G = C14.812- 1+4 order 448 = 26·7
36th non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.812- 1+4, C14.602+ 1+4, C28⋊Q8⋊31C2, C4⋊C4.105D14, (C2×D4).97D14, Dic7.Q8⋊26C2, C22⋊C4.26D14, C28.48D4⋊14C2, (C2×C28).178C23, (C2×C14).195C24, C28.17D4.8C2, (C22×C4).256D14, C2.62(D4⋊6D14), C22.D4.3D7, C22⋊Dic14⋊30C2, (D4×C14).133C22, C23.D14⋊29C2, Dic7⋊C4.40C22, C4⋊Dic7.226C22, (C22×C28).86C22, C22.216(C23×D7), C23.128(C22×D7), C23.D7.41C22, (C22×C14).220C23, C7⋊2(C22.57C24), (C2×Dic14).36C22, (C2×Dic7).100C23, (C4×Dic7).122C22, C23.18D14.3C2, C2.42(D4.10D14), (C22×Dic7).128C22, (C2×C4).59(C22×D7), (C7×C4⋊C4).175C22, (C7×C22⋊C4).50C22, (C7×C22.D4).3C2, SmallGroup(448,1104)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.812- 1+4
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=e2=a7b2, bab-1=cac-1=eae-1=a-1, ad=da, cbc-1=b-1, bd=db, ebe-1=a7b, dcd-1=a7c, ce=ec, ede-1=a7b2d >
Subgroups: 780 in 196 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C7, C2×C4, C2×C4 [×4], C2×C4 [×10], D4, Q8 [×3], C23 [×2], C14, C14 [×2], C14 [×2], C42 [×3], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4, C22×C4, C2×D4, C2×Q8 [×3], Dic7 [×8], C28 [×5], C2×C14, C2×C14 [×6], C22⋊Q8 [×4], C22.D4, C22.D4, C4.4D4, C42.C2 [×2], C42⋊2C2 [×4], C4⋊Q8 [×2], Dic14 [×3], C2×Dic7 [×8], C2×Dic7, C2×C28, C2×C28 [×4], C2×C28, C7×D4, C22×C14 [×2], C22.57C24, C4×Dic7, C4×Dic7 [×2], Dic7⋊C4 [×10], C4⋊Dic7 [×4], C23.D7, C23.D7 [×6], C7×C22⋊C4, C7×C22⋊C4 [×2], C7×C4⋊C4 [×2], C2×Dic14, C2×Dic14 [×2], C22×Dic7, C22×C28, D4×C14, C22⋊Dic14 [×2], C23.D14 [×4], C28⋊Q8 [×2], Dic7.Q8 [×2], C28.48D4 [×2], C23.18D14, C28.17D4, C7×C22.D4, C14.812- 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ 1+4, 2- 1+4 [×2], C22×D7 [×7], C22.57C24, C23×D7, D4⋊6D14, D4.10D14 [×2], C14.812- 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 30 206 80)(2 29 207 79)(3 42 208 78)(4 41 209 77)(5 40 210 76)(6 39 197 75)(7 38 198 74)(8 37 199 73)(9 36 200 72)(10 35 201 71)(11 34 202 84)(12 33 203 83)(13 32 204 82)(14 31 205 81)(15 140 58 110)(16 139 59 109)(17 138 60 108)(18 137 61 107)(19 136 62 106)(20 135 63 105)(21 134 64 104)(22 133 65 103)(23 132 66 102)(24 131 67 101)(25 130 68 100)(26 129 69 99)(27 128 70 112)(28 127 57 111)(43 118 191 216)(44 117 192 215)(45 116 193 214)(46 115 194 213)(47 114 195 212)(48 113 196 211)(49 126 183 224)(50 125 184 223)(51 124 185 222)(52 123 186 221)(53 122 187 220)(54 121 188 219)(55 120 189 218)(56 119 190 217)(85 170 166 152)(86 169 167 151)(87 182 168 150)(88 181 155 149)(89 180 156 148)(90 179 157 147)(91 178 158 146)(92 177 159 145)(93 176 160 144)(94 175 161 143)(95 174 162 142)(96 173 163 141)(97 172 164 154)(98 171 165 153)
(1 132 8 139)(2 131 9 138)(3 130 10 137)(4 129 11 136)(5 128 12 135)(6 127 13 134)(7 140 14 133)(15 31 22 38)(16 30 23 37)(17 29 24 36)(18 42 25 35)(19 41 26 34)(20 40 27 33)(21 39 28 32)(43 160 50 167)(44 159 51 166)(45 158 52 165)(46 157 53 164)(47 156 54 163)(48 155 55 162)(49 168 56 161)(57 82 64 75)(58 81 65 74)(59 80 66 73)(60 79 67 72)(61 78 68 71)(62 77 69 84)(63 76 70 83)(85 192 92 185)(86 191 93 184)(87 190 94 183)(88 189 95 196)(89 188 96 195)(90 187 97 194)(91 186 98 193)(99 202 106 209)(100 201 107 208)(101 200 108 207)(102 199 109 206)(103 198 110 205)(104 197 111 204)(105 210 112 203)(113 181 120 174)(114 180 121 173)(115 179 122 172)(116 178 123 171)(117 177 124 170)(118 176 125 169)(119 175 126 182)(141 212 148 219)(142 211 149 218)(143 224 150 217)(144 223 151 216)(145 222 152 215)(146 221 153 214)(147 220 154 213)
(1 124 199 215)(2 125 200 216)(3 126 201 217)(4 113 202 218)(5 114 203 219)(6 115 204 220)(7 116 205 221)(8 117 206 222)(9 118 207 223)(10 119 208 224)(11 120 209 211)(12 121 210 212)(13 122 197 213)(14 123 198 214)(15 98 65 158)(16 85 66 159)(17 86 67 160)(18 87 68 161)(19 88 69 162)(20 89 70 163)(21 90 57 164)(22 91 58 165)(23 92 59 166)(24 93 60 167)(25 94 61 168)(26 95 62 155)(27 96 63 156)(28 97 64 157)(29 184 72 43)(30 185 73 44)(31 186 74 45)(32 187 75 46)(33 188 76 47)(34 189 77 48)(35 190 78 49)(36 191 79 50)(37 192 80 51)(38 193 81 52)(39 194 82 53)(40 195 83 54)(41 196 84 55)(42 183 71 56)(99 142 136 181)(100 143 137 182)(101 144 138 169)(102 145 139 170)(103 146 140 171)(104 147 127 172)(105 148 128 173)(106 149 129 174)(107 150 130 175)(108 151 131 176)(109 152 132 177)(110 153 133 178)(111 154 134 179)(112 141 135 180)
(1 51 199 192)(2 50 200 191)(3 49 201 190)(4 48 202 189)(5 47 203 188)(6 46 204 187)(7 45 205 186)(8 44 206 185)(9 43 207 184)(10 56 208 183)(11 55 209 196)(12 54 210 195)(13 53 197 194)(14 52 198 193)(15 171 65 146)(16 170 66 145)(17 169 67 144)(18 182 68 143)(19 181 69 142)(20 180 70 141)(21 179 57 154)(22 178 58 153)(23 177 59 152)(24 176 60 151)(25 175 61 150)(26 174 62 149)(27 173 63 148)(28 172 64 147)(29 118 72 223)(30 117 73 222)(31 116 74 221)(32 115 75 220)(33 114 76 219)(34 113 77 218)(35 126 78 217)(36 125 79 216)(37 124 80 215)(38 123 81 214)(39 122 82 213)(40 121 83 212)(41 120 84 211)(42 119 71 224)(85 139 159 102)(86 138 160 101)(87 137 161 100)(88 136 162 99)(89 135 163 112)(90 134 164 111)(91 133 165 110)(92 132 166 109)(93 131 167 108)(94 130 168 107)(95 129 155 106)(96 128 156 105)(97 127 157 104)(98 140 158 103)
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), (1,30,206,80)(2,29,207,79)(3,42,208,78)(4,41,209,77)(5,40,210,76)(6,39,197,75)(7,38,198,74)(8,37,199,73)(9,36,200,72)(10,35,201,71)(11,34,202,84)(12,33,203,83)(13,32,204,82)(14,31,205,81)(15,140,58,110)(16,139,59,109)(17,138,60,108)(18,137,61,107)(19,136,62,106)(20,135,63,105)(21,134,64,104)(22,133,65,103)(23,132,66,102)(24,131,67,101)(25,130,68,100)(26,129,69,99)(27,128,70,112)(28,127,57,111)(43,118,191,216)(44,117,192,215)(45,116,193,214)(46,115,194,213)(47,114,195,212)(48,113,196,211)(49,126,183,224)(50,125,184,223)(51,124,185,222)(52,123,186,221)(53,122,187,220)(54,121,188,219)(55,120,189,218)(56,119,190,217)(85,170,166,152)(86,169,167,151)(87,182,168,150)(88,181,155,149)(89,180,156,148)(90,179,157,147)(91,178,158,146)(92,177,159,145)(93,176,160,144)(94,175,161,143)(95,174,162,142)(96,173,163,141)(97,172,164,154)(98,171,165,153), (1,132,8,139)(2,131,9,138)(3,130,10,137)(4,129,11,136)(5,128,12,135)(6,127,13,134)(7,140,14,133)(15,31,22,38)(16,30,23,37)(17,29,24,36)(18,42,25,35)(19,41,26,34)(20,40,27,33)(21,39,28,32)(43,160,50,167)(44,159,51,166)(45,158,52,165)(46,157,53,164)(47,156,54,163)(48,155,55,162)(49,168,56,161)(57,82,64,75)(58,81,65,74)(59,80,66,73)(60,79,67,72)(61,78,68,71)(62,77,69,84)(63,76,70,83)(85,192,92,185)(86,191,93,184)(87,190,94,183)(88,189,95,196)(89,188,96,195)(90,187,97,194)(91,186,98,193)(99,202,106,209)(100,201,107,208)(101,200,108,207)(102,199,109,206)(103,198,110,205)(104,197,111,204)(105,210,112,203)(113,181,120,174)(114,180,121,173)(115,179,122,172)(116,178,123,171)(117,177,124,170)(118,176,125,169)(119,175,126,182)(141,212,148,219)(142,211,149,218)(143,224,150,217)(144,223,151,216)(145,222,152,215)(146,221,153,214)(147,220,154,213), (1,124,199,215)(2,125,200,216)(3,126,201,217)(4,113,202,218)(5,114,203,219)(6,115,204,220)(7,116,205,221)(8,117,206,222)(9,118,207,223)(10,119,208,224)(11,120,209,211)(12,121,210,212)(13,122,197,213)(14,123,198,214)(15,98,65,158)(16,85,66,159)(17,86,67,160)(18,87,68,161)(19,88,69,162)(20,89,70,163)(21,90,57,164)(22,91,58,165)(23,92,59,166)(24,93,60,167)(25,94,61,168)(26,95,62,155)(27,96,63,156)(28,97,64,157)(29,184,72,43)(30,185,73,44)(31,186,74,45)(32,187,75,46)(33,188,76,47)(34,189,77,48)(35,190,78,49)(36,191,79,50)(37,192,80,51)(38,193,81,52)(39,194,82,53)(40,195,83,54)(41,196,84,55)(42,183,71,56)(99,142,136,181)(100,143,137,182)(101,144,138,169)(102,145,139,170)(103,146,140,171)(104,147,127,172)(105,148,128,173)(106,149,129,174)(107,150,130,175)(108,151,131,176)(109,152,132,177)(110,153,133,178)(111,154,134,179)(112,141,135,180), (1,51,199,192)(2,50,200,191)(3,49,201,190)(4,48,202,189)(5,47,203,188)(6,46,204,187)(7,45,205,186)(8,44,206,185)(9,43,207,184)(10,56,208,183)(11,55,209,196)(12,54,210,195)(13,53,197,194)(14,52,198,193)(15,171,65,146)(16,170,66,145)(17,169,67,144)(18,182,68,143)(19,181,69,142)(20,180,70,141)(21,179,57,154)(22,178,58,153)(23,177,59,152)(24,176,60,151)(25,175,61,150)(26,174,62,149)(27,173,63,148)(28,172,64,147)(29,118,72,223)(30,117,73,222)(31,116,74,221)(32,115,75,220)(33,114,76,219)(34,113,77,218)(35,126,78,217)(36,125,79,216)(37,124,80,215)(38,123,81,214)(39,122,82,213)(40,121,83,212)(41,120,84,211)(42,119,71,224)(85,139,159,102)(86,138,160,101)(87,137,161,100)(88,136,162,99)(89,135,163,112)(90,134,164,111)(91,133,165,110)(92,132,166,109)(93,131,167,108)(94,130,168,107)(95,129,155,106)(96,128,156,105)(97,127,157,104)(98,140,158,103)>;
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), (1,30,206,80)(2,29,207,79)(3,42,208,78)(4,41,209,77)(5,40,210,76)(6,39,197,75)(7,38,198,74)(8,37,199,73)(9,36,200,72)(10,35,201,71)(11,34,202,84)(12,33,203,83)(13,32,204,82)(14,31,205,81)(15,140,58,110)(16,139,59,109)(17,138,60,108)(18,137,61,107)(19,136,62,106)(20,135,63,105)(21,134,64,104)(22,133,65,103)(23,132,66,102)(24,131,67,101)(25,130,68,100)(26,129,69,99)(27,128,70,112)(28,127,57,111)(43,118,191,216)(44,117,192,215)(45,116,193,214)(46,115,194,213)(47,114,195,212)(48,113,196,211)(49,126,183,224)(50,125,184,223)(51,124,185,222)(52,123,186,221)(53,122,187,220)(54,121,188,219)(55,120,189,218)(56,119,190,217)(85,170,166,152)(86,169,167,151)(87,182,168,150)(88,181,155,149)(89,180,156,148)(90,179,157,147)(91,178,158,146)(92,177,159,145)(93,176,160,144)(94,175,161,143)(95,174,162,142)(96,173,163,141)(97,172,164,154)(98,171,165,153), (1,132,8,139)(2,131,9,138)(3,130,10,137)(4,129,11,136)(5,128,12,135)(6,127,13,134)(7,140,14,133)(15,31,22,38)(16,30,23,37)(17,29,24,36)(18,42,25,35)(19,41,26,34)(20,40,27,33)(21,39,28,32)(43,160,50,167)(44,159,51,166)(45,158,52,165)(46,157,53,164)(47,156,54,163)(48,155,55,162)(49,168,56,161)(57,82,64,75)(58,81,65,74)(59,80,66,73)(60,79,67,72)(61,78,68,71)(62,77,69,84)(63,76,70,83)(85,192,92,185)(86,191,93,184)(87,190,94,183)(88,189,95,196)(89,188,96,195)(90,187,97,194)(91,186,98,193)(99,202,106,209)(100,201,107,208)(101,200,108,207)(102,199,109,206)(103,198,110,205)(104,197,111,204)(105,210,112,203)(113,181,120,174)(114,180,121,173)(115,179,122,172)(116,178,123,171)(117,177,124,170)(118,176,125,169)(119,175,126,182)(141,212,148,219)(142,211,149,218)(143,224,150,217)(144,223,151,216)(145,222,152,215)(146,221,153,214)(147,220,154,213), (1,124,199,215)(2,125,200,216)(3,126,201,217)(4,113,202,218)(5,114,203,219)(6,115,204,220)(7,116,205,221)(8,117,206,222)(9,118,207,223)(10,119,208,224)(11,120,209,211)(12,121,210,212)(13,122,197,213)(14,123,198,214)(15,98,65,158)(16,85,66,159)(17,86,67,160)(18,87,68,161)(19,88,69,162)(20,89,70,163)(21,90,57,164)(22,91,58,165)(23,92,59,166)(24,93,60,167)(25,94,61,168)(26,95,62,155)(27,96,63,156)(28,97,64,157)(29,184,72,43)(30,185,73,44)(31,186,74,45)(32,187,75,46)(33,188,76,47)(34,189,77,48)(35,190,78,49)(36,191,79,50)(37,192,80,51)(38,193,81,52)(39,194,82,53)(40,195,83,54)(41,196,84,55)(42,183,71,56)(99,142,136,181)(100,143,137,182)(101,144,138,169)(102,145,139,170)(103,146,140,171)(104,147,127,172)(105,148,128,173)(106,149,129,174)(107,150,130,175)(108,151,131,176)(109,152,132,177)(110,153,133,178)(111,154,134,179)(112,141,135,180), (1,51,199,192)(2,50,200,191)(3,49,201,190)(4,48,202,189)(5,47,203,188)(6,46,204,187)(7,45,205,186)(8,44,206,185)(9,43,207,184)(10,56,208,183)(11,55,209,196)(12,54,210,195)(13,53,197,194)(14,52,198,193)(15,171,65,146)(16,170,66,145)(17,169,67,144)(18,182,68,143)(19,181,69,142)(20,180,70,141)(21,179,57,154)(22,178,58,153)(23,177,59,152)(24,176,60,151)(25,175,61,150)(26,174,62,149)(27,173,63,148)(28,172,64,147)(29,118,72,223)(30,117,73,222)(31,116,74,221)(32,115,75,220)(33,114,76,219)(34,113,77,218)(35,126,78,217)(36,125,79,216)(37,124,80,215)(38,123,81,214)(39,122,82,213)(40,121,83,212)(41,120,84,211)(42,119,71,224)(85,139,159,102)(86,138,160,101)(87,137,161,100)(88,136,162,99)(89,135,163,112)(90,134,164,111)(91,133,165,110)(92,132,166,109)(93,131,167,108)(94,130,168,107)(95,129,155,106)(96,128,156,105)(97,127,157,104)(98,140,158,103) );
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)], [(1,30,206,80),(2,29,207,79),(3,42,208,78),(4,41,209,77),(5,40,210,76),(6,39,197,75),(7,38,198,74),(8,37,199,73),(9,36,200,72),(10,35,201,71),(11,34,202,84),(12,33,203,83),(13,32,204,82),(14,31,205,81),(15,140,58,110),(16,139,59,109),(17,138,60,108),(18,137,61,107),(19,136,62,106),(20,135,63,105),(21,134,64,104),(22,133,65,103),(23,132,66,102),(24,131,67,101),(25,130,68,100),(26,129,69,99),(27,128,70,112),(28,127,57,111),(43,118,191,216),(44,117,192,215),(45,116,193,214),(46,115,194,213),(47,114,195,212),(48,113,196,211),(49,126,183,224),(50,125,184,223),(51,124,185,222),(52,123,186,221),(53,122,187,220),(54,121,188,219),(55,120,189,218),(56,119,190,217),(85,170,166,152),(86,169,167,151),(87,182,168,150),(88,181,155,149),(89,180,156,148),(90,179,157,147),(91,178,158,146),(92,177,159,145),(93,176,160,144),(94,175,161,143),(95,174,162,142),(96,173,163,141),(97,172,164,154),(98,171,165,153)], [(1,132,8,139),(2,131,9,138),(3,130,10,137),(4,129,11,136),(5,128,12,135),(6,127,13,134),(7,140,14,133),(15,31,22,38),(16,30,23,37),(17,29,24,36),(18,42,25,35),(19,41,26,34),(20,40,27,33),(21,39,28,32),(43,160,50,167),(44,159,51,166),(45,158,52,165),(46,157,53,164),(47,156,54,163),(48,155,55,162),(49,168,56,161),(57,82,64,75),(58,81,65,74),(59,80,66,73),(60,79,67,72),(61,78,68,71),(62,77,69,84),(63,76,70,83),(85,192,92,185),(86,191,93,184),(87,190,94,183),(88,189,95,196),(89,188,96,195),(90,187,97,194),(91,186,98,193),(99,202,106,209),(100,201,107,208),(101,200,108,207),(102,199,109,206),(103,198,110,205),(104,197,111,204),(105,210,112,203),(113,181,120,174),(114,180,121,173),(115,179,122,172),(116,178,123,171),(117,177,124,170),(118,176,125,169),(119,175,126,182),(141,212,148,219),(142,211,149,218),(143,224,150,217),(144,223,151,216),(145,222,152,215),(146,221,153,214),(147,220,154,213)], [(1,124,199,215),(2,125,200,216),(3,126,201,217),(4,113,202,218),(5,114,203,219),(6,115,204,220),(7,116,205,221),(8,117,206,222),(9,118,207,223),(10,119,208,224),(11,120,209,211),(12,121,210,212),(13,122,197,213),(14,123,198,214),(15,98,65,158),(16,85,66,159),(17,86,67,160),(18,87,68,161),(19,88,69,162),(20,89,70,163),(21,90,57,164),(22,91,58,165),(23,92,59,166),(24,93,60,167),(25,94,61,168),(26,95,62,155),(27,96,63,156),(28,97,64,157),(29,184,72,43),(30,185,73,44),(31,186,74,45),(32,187,75,46),(33,188,76,47),(34,189,77,48),(35,190,78,49),(36,191,79,50),(37,192,80,51),(38,193,81,52),(39,194,82,53),(40,195,83,54),(41,196,84,55),(42,183,71,56),(99,142,136,181),(100,143,137,182),(101,144,138,169),(102,145,139,170),(103,146,140,171),(104,147,127,172),(105,148,128,173),(106,149,129,174),(107,150,130,175),(108,151,131,176),(109,152,132,177),(110,153,133,178),(111,154,134,179),(112,141,135,180)], [(1,51,199,192),(2,50,200,191),(3,49,201,190),(4,48,202,189),(5,47,203,188),(6,46,204,187),(7,45,205,186),(8,44,206,185),(9,43,207,184),(10,56,208,183),(11,55,209,196),(12,54,210,195),(13,53,197,194),(14,52,198,193),(15,171,65,146),(16,170,66,145),(17,169,67,144),(18,182,68,143),(19,181,69,142),(20,180,70,141),(21,179,57,154),(22,178,58,153),(23,177,59,152),(24,176,60,151),(25,175,61,150),(26,174,62,149),(27,173,63,148),(28,172,64,147),(29,118,72,223),(30,117,73,222),(31,116,74,221),(32,115,75,220),(33,114,76,219),(34,113,77,218),(35,126,78,217),(36,125,79,216),(37,124,80,215),(38,123,81,214),(39,122,82,213),(40,121,83,212),(41,120,84,211),(42,119,71,224),(85,139,159,102),(86,138,160,101),(87,137,161,100),(88,136,162,99),(89,135,163,112),(90,134,164,111),(91,133,165,110),(92,132,166,109),(93,131,167,108),(94,130,168,107),(95,129,155,106),(96,128,156,105),(97,127,157,104),(98,140,158,103)])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4E | 4F | ··· | 4M | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | D14 | 2+ 1+4 | 2- 1+4 | D4⋊6D14 | D4.10D14 |
| kernel | C14.812- 1+4 | C22⋊Dic14 | C23.D14 | C28⋊Q8 | Dic7.Q8 | C28.48D4 | C23.18D14 | C28.17D4 | C7×C22.D4 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 4 | 2 | 2 | 2 | 1 | 1 | 1 | 3 | 9 | 6 | 3 | 3 | 1 | 2 | 6 | 12 |
Matrix representation of C14.812- 1+4 ►in GL8(𝔽29)
| 6 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 13 |
| 9 | 20 | 13 | 22 | 0 | 0 | 0 | 0 |
| 15 | 7 | 14 | 27 | 0 | 0 | 0 | 0 |
| 5 | 24 | 20 | 22 | 0 | 0 | 0 | 0 |
| 10 | 24 | 11 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 23 | 12 | 24 |
| 0 | 0 | 0 | 0 | 0 | 8 | 26 | 26 |
| 0 | 0 | 0 | 0 | 12 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 12 | 21 | 21 |
| 20 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 10 | 0 | 0 | 0 | 0 |
| 24 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 24 | 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 20 | 0 | 7 |
| 0 | 0 | 0 | 0 | 28 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 9 |
| 28 | 1 | 21 | 17 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 | 1 | 2 |
| 0 | 0 | 0 | 0 | 9 | 0 | 28 | 28 |
| 9 | 20 | 1 | 5 | 0 | 0 | 0 | 0 |
| 15 | 7 | 19 | 10 | 0 | 0 | 0 | 0 |
| 5 | 24 | 20 | 22 | 0 | 0 | 0 | 0 |
| 10 | 24 | 11 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 21 | 3 | 3 |
| 0 | 0 | 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 17 | 8 | 8 |
G:=sub<GL(8,GF(29))| [6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,4,0,5,0,0,0,0,0,0,13,0,5,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,28,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13],[9,15,5,10,0,0,0,0,20,7,24,24,0,0,0,0,13,14,20,11,0,0,0,0,22,27,22,22,0,0,0,0,0,0,0,0,0,0,12,17,0,0,0,0,23,8,9,12,0,0,0,0,12,26,0,21,0,0,0,0,24,26,0,21],[20,0,24,0,0,0,0,0,0,7,0,24,0,0,0,0,28,0,9,0,0,0,0,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,14,20,21,9,0,0,0,0,1,0,0,0,0,0,0,0,2,7,0,9],[28,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,21,0,1,0,0,0,0,0,17,26,28,28,0,0,0,0,0,0,0,0,1,7,0,9,0,0,0,0,8,28,14,0,0,0,0,0,0,0,1,28,0,0,0,0,0,0,2,28],[9,15,5,10,0,0,0,0,20,7,24,24,0,0,0,0,1,19,20,11,0,0,0,0,5,10,22,22,0,0,0,0,0,0,0,0,0,0,17,12,0,0,0,0,0,21,0,17,0,0,0,0,17,3,0,8,0,0,0,0,0,3,0,8] >;
C14.812- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{81}2_-^{1+4} % in TeX
G:=Group("C14.81ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1104);
// by ID
G=gap.SmallGroup(448,1104);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,219,184,1571,570,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=a^7,d^2=e^2=a^7*b^2,b*a*b^-1=c*a*c^-1=e*a*e^-1=a^-1,a*d=d*a,c*b*c^-1=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=a^7*b^2*d>;
// generators/relations