,194,172),(99,131,213,126),(100,130,214,125),(101,129,215,124),(102,128,216,123),(103,127,217,122),(104,140,218,121),(105,139,219,120),(106,138,220,119),(107,137,221,118),(108,136,222,117),(109,135,223,116),(110,134,224,115),(111,133,211,114),(112,132,212,113)], [(1,108),(2,109),(3,110),(4,111),(5,112),(6,99),(7,100),(8,101),(9,102),(10,103),(11,104),(12,105),(13,106),(14,107),(15,212),(16,213),(17,214),(18,215),(19,216),(20,217),(21,218),(22,219),(23,220),(24,221),(25,222),(26,223),(27,224),(28,211),(29,177),(30,178),(31,179),(32,180),(33,181),(34,182),(35,169),(36,170),(37,171),(38,172),(39,173),(40,174),(41,175),(42,176),(43,189),(44,190),(45,191),(46,192),(47,193),(48,194),(49,195),(50,196),(51,183),(52,184),(53,185),(54,186),(55,187),(56,188),(57,72),(58,73),(59,74),(60,75),(61,76),(62,77),(63,78),(64,79),(65,80),(66,81),(67,82),(68,83),(69,84),(70,71),(85,143),(86,144),(87,145),(88,146),(89,147),(90,148),(91,149),(92,150),(93,151),(94,152),(95,153),(96,154),(97,141),(98,142),(113,158),(114,159),(115,160),(116,161),(117,162),(118,163),(119,164),(120,165),(121,166),(122,167),(123,168),(124,155),(125,156),(126,157),(127,209),(128,210),(129,197),(130,198),(131,199),(132,200),(133,201),(134,202),(135,203),(136,204),(137,205),(138,206),(139,207),(140,208)], [(1,204,25,162),(2,203,26,161),(3,202,27,160),(4,201,28,159),(5,200,15,158),(6,199,16,157),(7,198,17,156),(8,197,18,155),(9,210,19,168),(10,209,20,167),(11,208,21,166),(12,207,22,165),(13,206,23,164),(14,205,24,163),(29,80,93,43),(30,79,94,56),(31,78,95,55),(32,77,96,54),(33,76,97,53),(34,75,98,52),(35,74,85,51),(36,73,86,50),(37,72,87,49),(38,71,88,48),(39,84,89,47),(40,83,90,46),(41,82,91,45),(42,81,92,44),(57,152,195,178),(58,151,196,177),(59,150,183,176),(60,149,184,175),(61,148,185,174),(62,147,186,173),(63,146,187,172),(64,145,188,171),(65,144,189,170),(66,143,190,169),(67,142,191,182),(68,141,192,181),(69,154,193,180),(70,153,194,179),(99,138,213,119),(100,137,214,118),(101,136,215,117),(102,135,216,116),(103,134,217,115),(104,133,218,114),(105,132,219,113),(106,131,220,126),(107,130,221,125),(108,129,222,124),(109,128,223,123),(110,127,224,122),(111,140,211,121),(112,139,212,120)], [(1,67,18,184),(2,68,19,185),(3,69,20,186),(4,70,21,187),(5,57,22,188),(6,58,23,189),(7,59,24,190),(8,60,25,191),(9,61,26,192),(10,62,27,193),(11,63,28,194),(12,64,15,195),(13,65,16,196),(14,66,17,183),(29,119,86,131),(30,120,87,132),(31,121,88,133),(32,122,89,134),(33,123,90,135),(34,124,91,136),(35,125,92,137),(36,126,93,138),(37,113,94,139),(38,114,95,140),(39,115,96,127),(40,116,97,128),(41,117,98,129),(42,118,85,130),(43,99,73,220),(44,100,74,221),(45,101,75,222),(46,102,76,223),(47,103,77,224),(48,104,78,211),(49,105,79,212),(50,106,80,213),(51,107,81,214),(52,108,82,215),(53,109,83,216),(54,110,84,217),(55,111,71,218),(56,112,72,219),(141,210,174,161),(142,197,175,162),(143,198,176,163),(144,199,177,164),(145,200,178,165),(146,201,179,166),(147,202,180,167),(148,203,181,168),(149,204,182,155),(150,205,169,156),(151,206,170,157),(152,207,171,158),(153,208,172,159),(154,209,173,160)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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