G = C14.1052- 1+4 order 448 = 26·7
60th non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.1052- 1+4, (C2×C28).220D4, C28.429(C2×D4), (C2×D4).235D14, (C2×Q8).193D14, Dic7⋊Q8⋊32C2, C28.48D4⋊48C2, C28.17D4⋊30C2, (C2×C28).650C23, (C2×C14).311C24, C14.163(C22×D4), (C22×C4).285D14, (C22×Dic14)⋊22C2, (D4×C14).314C22, Dic7⋊C4.92C22, C4⋊Dic7.320C22, (Q8×C14).240C22, C22.322(C23×D7), C23.208(C22×D7), C23.18D14⋊32C2, C23.21D14⋊35C2, (C22×C14).237C23, (C22×C28).320C22, C7⋊7(C23.38C23), (C4×Dic7).174C22, (C2×Dic7).161C23, C2.69(D4.10D14), C23.D7.133C22, (C2×Dic14).310C22, (C22×Dic7).166C22, C4.32(C2×C7⋊D4), (C2×C4○D4).11D7, (C2×C14).79(C2×D4), (C14×C4○D4).12C2, (C2×C4).97(C7⋊D4), C2.36(C22×C7⋊D4), C22.22(C2×C7⋊D4), (C2×C4).249(C22×D7), SmallGroup(448,1278)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.1052- 1+4
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=b2, e2=a7b2, bab-1=cac-1=eae-1=a-1, ad=da, cbc-1=a7b-1, dbd-1=ebe-1=a7b, dcd-1=a7c, ce=ec, ede-1=a7b2d >
Subgroups: 980 in 270 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×Q8, C2×C4○D4, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C23.38C23, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C23.D7, C2×Dic14, C2×Dic14, C22×Dic7, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C28.48D4, C23.21D14, C23.18D14, C28.17D4, Dic7⋊Q8, C22×Dic14, C14×C4○D4, C14.1052- 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2- 1+4, C7⋊D4, C22×D7, C23.38C23, C2×C7⋊D4, C23×D7, D4.10D14, C22×C7⋊D4, C14.1052- 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)
(1 185 87 80)(2 184 88 79)(3 183 89 78)(4 196 90 77)(5 195 91 76)(6 194 92 75)(7 193 93 74)(8 192 94 73)(9 191 95 72)(10 190 96 71)(11 189 97 84)(12 188 98 83)(13 187 85 82)(14 186 86 81)(15 29 211 55)(16 42 212 54)(17 41 213 53)(18 40 214 52)(19 39 215 51)(20 38 216 50)(21 37 217 49)(22 36 218 48)(23 35 219 47)(24 34 220 46)(25 33 221 45)(26 32 222 44)(27 31 223 43)(28 30 224 56)(57 200 165 172)(58 199 166 171)(59 198 167 170)(60 197 168 169)(61 210 155 182)(62 209 156 181)(63 208 157 180)(64 207 158 179)(65 206 159 178)(66 205 160 177)(67 204 161 176)(68 203 162 175)(69 202 163 174)(70 201 164 173)(99 122 150 127)(100 121 151 140)(101 120 152 139)(102 119 153 138)(103 118 154 137)(104 117 141 136)(105 116 142 135)(106 115 143 134)(107 114 144 133)(108 113 145 132)(109 126 146 131)(110 125 147 130)(111 124 148 129)(112 123 149 128)
(1 30 8 37)(2 29 9 36)(3 42 10 35)(4 41 11 34)(5 40 12 33)(6 39 13 32)(7 38 14 31)(15 184 22 191)(16 183 23 190)(17 196 24 189)(18 195 25 188)(19 194 26 187)(20 193 27 186)(21 192 28 185)(43 93 50 86)(44 92 51 85)(45 91 52 98)(46 90 53 97)(47 89 54 96)(48 88 55 95)(49 87 56 94)(57 124 64 117)(58 123 65 116)(59 122 66 115)(60 121 67 114)(61 120 68 113)(62 119 69 126)(63 118 70 125)(71 212 78 219)(72 211 79 218)(73 224 80 217)(74 223 81 216)(75 222 82 215)(76 221 83 214)(77 220 84 213)(99 198 106 205)(100 197 107 204)(101 210 108 203)(102 209 109 202)(103 208 110 201)(104 207 111 200)(105 206 112 199)(127 160 134 167)(128 159 135 166)(129 158 136 165)(130 157 137 164)(131 156 138 163)(132 155 139 162)(133 168 140 161)(141 179 148 172)(142 178 149 171)(143 177 150 170)(144 176 151 169)(145 175 152 182)(146 174 153 181)(147 173 154 180)
(1 133 87 114)(2 134 88 115)(3 135 89 116)(4 136 90 117)(5 137 91 118)(6 138 92 119)(7 139 93 120)(8 140 94 121)(9 127 95 122)(10 128 96 123)(11 129 97 124)(12 130 98 125)(13 131 85 126)(14 132 86 113)(15 198 211 170)(16 199 212 171)(17 200 213 172)(18 201 214 173)(19 202 215 174)(20 203 216 175)(21 204 217 176)(22 205 218 177)(23 206 219 178)(24 207 220 179)(25 208 221 180)(26 209 222 181)(27 210 223 182)(28 197 224 169)(29 160 55 66)(30 161 56 67)(31 162 43 68)(32 163 44 69)(33 164 45 70)(34 165 46 57)(35 166 47 58)(36 167 48 59)(37 168 49 60)(38 155 50 61)(39 156 51 62)(40 157 52 63)(41 158 53 64)(42 159 54 65)(71 142 190 105)(72 143 191 106)(73 144 192 107)(74 145 193 108)(75 146 194 109)(76 147 195 110)(77 148 196 111)(78 149 183 112)(79 150 184 99)(80 151 185 100)(81 152 186 101)(82 153 187 102)(83 154 188 103)(84 141 189 104)
(1 151 94 107)(2 150 95 106)(3 149 96 105)(4 148 97 104)(5 147 98 103)(6 146 85 102)(7 145 86 101)(8 144 87 100)(9 143 88 99)(10 142 89 112)(11 141 90 111)(12 154 91 110)(13 153 92 109)(14 152 93 108)(15 160 218 59)(16 159 219 58)(17 158 220 57)(18 157 221 70)(19 156 222 69)(20 155 223 68)(21 168 224 67)(22 167 211 66)(23 166 212 65)(24 165 213 64)(25 164 214 63)(26 163 215 62)(27 162 216 61)(28 161 217 60)(29 170 48 205)(30 169 49 204)(31 182 50 203)(32 181 51 202)(33 180 52 201)(34 179 53 200)(35 178 54 199)(36 177 55 198)(37 176 56 197)(38 175 43 210)(39 174 44 209)(40 173 45 208)(41 172 46 207)(42 171 47 206)(71 123 183 135)(72 122 184 134)(73 121 185 133)(74 120 186 132)(75 119 187 131)(76 118 188 130)(77 117 189 129)(78 116 190 128)(79 115 191 127)(80 114 192 140)(81 113 193 139)(82 126 194 138)(83 125 195 137)(84 124 196 136)
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), (1,185,87,80)(2,184,88,79)(3,183,89,78)(4,196,90,77)(5,195,91,76)(6,194,92,75)(7,193,93,74)(8,192,94,73)(9,191,95,72)(10,190,96,71)(11,189,97,84)(12,188,98,83)(13,187,85,82)(14,186,86,81)(15,29,211,55)(16,42,212,54)(17,41,213,53)(18,40,214,52)(19,39,215,51)(20,38,216,50)(21,37,217,49)(22,36,218,48)(23,35,219,47)(24,34,220,46)(25,33,221,45)(26,32,222,44)(27,31,223,43)(28,30,224,56)(57,200,165,172)(58,199,166,171)(59,198,167,170)(60,197,168,169)(61,210,155,182)(62,209,156,181)(63,208,157,180)(64,207,158,179)(65,206,159,178)(66,205,160,177)(67,204,161,176)(68,203,162,175)(69,202,163,174)(70,201,164,173)(99,122,150,127)(100,121,151,140)(101,120,152,139)(102,119,153,138)(103,118,154,137)(104,117,141,136)(105,116,142,135)(106,115,143,134)(107,114,144,133)(108,113,145,132)(109,126,146,131)(110,125,147,130)(111,124,148,129)(112,123,149,128), (1,30,8,37)(2,29,9,36)(3,42,10,35)(4,41,11,34)(5,40,12,33)(6,39,13,32)(7,38,14,31)(15,184,22,191)(16,183,23,190)(17,196,24,189)(18,195,25,188)(19,194,26,187)(20,193,27,186)(21,192,28,185)(43,93,50,86)(44,92,51,85)(45,91,52,98)(46,90,53,97)(47,89,54,96)(48,88,55,95)(49,87,56,94)(57,124,64,117)(58,123,65,116)(59,122,66,115)(60,121,67,114)(61,120,68,113)(62,119,69,126)(63,118,70,125)(71,212,78,219)(72,211,79,218)(73,224,80,217)(74,223,81,216)(75,222,82,215)(76,221,83,214)(77,220,84,213)(99,198,106,205)(100,197,107,204)(101,210,108,203)(102,209,109,202)(103,208,110,201)(104,207,111,200)(105,206,112,199)(127,160,134,167)(128,159,135,166)(129,158,136,165)(130,157,137,164)(131,156,138,163)(132,155,139,162)(133,168,140,161)(141,179,148,172)(142,178,149,171)(143,177,150,170)(144,176,151,169)(145,175,152,182)(146,174,153,181)(147,173,154,180), (1,133,87,114)(2,134,88,115)(3,135,89,116)(4,136,90,117)(5,137,91,118)(6,138,92,119)(7,139,93,120)(8,140,94,121)(9,127,95,122)(10,128,96,123)(11,129,97,124)(12,130,98,125)(13,131,85,126)(14,132,86,113)(15,198,211,170)(16,199,212,171)(17,200,213,172)(18,201,214,173)(19,202,215,174)(20,203,216,175)(21,204,217,176)(22,205,218,177)(23,206,219,178)(24,207,220,179)(25,208,221,180)(26,209,222,181)(27,210,223,182)(28,197,224,169)(29,160,55,66)(30,161,56,67)(31,162,43,68)(32,163,44,69)(33,164,45,70)(34,165,46,57)(35,166,47,58)(36,167,48,59)(37,168,49,60)(38,155,50,61)(39,156,51,62)(40,157,52,63)(41,158,53,64)(42,159,54,65)(71,142,190,105)(72,143,191,106)(73,144,192,107)(74,145,193,108)(75,146,194,109)(76,147,195,110)(77,148,196,111)(78,149,183,112)(79,150,184,99)(80,151,185,100)(81,152,186,101)(82,153,187,102)(83,154,188,103)(84,141,189,104), (1,151,94,107)(2,150,95,106)(3,149,96,105)(4,148,97,104)(5,147,98,103)(6,146,85,102)(7,145,86,101)(8,144,87,100)(9,143,88,99)(10,142,89,112)(11,141,90,111)(12,154,91,110)(13,153,92,109)(14,152,93,108)(15,160,218,59)(16,159,219,58)(17,158,220,57)(18,157,221,70)(19,156,222,69)(20,155,223,68)(21,168,224,67)(22,167,211,66)(23,166,212,65)(24,165,213,64)(25,164,214,63)(26,163,215,62)(27,162,216,61)(28,161,217,60)(29,170,48,205)(30,169,49,204)(31,182,50,203)(32,181,51,202)(33,180,52,201)(34,179,53,200)(35,178,54,199)(36,177,55,198)(37,176,56,197)(38,175,43,210)(39,174,44,209)(40,173,45,208)(41,172,46,207)(42,171,47,206)(71,123,183,135)(72,122,184,134)(73,121,185,133)(74,120,186,132)(75,119,187,131)(76,118,188,130)(77,117,189,129)(78,116,190,128)(79,115,191,127)(80,114,192,140)(81,113,193,139)(82,126,194,138)(83,125,195,137)(84,124,196,136)>;
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), (1,185,87,80)(2,184,88,79)(3,183,89,78)(4,196,90,77)(5,195,91,76)(6,194,92,75)(7,193,93,74)(8,192,94,73)(9,191,95,72)(10,190,96,71)(11,189,97,84)(12,188,98,83)(13,187,85,82)(14,186,86,81)(15,29,211,55)(16,42,212,54)(17,41,213,53)(18,40,214,52)(19,39,215,51)(20,38,216,50)(21,37,217,49)(22,36,218,48)(23,35,219,47)(24,34,220,46)(25,33,221,45)(26,32,222,44)(27,31,223,43)(28,30,224,56)(57,200,165,172)(58,199,166,171)(59,198,167,170)(60,197,168,169)(61,210,155,182)(62,209,156,181)(63,208,157,180)(64,207,158,179)(65,206,159,178)(66,205,160,177)(67,204,161,176)(68,203,162,175)(69,202,163,174)(70,201,164,173)(99,122,150,127)(100,121,151,140)(101,120,152,139)(102,119,153,138)(103,118,154,137)(104,117,141,136)(105,116,142,135)(106,115,143,134)(107,114,144,133)(108,113,145,132)(109,126,146,131)(110,125,147,130)(111,124,148,129)(112,123,149,128), (1,30,8,37)(2,29,9,36)(3,42,10,35)(4,41,11,34)(5,40,12,33)(6,39,13,32)(7,38,14,31)(15,184,22,191)(16,183,23,190)(17,196,24,189)(18,195,25,188)(19,194,26,187)(20,193,27,186)(21,192,28,185)(43,93,50,86)(44,92,51,85)(45,91,52,98)(46,90,53,97)(47,89,54,96)(48,88,55,95)(49,87,56,94)(57,124,64,117)(58,123,65,116)(59,122,66,115)(60,121,67,114)(61,120,68,113)(62,119,69,126)(63,118,70,125)(71,212,78,219)(72,211,79,218)(73,224,80,217)(74,223,81,216)(75,222,82,215)(76,221,83,214)(77,220,84,213)(99,198,106,205)(100,197,107,204)(101,210,108,203)(102,209,109,202)(103,208,110,201)(104,207,111,200)(105,206,112,199)(127,160,134,167)(128,159,135,166)(129,158,136,165)(130,157,137,164)(131,156,138,163)(132,155,139,162)(133,168,140,161)(141,179,148,172)(142,178,149,171)(143,177,150,170)(144,176,151,169)(145,175,152,182)(146,174,153,181)(147,173,154,180), (1,133,87,114)(2,134,88,115)(3,135,89,116)(4,136,90,117)(5,137,91,118)(6,138,92,119)(7,139,93,120)(8,140,94,121)(9,127,95,122)(10,128,96,123)(11,129,97,124)(12,130,98,125)(13,131,85,126)(14,132,86,113)(15,198,211,170)(16,199,212,171)(17,200,213,172)(18,201,214,173)(19,202,215,174)(20,203,216,175)(21,204,217,176)(22,205,218,177)(23,206,219,178)(24,207,220,179)(25,208,221,180)(26,209,222,181)(27,210,223,182)(28,197,224,169)(29,160,55,66)(30,161,56,67)(31,162,43,68)(32,163,44,69)(33,164,45,70)(34,165,46,57)(35,166,47,58)(36,167,48,59)(37,168,49,60)(38,155,50,61)(39,156,51,62)(40,157,52,63)(41,158,53,64)(42,159,54,65)(71,142,190,105)(72,143,191,106)(73,144,192,107)(74,145,193,108)(75,146,194,109)(76,147,195,110)(77,148,196,111)(78,149,183,112)(79,150,184,99)(80,151,185,100)(81,152,186,101)(82,153,187,102)(83,154,188,103)(84,141,189,104), (1,151,94,107)(2,150,95,106)(3,149,96,105)(4,148,97,104)(5,147,98,103)(6,146,85,102)(7,145,86,101)(8,144,87,100)(9,143,88,99)(10,142,89,112)(11,141,90,111)(12,154,91,110)(13,153,92,109)(14,152,93,108)(15,160,218,59)(16,159,219,58)(17,158,220,57)(18,157,221,70)(19,156,222,69)(20,155,223,68)(21,168,224,67)(22,167,211,66)(23,166,212,65)(24,165,213,64)(25,164,214,63)(26,163,215,62)(27,162,216,61)(28,161,217,60)(29,170,48,205)(30,169,49,204)(31,182,50,203)(32,181,51,202)(33,180,52,201)(34,179,53,200)(35,178,54,199)(36,177,55,198)(37,176,56,197)(38,175,43,210)(39,174,44,209)(40,173,45,208)(41,172,46,207)(42,171,47,206)(71,123,183,135)(72,122,184,134)(73,121,185,133)(74,120,186,132)(75,119,187,131)(76,118,188,130)(77,117,189,129)(78,116,190,128)(79,115,191,127)(80,114,192,140)(81,113,193,139)(82,126,194,138)(83,125,195,137)(84,124,196,136) );
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)], [(1,185,87,80),(2,184,88,79),(3,183,89,78),(4,196,90,77),(5,195,91,76),(6,194,92,75),(7,193,93,74),(8,192,94,73),(9,191,95,72),(10,190,96,71),(11,189,97,84),(12,188,98,83),(13,187,85,82),(14,186,86,81),(15,29,211,55),(16,42,212,54),(17,41,213,53),(18,40,214,52),(19,39,215,51),(20,38,216,50),(21,37,217,49),(22,36,218,48),(23,35,219,47),(24,34,220,46),(25,33,221,45),(26,32,222,44),(27,31,223,43),(28,30,224,56),(57,200,165,172),(58,199,166,171),(59,198,167,170),(60,197,168,169),(61,210,155,182),(62,209,156,181),(63,208,157,180),(64,207,158,179),(65,206,159,178),(66,205,160,177),(67,204,161,176),(68,203,162,175),(69,202,163,174),(70,201,164,173),(99,122,150,127),(100,121,151,140),(101,120,152,139),(102,119,153,138),(103,118,154,137),(104,117,141,136),(105,116,142,135),(106,115,143,134),(107,114,144,133),(108,113,145,132),(109,126,146,131),(110,125,147,130),(111,124,148,129),(112,123,149,128)], [(1,30,8,37),(2,29,9,36),(3,42,10,35),(4,41,11,34),(5,40,12,33),(6,39,13,32),(7,38,14,31),(15,184,22,191),(16,183,23,190),(17,196,24,189),(18,195,25,188),(19,194,26,187),(20,193,27,186),(21,192,28,185),(43,93,50,86),(44,92,51,85),(45,91,52,98),(46,90,53,97),(47,89,54,96),(48,88,55,95),(49,87,56,94),(57,124,64,117),(58,123,65,116),(59,122,66,115),(60,121,67,114),(61,120,68,113),(62,119,69,126),(63,118,70,125),(71,212,78,219),(72,211,79,218),(73,224,80,217),(74,223,81,216),(75,222,82,215),(76,221,83,214),(77,220,84,213),(99,198,106,205),(100,197,107,204),(101,210,108,203),(102,209,109,202),(103,208,110,201),(104,207,111,200),(105,206,112,199),(127,160,134,167),(128,159,135,166),(129,158,136,165),(130,157,137,164),(131,156,138,163),(132,155,139,162),(133,168,140,161),(141,179,148,172),(142,178,149,171),(143,177,150,170),(144,176,151,169),(145,175,152,182),(146,174,153,181),(147,173,154,180)], [(1,133,87,114),(2,134,88,115),(3,135,89,116),(4,136,90,117),(5,137,91,118),(6,138,92,119),(7,139,93,120),(8,140,94,121),(9,127,95,122),(10,128,96,123),(11,129,97,124),(12,130,98,125),(13,131,85,126),(14,132,86,113),(15,198,211,170),(16,199,212,171),(17,200,213,172),(18,201,214,173),(19,202,215,174),(20,203,216,175),(21,204,217,176),(22,205,218,177),(23,206,219,178),(24,207,220,179),(25,208,221,180),(26,209,222,181),(27,210,223,182),(28,197,224,169),(29,160,55,66),(30,161,56,67),(31,162,43,68),(32,163,44,69),(33,164,45,70),(34,165,46,57),(35,166,47,58),(36,167,48,59),(37,168,49,60),(38,155,50,61),(39,156,51,62),(40,157,52,63),(41,158,53,64),(42,159,54,65),(71,142,190,105),(72,143,191,106),(73,144,192,107),(74,145,193,108),(75,146,194,109),(76,147,195,110),(77,148,196,111),(78,149,183,112),(79,150,184,99),(80,151,185,100),(81,152,186,101),(82,153,187,102),(83,154,188,103),(84,141,189,104)], [(1,151,94,107),(2,150,95,106),(3,149,96,105),(4,148,97,104),(5,147,98,103),(6,146,85,102),(7,145,86,101),(8,144,87,100),(9,143,88,99),(10,142,89,112),(11,141,90,111),(12,154,91,110),(13,153,92,109),(14,152,93,108),(15,160,218,59),(16,159,219,58),(17,158,220,57),(18,157,221,70),(19,156,222,69),(20,155,223,68),(21,168,224,67),(22,167,211,66),(23,166,212,65),(24,165,213,64),(25,164,214,63),(26,163,215,62),(27,162,216,61),(28,161,217,60),(29,170,48,205),(30,169,49,204),(31,182,50,203),(32,181,51,202),(33,180,52,201),(34,179,53,200),(35,178,54,199),(36,177,55,198),(37,176,56,197),(38,175,43,210),(39,174,44,209),(40,173,45,208),(41,172,46,207),(42,171,47,206),(71,123,183,135),(72,122,184,134),(73,121,185,133),(74,120,186,132),(75,119,187,131),(76,118,188,130),(77,117,189,129),(78,116,190,128),(79,115,191,127),(80,114,192,140),(81,113,193,139),(82,126,194,138),(83,125,195,137),(84,124,196,136)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 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 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | 2- 1+4 | D4.10D14 |
| kernel | C14.1052- 1+4 | C28.48D4 | C23.21D14 | C23.18D14 | C28.17D4 | Dic7⋊Q8 | C22×Dic14 | C14×C4○D4 | C2×C28 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C14 | C2 |
| # reps | 1 | 4 | 1 | 4 | 2 | 2 | 1 | 1 | 4 | 3 | 9 | 9 | 3 | 24 | 2 | 12 |
Matrix representation of C14.1052- 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 26 | 3 | 0 | 0 |
| 0 | 0 | 10 | 0 | 10 | 10 |
| 0 | 0 | 22 | 19 | 19 | 22 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 8 | 0 | 0 |
| 0 | 0 | 3 | 27 | 0 | 0 |
| 0 | 0 | 21 | 0 | 8 | 21 |
| 0 | 0 | 19 | 21 | 19 | 21 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 25 | 28 | 21 |
| 0 | 0 | 20 | 22 | 13 | 22 |
| 0 | 0 | 1 | 20 | 4 | 4 |
| 0 | 0 | 8 | 16 | 4 | 4 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 1 |
| 0 | 0 | 7 | 28 | 27 | 7 |
| 0 | 0 | 26 | 17 | 1 | 0 |
| 0 | 0 | 25 | 17 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 21 | 11 | 0 |
| 0 | 0 | 15 | 2 | 1 | 18 |
| 0 | 0 | 21 | 0 | 8 | 21 |
| 0 | 0 | 23 | 8 | 8 | 21 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,26,10,22,0,0,10,3,0,19,0,0,0,0,10,19,0,0,0,0,10,22],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,3,21,19,0,0,8,27,0,21,0,0,0,0,8,19,0,0,0,0,21,21],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,28,20,1,8,0,0,25,22,20,16,0,0,28,13,4,4,0,0,21,22,4,4],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,26,25,0,0,0,28,17,17,0,0,28,27,1,1,0,0,1,7,0,0],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,27,15,21,23,0,0,21,2,0,8,0,0,11,1,8,8,0,0,0,18,21,21] >;
C14.1052- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{105}2_-^{1+4} % in TeX
G:=Group("C14.105ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1278);
// by ID
G=gap.SmallGroup(448,1278);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,184,675,570,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=a^7,d^2=b^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=a^7*b^-1,d*b*d^-1=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