G = C14.12- 1+4 order 448 = 26·7
1st non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.12- 1+4, C14.72+ 1+4, C28⋊Q8⋊8C2, (C2×C28)⋊3Q8, (C2×C4)⋊3Dic14, C28.67(C2×Q8), C4⋊C4.261D14, C4.Dic14⋊8C2, C14.9(C22×Q8), (C2×C14).44C24, C4.32(C2×Dic14), (C2×C28).135C23, C28.48D4.6C2, (C22×C4).175D14, Dic7⋊C4.1C22, C2.11(D4⋊6D14), C22.82(C23×D7), C4⋊Dic7.358C22, (C22×C28).74C22, (C2×Dic7).14C23, (C4×Dic7).62C22, C2.11(C22×Dic14), C22.11(C2×Dic14), C23.223(C22×D7), C23.D7.86C22, (C22×C14).393C23, C2.5(Q8.10D14), C7⋊1(C23.41C23), (C2×Dic14).21C22, C23.21D14.20C2, (C2×C4⋊C4).27D7, (C14×C4⋊C4).20C2, (C2×C14).52(C2×Q8), (C7×C4⋊C4).293C22, (C2×C4).570(C22×D7), SmallGroup(448,953)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.12- 1+4
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=e2=b2, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=a7b-1, dbd-1=ebe-1=a7b, dcd-1=ece-1=a7c, ede-1=a7b2d >
Subgroups: 772 in 206 conjugacy classes, 111 normal (17 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×2], C7, C2×C4 [×10], C2×C4 [×10], Q8 [×4], C23, C14 [×3], C14 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic7 [×8], C28 [×4], C28 [×4], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8 [×4], Dic14 [×4], C2×Dic7 [×8], C2×C28 [×10], C2×C28 [×2], C22×C14, C23.41C23, C4×Dic7 [×4], Dic7⋊C4 [×8], C4⋊Dic7 [×8], C23.D7 [×4], C7×C4⋊C4 [×4], C2×Dic14 [×4], C22×C28, C22×C28 [×2], C28⋊Q8 [×4], C4.Dic14 [×4], C28.48D4 [×4], C23.21D14 [×2], C14×C4⋊C4, C14.12- 1+4
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D7, C2×Q8 [×6], C24, D14 [×7], C22×Q8, 2+ 1+4, 2- 1+4, Dic14 [×4], C22×D7 [×7], C23.41C23, C2×Dic14 [×6], C23×D7, C22×Dic14, D4⋊6D14, Q8.10D14, C14.12- 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 23 193 61)(2 22 194 60)(3 21 195 59)(4 20 196 58)(5 19 183 57)(6 18 184 70)(7 17 185 69)(8 16 186 68)(9 15 187 67)(10 28 188 66)(11 27 189 65)(12 26 190 64)(13 25 191 63)(14 24 192 62)(29 94 121 151)(30 93 122 150)(31 92 123 149)(32 91 124 148)(33 90 125 147)(34 89 126 146)(35 88 113 145)(36 87 114 144)(37 86 115 143)(38 85 116 142)(39 98 117 141)(40 97 118 154)(41 96 119 153)(42 95 120 152)(43 175 218 158)(44 174 219 157)(45 173 220 156)(46 172 221 155)(47 171 222 168)(48 170 223 167)(49 169 224 166)(50 182 211 165)(51 181 212 164)(52 180 213 163)(53 179 214 162)(54 178 215 161)(55 177 216 160)(56 176 217 159)(71 105 201 139)(72 104 202 138)(73 103 203 137)(74 102 204 136)(75 101 205 135)(76 100 206 134)(77 99 207 133)(78 112 208 132)(79 111 209 131)(80 110 210 130)(81 109 197 129)(82 108 198 128)(83 107 199 127)(84 106 200 140)
(1 68 8 61)(2 67 9 60)(3 66 10 59)(4 65 11 58)(5 64 12 57)(6 63 13 70)(7 62 14 69)(15 187 22 194)(16 186 23 193)(17 185 24 192)(18 184 25 191)(19 183 26 190)(20 196 27 189)(21 195 28 188)(29 144 36 151)(30 143 37 150)(31 142 38 149)(32 141 39 148)(33 154 40 147)(34 153 41 146)(35 152 42 145)(43 165 50 158)(44 164 51 157)(45 163 52 156)(46 162 53 155)(47 161 54 168)(48 160 55 167)(49 159 56 166)(71 105 78 112)(72 104 79 111)(73 103 80 110)(74 102 81 109)(75 101 82 108)(76 100 83 107)(77 99 84 106)(85 116 92 123)(86 115 93 122)(87 114 94 121)(88 113 95 120)(89 126 96 119)(90 125 97 118)(91 124 98 117)(127 206 134 199)(128 205 135 198)(129 204 136 197)(130 203 137 210)(131 202 138 209)(132 201 139 208)(133 200 140 207)(169 224 176 217)(170 223 177 216)(171 222 178 215)(172 221 179 214)(173 220 180 213)(174 219 181 212)(175 218 182 211)
(1 120 193 42)(2 121 194 29)(3 122 195 30)(4 123 196 31)(5 124 183 32)(6 125 184 33)(7 126 185 34)(8 113 186 35)(9 114 187 36)(10 115 188 37)(11 116 189 38)(12 117 190 39)(13 118 191 40)(14 119 192 41)(15 151 67 94)(16 152 68 95)(17 153 69 96)(18 154 70 97)(19 141 57 98)(20 142 58 85)(21 143 59 86)(22 144 60 87)(23 145 61 88)(24 146 62 89)(25 147 63 90)(26 148 64 91)(27 149 65 92)(28 150 66 93)(43 140 218 106)(44 127 219 107)(45 128 220 108)(46 129 221 109)(47 130 222 110)(48 131 223 111)(49 132 224 112)(50 133 211 99)(51 134 212 100)(52 135 213 101)(53 136 214 102)(54 137 215 103)(55 138 216 104)(56 139 217 105)(71 166 201 169)(72 167 202 170)(73 168 203 171)(74 155 204 172)(75 156 205 173)(76 157 206 174)(77 158 207 175)(78 159 208 176)(79 160 209 177)(80 161 210 178)(81 162 197 179)(82 163 198 180)(83 164 199 181)(84 165 200 182)
(1 47 193 222)(2 48 194 223)(3 49 195 224)(4 50 196 211)(5 51 183 212)(6 52 184 213)(7 53 185 214)(8 54 186 215)(9 55 187 216)(10 56 188 217)(11 43 189 218)(12 44 190 219)(13 45 191 220)(14 46 192 221)(15 170 67 167)(16 171 68 168)(17 172 69 155)(18 173 70 156)(19 174 57 157)(20 175 58 158)(21 176 59 159)(22 177 60 160)(23 178 61 161)(24 179 62 162)(25 180 63 163)(26 181 64 164)(27 182 65 165)(28 169 66 166)(29 138 121 104)(30 139 122 105)(31 140 123 106)(32 127 124 107)(33 128 125 108)(34 129 126 109)(35 130 113 110)(36 131 114 111)(37 132 115 112)(38 133 116 99)(39 134 117 100)(40 135 118 101)(41 136 119 102)(42 137 120 103)(71 143 201 86)(72 144 202 87)(73 145 203 88)(74 146 204 89)(75 147 205 90)(76 148 206 91)(77 149 207 92)(78 150 208 93)(79 151 209 94)(80 152 210 95)(81 153 197 96)(82 154 198 97)(83 141 199 98)(84 142 200 85)
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), (1,23,193,61)(2,22,194,60)(3,21,195,59)(4,20,196,58)(5,19,183,57)(6,18,184,70)(7,17,185,69)(8,16,186,68)(9,15,187,67)(10,28,188,66)(11,27,189,65)(12,26,190,64)(13,25,191,63)(14,24,192,62)(29,94,121,151)(30,93,122,150)(31,92,123,149)(32,91,124,148)(33,90,125,147)(34,89,126,146)(35,88,113,145)(36,87,114,144)(37,86,115,143)(38,85,116,142)(39,98,117,141)(40,97,118,154)(41,96,119,153)(42,95,120,152)(43,175,218,158)(44,174,219,157)(45,173,220,156)(46,172,221,155)(47,171,222,168)(48,170,223,167)(49,169,224,166)(50,182,211,165)(51,181,212,164)(52,180,213,163)(53,179,214,162)(54,178,215,161)(55,177,216,160)(56,176,217,159)(71,105,201,139)(72,104,202,138)(73,103,203,137)(74,102,204,136)(75,101,205,135)(76,100,206,134)(77,99,207,133)(78,112,208,132)(79,111,209,131)(80,110,210,130)(81,109,197,129)(82,108,198,128)(83,107,199,127)(84,106,200,140), (1,68,8,61)(2,67,9,60)(3,66,10,59)(4,65,11,58)(5,64,12,57)(6,63,13,70)(7,62,14,69)(15,187,22,194)(16,186,23,193)(17,185,24,192)(18,184,25,191)(19,183,26,190)(20,196,27,189)(21,195,28,188)(29,144,36,151)(30,143,37,150)(31,142,38,149)(32,141,39,148)(33,154,40,147)(34,153,41,146)(35,152,42,145)(43,165,50,158)(44,164,51,157)(45,163,52,156)(46,162,53,155)(47,161,54,168)(48,160,55,167)(49,159,56,166)(71,105,78,112)(72,104,79,111)(73,103,80,110)(74,102,81,109)(75,101,82,108)(76,100,83,107)(77,99,84,106)(85,116,92,123)(86,115,93,122)(87,114,94,121)(88,113,95,120)(89,126,96,119)(90,125,97,118)(91,124,98,117)(127,206,134,199)(128,205,135,198)(129,204,136,197)(130,203,137,210)(131,202,138,209)(132,201,139,208)(133,200,140,207)(169,224,176,217)(170,223,177,216)(171,222,178,215)(172,221,179,214)(173,220,180,213)(174,219,181,212)(175,218,182,211), (1,120,193,42)(2,121,194,29)(3,122,195,30)(4,123,196,31)(5,124,183,32)(6,125,184,33)(7,126,185,34)(8,113,186,35)(9,114,187,36)(10,115,188,37)(11,116,189,38)(12,117,190,39)(13,118,191,40)(14,119,192,41)(15,151,67,94)(16,152,68,95)(17,153,69,96)(18,154,70,97)(19,141,57,98)(20,142,58,85)(21,143,59,86)(22,144,60,87)(23,145,61,88)(24,146,62,89)(25,147,63,90)(26,148,64,91)(27,149,65,92)(28,150,66,93)(43,140,218,106)(44,127,219,107)(45,128,220,108)(46,129,221,109)(47,130,222,110)(48,131,223,111)(49,132,224,112)(50,133,211,99)(51,134,212,100)(52,135,213,101)(53,136,214,102)(54,137,215,103)(55,138,216,104)(56,139,217,105)(71,166,201,169)(72,167,202,170)(73,168,203,171)(74,155,204,172)(75,156,205,173)(76,157,206,174)(77,158,207,175)(78,159,208,176)(79,160,209,177)(80,161,210,178)(81,162,197,179)(82,163,198,180)(83,164,199,181)(84,165,200,182), (1,47,193,222)(2,48,194,223)(3,49,195,224)(4,50,196,211)(5,51,183,212)(6,52,184,213)(7,53,185,214)(8,54,186,215)(9,55,187,216)(10,56,188,217)(11,43,189,218)(12,44,190,219)(13,45,191,220)(14,46,192,221)(15,170,67,167)(16,171,68,168)(17,172,69,155)(18,173,70,156)(19,174,57,157)(20,175,58,158)(21,176,59,159)(22,177,60,160)(23,178,61,161)(24,179,62,162)(25,180,63,163)(26,181,64,164)(27,182,65,165)(28,169,66,166)(29,138,121,104)(30,139,122,105)(31,140,123,106)(32,127,124,107)(33,128,125,108)(34,129,126,109)(35,130,113,110)(36,131,114,111)(37,132,115,112)(38,133,116,99)(39,134,117,100)(40,135,118,101)(41,136,119,102)(42,137,120,103)(71,143,201,86)(72,144,202,87)(73,145,203,88)(74,146,204,89)(75,147,205,90)(76,148,206,91)(77,149,207,92)(78,150,208,93)(79,151,209,94)(80,152,210,95)(81,153,197,96)(82,154,198,97)(83,141,199,98)(84,142,200,85)>;
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), (1,23,193,61)(2,22,194,60)(3,21,195,59)(4,20,196,58)(5,19,183,57)(6,18,184,70)(7,17,185,69)(8,16,186,68)(9,15,187,67)(10,28,188,66)(11,27,189,65)(12,26,190,64)(13,25,191,63)(14,24,192,62)(29,94,121,151)(30,93,122,150)(31,92,123,149)(32,91,124,148)(33,90,125,147)(34,89,126,146)(35,88,113,145)(36,87,114,144)(37,86,115,143)(38,85,116,142)(39,98,117,141)(40,97,118,154)(41,96,119,153)(42,95,120,152)(43,175,218,158)(44,174,219,157)(45,173,220,156)(46,172,221,155)(47,171,222,168)(48,170,223,167)(49,169,224,166)(50,182,211,165)(51,181,212,164)(52,180,213,163)(53,179,214,162)(54,178,215,161)(55,177,216,160)(56,176,217,159)(71,105,201,139)(72,104,202,138)(73,103,203,137)(74,102,204,136)(75,101,205,135)(76,100,206,134)(77,99,207,133)(78,112,208,132)(79,111,209,131)(80,110,210,130)(81,109,197,129)(82,108,198,128)(83,107,199,127)(84,106,200,140), (1,68,8,61)(2,67,9,60)(3,66,10,59)(4,65,11,58)(5,64,12,57)(6,63,13,70)(7,62,14,69)(15,187,22,194)(16,186,23,193)(17,185,24,192)(18,184,25,191)(19,183,26,190)(20,196,27,189)(21,195,28,188)(29,144,36,151)(30,143,37,150)(31,142,38,149)(32,141,39,148)(33,154,40,147)(34,153,41,146)(35,152,42,145)(43,165,50,158)(44,164,51,157)(45,163,52,156)(46,162,53,155)(47,161,54,168)(48,160,55,167)(49,159,56,166)(71,105,78,112)(72,104,79,111)(73,103,80,110)(74,102,81,109)(75,101,82,108)(76,100,83,107)(77,99,84,106)(85,116,92,123)(86,115,93,122)(87,114,94,121)(88,113,95,120)(89,126,96,119)(90,125,97,118)(91,124,98,117)(127,206,134,199)(128,205,135,198)(129,204,136,197)(130,203,137,210)(131,202,138,209)(132,201,139,208)(133,200,140,207)(169,224,176,217)(170,223,177,216)(171,222,178,215)(172,221,179,214)(173,220,180,213)(174,219,181,212)(175,218,182,211), (1,120,193,42)(2,121,194,29)(3,122,195,30)(4,123,196,31)(5,124,183,32)(6,125,184,33)(7,126,185,34)(8,113,186,35)(9,114,187,36)(10,115,188,37)(11,116,189,38)(12,117,190,39)(13,118,191,40)(14,119,192,41)(15,151,67,94)(16,152,68,95)(17,153,69,96)(18,154,70,97)(19,141,57,98)(20,142,58,85)(21,143,59,86)(22,144,60,87)(23,145,61,88)(24,146,62,89)(25,147,63,90)(26,148,64,91)(27,149,65,92)(28,150,66,93)(43,140,218,106)(44,127,219,107)(45,128,220,108)(46,129,221,109)(47,130,222,110)(48,131,223,111)(49,132,224,112)(50,133,211,99)(51,134,212,100)(52,135,213,101)(53,136,214,102)(54,137,215,103)(55,138,216,104)(56,139,217,105)(71,166,201,169)(72,167,202,170)(73,168,203,171)(74,155,204,172)(75,156,205,173)(76,157,206,174)(77,158,207,175)(78,159,208,176)(79,160,209,177)(80,161,210,178)(81,162,197,179)(82,163,198,180)(83,164,199,181)(84,165,200,182), (1,47,193,222)(2,48,194,223)(3,49,195,224)(4,50,196,211)(5,51,183,212)(6,52,184,213)(7,53,185,214)(8,54,186,215)(9,55,187,216)(10,56,188,217)(11,43,189,218)(12,44,190,219)(13,45,191,220)(14,46,192,221)(15,170,67,167)(16,171,68,168)(17,172,69,155)(18,173,70,156)(19,174,57,157)(20,175,58,158)(21,176,59,159)(22,177,60,160)(23,178,61,161)(24,179,62,162)(25,180,63,163)(26,181,64,164)(27,182,65,165)(28,169,66,166)(29,138,121,104)(30,139,122,105)(31,140,123,106)(32,127,124,107)(33,128,125,108)(34,129,126,109)(35,130,113,110)(36,131,114,111)(37,132,115,112)(38,133,116,99)(39,134,117,100)(40,135,118,101)(41,136,119,102)(42,137,120,103)(71,143,201,86)(72,144,202,87)(73,145,203,88)(74,146,204,89)(75,147,205,90)(76,148,206,91)(77,149,207,92)(78,150,208,93)(79,151,209,94)(80,152,210,95)(81,153,197,96)(82,154,198,97)(83,141,199,98)(84,142,200,85) );
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)], [(1,23,193,61),(2,22,194,60),(3,21,195,59),(4,20,196,58),(5,19,183,57),(6,18,184,70),(7,17,185,69),(8,16,186,68),(9,15,187,67),(10,28,188,66),(11,27,189,65),(12,26,190,64),(13,25,191,63),(14,24,192,62),(29,94,121,151),(30,93,122,150),(31,92,123,149),(32,91,124,148),(33,90,125,147),(34,89,126,146),(35,88,113,145),(36,87,114,144),(37,86,115,143),(38,85,116,142),(39,98,117,141),(40,97,118,154),(41,96,119,153),(42,95,120,152),(43,175,218,158),(44,174,219,157),(45,173,220,156),(46,172,221,155),(47,171,222,168),(48,170,223,167),(49,169,224,166),(50,182,211,165),(51,181,212,164),(52,180,213,163),(53,179,214,162),(54,178,215,161),(55,177,216,160),(56,176,217,159),(71,105,201,139),(72,104,202,138),(73,103,203,137),(74,102,204,136),(75,101,205,135),(76,100,206,134),(77,99,207,133),(78,112,208,132),(79,111,209,131),(80,110,210,130),(81,109,197,129),(82,108,198,128),(83,107,199,127),(84,106,200,140)], [(1,68,8,61),(2,67,9,60),(3,66,10,59),(4,65,11,58),(5,64,12,57),(6,63,13,70),(7,62,14,69),(15,187,22,194),(16,186,23,193),(17,185,24,192),(18,184,25,191),(19,183,26,190),(20,196,27,189),(21,195,28,188),(29,144,36,151),(30,143,37,150),(31,142,38,149),(32,141,39,148),(33,154,40,147),(34,153,41,146),(35,152,42,145),(43,165,50,158),(44,164,51,157),(45,163,52,156),(46,162,53,155),(47,161,54,168),(48,160,55,167),(49,159,56,166),(71,105,78,112),(72,104,79,111),(73,103,80,110),(74,102,81,109),(75,101,82,108),(76,100,83,107),(77,99,84,106),(85,116,92,123),(86,115,93,122),(87,114,94,121),(88,113,95,120),(89,126,96,119),(90,125,97,118),(91,124,98,117),(127,206,134,199),(128,205,135,198),(129,204,136,197),(130,203,137,210),(131,202,138,209),(132,201,139,208),(133,200,140,207),(169,224,176,217),(170,223,177,216),(171,222,178,215),(172,221,179,214),(173,220,180,213),(174,219,181,212),(175,218,182,211)], [(1,120,193,42),(2,121,194,29),(3,122,195,30),(4,123,196,31),(5,124,183,32),(6,125,184,33),(7,126,185,34),(8,113,186,35),(9,114,187,36),(10,115,188,37),(11,116,189,38),(12,117,190,39),(13,118,191,40),(14,119,192,41),(15,151,67,94),(16,152,68,95),(17,153,69,96),(18,154,70,97),(19,141,57,98),(20,142,58,85),(21,143,59,86),(22,144,60,87),(23,145,61,88),(24,146,62,89),(25,147,63,90),(26,148,64,91),(27,149,65,92),(28,150,66,93),(43,140,218,106),(44,127,219,107),(45,128,220,108),(46,129,221,109),(47,130,222,110),(48,131,223,111),(49,132,224,112),(50,133,211,99),(51,134,212,100),(52,135,213,101),(53,136,214,102),(54,137,215,103),(55,138,216,104),(56,139,217,105),(71,166,201,169),(72,167,202,170),(73,168,203,171),(74,155,204,172),(75,156,205,173),(76,157,206,174),(77,158,207,175),(78,159,208,176),(79,160,209,177),(80,161,210,178),(81,162,197,179),(82,163,198,180),(83,164,199,181),(84,165,200,182)], [(1,47,193,222),(2,48,194,223),(3,49,195,224),(4,50,196,211),(5,51,183,212),(6,52,184,213),(7,53,185,214),(8,54,186,215),(9,55,187,216),(10,56,188,217),(11,43,189,218),(12,44,190,219),(13,45,191,220),(14,46,192,221),(15,170,67,167),(16,171,68,168),(17,172,69,155),(18,173,70,156),(19,174,57,157),(20,175,58,158),(21,176,59,159),(22,177,60,160),(23,178,61,161),(24,179,62,162),(25,180,63,163),(26,181,64,164),(27,182,65,165),(28,169,66,166),(29,138,121,104),(30,139,122,105),(31,140,123,106),(32,127,124,107),(33,128,125,108),(34,129,126,109),(35,130,113,110),(36,131,114,111),(37,132,115,112),(38,133,116,99),(39,134,117,100),(40,135,118,101),(41,136,119,102),(42,137,120,103),(71,143,201,86),(72,144,202,87),(73,145,203,88),(74,146,204,89),(75,147,205,90),(76,148,206,91),(77,149,207,92),(78,150,208,93),(79,151,209,94),(80,152,210,95),(81,153,197,96),(82,154,198,97),(83,141,199,98),(84,142,200,85)])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | Q8 | D7 | D14 | D14 | Dic14 | 2+ 1+4 | 2- 1+4 | D4⋊6D14 | Q8.10D14 |
| kernel | C14.12- 1+4 | C28⋊Q8 | C4.Dic14 | C28.48D4 | C23.21D14 | C14×C4⋊C4 | C2×C28 | C2×C4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C14 | C14 | C2 | C2 |
| # reps | 1 | 4 | 4 | 4 | 2 | 1 | 4 | 3 | 12 | 9 | 24 | 1 | 1 | 6 | 6 |
Matrix representation of C14.12- 1+4 ►in GL6(𝔽29)
| 22 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 8 | 17 |
| 0 | 0 | 20 | 0 | 22 | 23 |
| 0 | 0 | 0 | 8 | 15 | 21 |
| 0 | 0 | 20 | 15 | 14 | 8 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 8 | 17 |
| 0 | 0 | 20 | 0 | 22 | 23 |
| 0 | 0 | 8 | 21 | 15 | 21 |
| 0 | 0 | 1 | 14 | 14 | 8 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 12 | 0 | 0 |
| 0 | 0 | 9 | 6 | 0 | 0 |
| 0 | 0 | 0 | 8 | 15 | 21 |
| 0 | 0 | 9 | 21 | 21 | 14 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 13 | 0 | 0 |
| 0 | 0 | 11 | 1 | 0 | 0 |
| 0 | 0 | 18 | 28 | 0 | 28 |
| 0 | 0 | 0 | 1 | 1 | 0 |
G:=sub<GL(6,GF(29))| [22,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,17,0,0,0,0,17,0,0,0,0,0,0,0,6,20,0,20,0,0,0,0,8,15,0,0,8,22,15,14,0,0,17,23,21,8],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,6,20,8,1,0,0,0,0,21,14,0,0,8,22,15,14,0,0,17,23,21,8],[12,0,0,0,0,0,0,17,0,0,0,0,0,0,23,9,0,9,0,0,12,6,8,21,0,0,0,0,15,21,0,0,0,0,21,14],[12,0,0,0,0,0,0,17,0,0,0,0,0,0,28,11,18,0,0,0,13,1,28,1,0,0,0,0,0,1,0,0,0,0,28,0] >;
C14.12- 1+4 in GAP, Magma, Sage, TeX
C_{14}._12_-^{1+4} % in TeX
G:=Group("C14.1ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,953);
// by ID
G=gap.SmallGroup(448,953);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,758,675,570,80,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=a^7,d^2=e^2=b^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=a^7*b^-1,d*b*d^-1=e*b*e^-1=a^7*b,d*c*d^-1=e*c*e^-1=a^7*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations