G = C14.52- 1+4 order 448 = 26·7
5th non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.52- 1+4, C4⋊C4.306D14, Dic7.Q8⋊2C2, C28.3Q8⋊9C2, D14⋊2Q8⋊10C2, C4.93(C4○D28), (C2×C14).57C24, Dic7⋊3Q8⋊10C2, C28.195(C4○D4), C28.48D4⋊28C2, (C2×C28).618C23, (C22×C4).182D14, C22.91(C23×D7), D14⋊C4.141C22, C23.21D14⋊4C2, C4⋊Dic7.193C22, (C2×Dic7).18C23, (C4×Dic7).65C22, (C22×D7).15C23, C23.228(C22×D7), C22.24(D4⋊2D7), Dic7⋊C4.150C22, (C22×C14).406C23, (C22×C28).219C22, C2.8(Q8.10D14), C7⋊1(C22.46C24), C23.23D14.1C2, C23.D7.142C22, (C2×Dic14).142C22, (C2×C4⋊C4)⋊22D7, (C14×C4⋊C4)⋊19C2, C4⋊C4⋊7D7⋊9C2, C4⋊C4⋊D7⋊1C2, (C4×C7⋊D4).4C2, C2.26(C2×C4○D28), C14.24(C2×C4○D4), (C2×C4×D7).56C22, C2.17(C2×D4⋊2D7), (C7×C4⋊C4).298C22, (C2×C4).145(C22×D7), (C2×C7⋊D4).94C22, (C2×C14).173(C4○D4), SmallGroup(448,966)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.52- 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=a7b, be=eb, cd=dc, ce=ec, ede-1=a7b2d >
Subgroups: 804 in 214 conjugacy classes, 99 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C22×D7, C22×C14, C22.46C24, C4×Dic7, C4×Dic7, Dic7⋊C4, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, D14⋊C4, D14⋊C4, C23.D7, C23.D7, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×C7⋊D4, C22×C28, C22×C28, Dic7⋊3Q8, Dic7.Q8, C28.3Q8, C4⋊C4⋊7D7, D14⋊2Q8, C4⋊C4⋊D7, C28.48D4, C23.21D14, C4×C7⋊D4, C23.23D14, C14×C4⋊C4, C14.52- 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2- 1+4, C22×D7, C22.46C24, C4○D28, D4⋊2D7, C23×D7, C2×C4○D28, C2×D4⋊2D7, Q8.10D14, C14.52- 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)
(1 140 183 71)(2 139 184 84)(3 138 185 83)(4 137 186 82)(5 136 187 81)(6 135 188 80)(7 134 189 79)(8 133 190 78)(9 132 191 77)(10 131 192 76)(11 130 193 75)(12 129 194 74)(13 128 195 73)(14 127 196 72)(15 103 92 38)(16 102 93 37)(17 101 94 36)(18 100 95 35)(19 99 96 34)(20 112 97 33)(21 111 98 32)(22 110 85 31)(23 109 86 30)(24 108 87 29)(25 107 88 42)(26 106 89 41)(27 105 90 40)(28 104 91 39)(43 160 152 60)(44 159 153 59)(45 158 154 58)(46 157 141 57)(47 156 142 70)(48 155 143 69)(49 168 144 68)(50 167 145 67)(51 166 146 66)(52 165 147 65)(53 164 148 64)(54 163 149 63)(55 162 150 62)(56 161 151 61)(113 219 209 179)(114 218 210 178)(115 217 197 177)(116 216 198 176)(117 215 199 175)(118 214 200 174)(119 213 201 173)(120 212 202 172)(121 211 203 171)(122 224 204 170)(123 223 205 169)(124 222 206 182)(125 221 207 181)(126 220 208 180)
(1 190)(2 191)(3 192)(4 193)(5 194)(6 195)(7 196)(8 183)(9 184)(10 185)(11 186)(12 187)(13 188)(14 189)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 99)(42 100)(43 145)(44 146)(45 147)(46 148)(47 149)(48 150)(49 151)(50 152)(51 153)(52 154)(53 141)(54 142)(55 143)(56 144)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(133 140)(155 162)(156 163)(157 164)(158 165)(159 166)(160 167)(161 168)(169 216)(170 217)(171 218)(172 219)(173 220)(174 221)(175 222)(176 223)(177 224)(178 211)(179 212)(180 213)(181 214)(182 215)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)
(1 171 183 211)(2 172 184 212)(3 173 185 213)(4 174 186 214)(5 175 187 215)(6 176 188 216)(7 177 189 217)(8 178 190 218)(9 179 191 219)(10 180 192 220)(11 181 193 221)(12 182 194 222)(13 169 195 223)(14 170 196 224)(15 164 92 64)(16 165 93 65)(17 166 94 66)(18 167 95 67)(19 168 96 68)(20 155 97 69)(21 156 98 70)(22 157 85 57)(23 158 86 58)(24 159 87 59)(25 160 88 60)(26 161 89 61)(27 162 90 62)(28 163 91 63)(29 51 108 146)(30 52 109 147)(31 53 110 148)(32 54 111 149)(33 55 112 150)(34 56 99 151)(35 43 100 152)(36 44 101 153)(37 45 102 154)(38 46 103 141)(39 47 104 142)(40 48 105 143)(41 49 106 144)(42 50 107 145)(71 210 140 114)(72 197 127 115)(73 198 128 116)(74 199 129 117)(75 200 130 118)(76 201 131 119)(77 202 132 120)(78 203 133 121)(79 204 134 122)(80 205 135 123)(81 206 136 124)(82 207 137 125)(83 208 138 126)(84 209 139 113)
(1 111 190 39)(2 112 191 40)(3 99 192 41)(4 100 193 42)(5 101 194 29)(6 102 195 30)(7 103 196 31)(8 104 183 32)(9 105 184 33)(10 106 185 34)(11 107 186 35)(12 108 187 36)(13 109 188 37)(14 110 189 38)(15 127 85 79)(16 128 86 80)(17 129 87 81)(18 130 88 82)(19 131 89 83)(20 132 90 84)(21 133 91 71)(22 134 92 72)(23 135 93 73)(24 136 94 74)(25 137 95 75)(26 138 96 76)(27 139 97 77)(28 140 98 78)(43 214 145 181)(44 215 146 182)(45 216 147 169)(46 217 148 170)(47 218 149 171)(48 219 150 172)(49 220 151 173)(50 221 152 174)(51 222 153 175)(52 223 154 176)(53 224 141 177)(54 211 142 178)(55 212 143 179)(56 213 144 180)(57 115 164 204)(58 116 165 205)(59 117 166 206)(60 118 167 207)(61 119 168 208)(62 120 155 209)(63 121 156 210)(64 122 157 197)(65 123 158 198)(66 124 159 199)(67 125 160 200)(68 126 161 201)(69 113 162 202)(70 114 163 203)
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), (1,140,183,71)(2,139,184,84)(3,138,185,83)(4,137,186,82)(5,136,187,81)(6,135,188,80)(7,134,189,79)(8,133,190,78)(9,132,191,77)(10,131,192,76)(11,130,193,75)(12,129,194,74)(13,128,195,73)(14,127,196,72)(15,103,92,38)(16,102,93,37)(17,101,94,36)(18,100,95,35)(19,99,96,34)(20,112,97,33)(21,111,98,32)(22,110,85,31)(23,109,86,30)(24,108,87,29)(25,107,88,42)(26,106,89,41)(27,105,90,40)(28,104,91,39)(43,160,152,60)(44,159,153,59)(45,158,154,58)(46,157,141,57)(47,156,142,70)(48,155,143,69)(49,168,144,68)(50,167,145,67)(51,166,146,66)(52,165,147,65)(53,164,148,64)(54,163,149,63)(55,162,150,62)(56,161,151,61)(113,219,209,179)(114,218,210,178)(115,217,197,177)(116,216,198,176)(117,215,199,175)(118,214,200,174)(119,213,201,173)(120,212,202,172)(121,211,203,171)(122,224,204,170)(123,223,205,169)(124,222,206,182)(125,221,207,181)(126,220,208,180), (1,190)(2,191)(3,192)(4,193)(5,194)(6,195)(7,196)(8,183)(9,184)(10,185)(11,186)(12,187)(13,188)(14,189)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,99)(42,100)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,141)(54,142)(55,143)(56,144)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(169,216)(170,217)(171,218)(172,219)(173,220)(174,221)(175,222)(176,223)(177,224)(178,211)(179,212)(180,213)(181,214)(182,215)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210), (1,171,183,211)(2,172,184,212)(3,173,185,213)(4,174,186,214)(5,175,187,215)(6,176,188,216)(7,177,189,217)(8,178,190,218)(9,179,191,219)(10,180,192,220)(11,181,193,221)(12,182,194,222)(13,169,195,223)(14,170,196,224)(15,164,92,64)(16,165,93,65)(17,166,94,66)(18,167,95,67)(19,168,96,68)(20,155,97,69)(21,156,98,70)(22,157,85,57)(23,158,86,58)(24,159,87,59)(25,160,88,60)(26,161,89,61)(27,162,90,62)(28,163,91,63)(29,51,108,146)(30,52,109,147)(31,53,110,148)(32,54,111,149)(33,55,112,150)(34,56,99,151)(35,43,100,152)(36,44,101,153)(37,45,102,154)(38,46,103,141)(39,47,104,142)(40,48,105,143)(41,49,106,144)(42,50,107,145)(71,210,140,114)(72,197,127,115)(73,198,128,116)(74,199,129,117)(75,200,130,118)(76,201,131,119)(77,202,132,120)(78,203,133,121)(79,204,134,122)(80,205,135,123)(81,206,136,124)(82,207,137,125)(83,208,138,126)(84,209,139,113), (1,111,190,39)(2,112,191,40)(3,99,192,41)(4,100,193,42)(5,101,194,29)(6,102,195,30)(7,103,196,31)(8,104,183,32)(9,105,184,33)(10,106,185,34)(11,107,186,35)(12,108,187,36)(13,109,188,37)(14,110,189,38)(15,127,85,79)(16,128,86,80)(17,129,87,81)(18,130,88,82)(19,131,89,83)(20,132,90,84)(21,133,91,71)(22,134,92,72)(23,135,93,73)(24,136,94,74)(25,137,95,75)(26,138,96,76)(27,139,97,77)(28,140,98,78)(43,214,145,181)(44,215,146,182)(45,216,147,169)(46,217,148,170)(47,218,149,171)(48,219,150,172)(49,220,151,173)(50,221,152,174)(51,222,153,175)(52,223,154,176)(53,224,141,177)(54,211,142,178)(55,212,143,179)(56,213,144,180)(57,115,164,204)(58,116,165,205)(59,117,166,206)(60,118,167,207)(61,119,168,208)(62,120,155,209)(63,121,156,210)(64,122,157,197)(65,123,158,198)(66,124,159,199)(67,125,160,200)(68,126,161,201)(69,113,162,202)(70,114,163,203)>;
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), (1,140,183,71)(2,139,184,84)(3,138,185,83)(4,137,186,82)(5,136,187,81)(6,135,188,80)(7,134,189,79)(8,133,190,78)(9,132,191,77)(10,131,192,76)(11,130,193,75)(12,129,194,74)(13,128,195,73)(14,127,196,72)(15,103,92,38)(16,102,93,37)(17,101,94,36)(18,100,95,35)(19,99,96,34)(20,112,97,33)(21,111,98,32)(22,110,85,31)(23,109,86,30)(24,108,87,29)(25,107,88,42)(26,106,89,41)(27,105,90,40)(28,104,91,39)(43,160,152,60)(44,159,153,59)(45,158,154,58)(46,157,141,57)(47,156,142,70)(48,155,143,69)(49,168,144,68)(50,167,145,67)(51,166,146,66)(52,165,147,65)(53,164,148,64)(54,163,149,63)(55,162,150,62)(56,161,151,61)(113,219,209,179)(114,218,210,178)(115,217,197,177)(116,216,198,176)(117,215,199,175)(118,214,200,174)(119,213,201,173)(120,212,202,172)(121,211,203,171)(122,224,204,170)(123,223,205,169)(124,222,206,182)(125,221,207,181)(126,220,208,180), (1,190)(2,191)(3,192)(4,193)(5,194)(6,195)(7,196)(8,183)(9,184)(10,185)(11,186)(12,187)(13,188)(14,189)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,109)(38,110)(39,111)(40,112)(41,99)(42,100)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,141)(54,142)(55,143)(56,144)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(169,216)(170,217)(171,218)(172,219)(173,220)(174,221)(175,222)(176,223)(177,224)(178,211)(179,212)(180,213)(181,214)(182,215)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210), (1,171,183,211)(2,172,184,212)(3,173,185,213)(4,174,186,214)(5,175,187,215)(6,176,188,216)(7,177,189,217)(8,178,190,218)(9,179,191,219)(10,180,192,220)(11,181,193,221)(12,182,194,222)(13,169,195,223)(14,170,196,224)(15,164,92,64)(16,165,93,65)(17,166,94,66)(18,167,95,67)(19,168,96,68)(20,155,97,69)(21,156,98,70)(22,157,85,57)(23,158,86,58)(24,159,87,59)(25,160,88,60)(26,161,89,61)(27,162,90,62)(28,163,91,63)(29,51,108,146)(30,52,109,147)(31,53,110,148)(32,54,111,149)(33,55,112,150)(34,56,99,151)(35,43,100,152)(36,44,101,153)(37,45,102,154)(38,46,103,141)(39,47,104,142)(40,48,105,143)(41,49,106,144)(42,50,107,145)(71,210,140,114)(72,197,127,115)(73,198,128,116)(74,199,129,117)(75,200,130,118)(76,201,131,119)(77,202,132,120)(78,203,133,121)(79,204,134,122)(80,205,135,123)(81,206,136,124)(82,207,137,125)(83,208,138,126)(84,209,139,113), (1,111,190,39)(2,112,191,40)(3,99,192,41)(4,100,193,42)(5,101,194,29)(6,102,195,30)(7,103,196,31)(8,104,183,32)(9,105,184,33)(10,106,185,34)(11,107,186,35)(12,108,187,36)(13,109,188,37)(14,110,189,38)(15,127,85,79)(16,128,86,80)(17,129,87,81)(18,130,88,82)(19,131,89,83)(20,132,90,84)(21,133,91,71)(22,134,92,72)(23,135,93,73)(24,136,94,74)(25,137,95,75)(26,138,96,76)(27,139,97,77)(28,140,98,78)(43,214,145,181)(44,215,146,182)(45,216,147,169)(46,217,148,170)(47,218,149,171)(48,219,150,172)(49,220,151,173)(50,221,152,174)(51,222,153,175)(52,223,154,176)(53,224,141,177)(54,211,142,178)(55,212,143,179)(56,213,144,180)(57,115,164,204)(58,116,165,205)(59,117,166,206)(60,118,167,207)(61,119,168,208)(62,120,155,209)(63,121,156,210)(64,122,157,197)(65,123,158,198)(66,124,159,199)(67,125,160,200)(68,126,161,201)(69,113,162,202)(70,114,163,203) );
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)], [(1,140,183,71),(2,139,184,84),(3,138,185,83),(4,137,186,82),(5,136,187,81),(6,135,188,80),(7,134,189,79),(8,133,190,78),(9,132,191,77),(10,131,192,76),(11,130,193,75),(12,129,194,74),(13,128,195,73),(14,127,196,72),(15,103,92,38),(16,102,93,37),(17,101,94,36),(18,100,95,35),(19,99,96,34),(20,112,97,33),(21,111,98,32),(22,110,85,31),(23,109,86,30),(24,108,87,29),(25,107,88,42),(26,106,89,41),(27,105,90,40),(28,104,91,39),(43,160,152,60),(44,159,153,59),(45,158,154,58),(46,157,141,57),(47,156,142,70),(48,155,143,69),(49,168,144,68),(50,167,145,67),(51,166,146,66),(52,165,147,65),(53,164,148,64),(54,163,149,63),(55,162,150,62),(56,161,151,61),(113,219,209,179),(114,218,210,178),(115,217,197,177),(116,216,198,176),(117,215,199,175),(118,214,200,174),(119,213,201,173),(120,212,202,172),(121,211,203,171),(122,224,204,170),(123,223,205,169),(124,222,206,182),(125,221,207,181),(126,220,208,180)], [(1,190),(2,191),(3,192),(4,193),(5,194),(6,195),(7,196),(8,183),(9,184),(10,185),(11,186),(12,187),(13,188),(14,189),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,109),(38,110),(39,111),(40,112),(41,99),(42,100),(43,145),(44,146),(45,147),(46,148),(47,149),(48,150),(49,151),(50,152),(51,153),(52,154),(53,141),(54,142),(55,143),(56,144),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(113,120),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(127,134),(128,135),(129,136),(130,137),(131,138),(132,139),(133,140),(155,162),(156,163),(157,164),(158,165),(159,166),(160,167),(161,168),(169,216),(170,217),(171,218),(172,219),(173,220),(174,221),(175,222),(176,223),(177,224),(178,211),(179,212),(180,213),(181,214),(182,215),(197,204),(198,205),(199,206),(200,207),(201,208),(202,209),(203,210)], [(1,171,183,211),(2,172,184,212),(3,173,185,213),(4,174,186,214),(5,175,187,215),(6,176,188,216),(7,177,189,217),(8,178,190,218),(9,179,191,219),(10,180,192,220),(11,181,193,221),(12,182,194,222),(13,169,195,223),(14,170,196,224),(15,164,92,64),(16,165,93,65),(17,166,94,66),(18,167,95,67),(19,168,96,68),(20,155,97,69),(21,156,98,70),(22,157,85,57),(23,158,86,58),(24,159,87,59),(25,160,88,60),(26,161,89,61),(27,162,90,62),(28,163,91,63),(29,51,108,146),(30,52,109,147),(31,53,110,148),(32,54,111,149),(33,55,112,150),(34,56,99,151),(35,43,100,152),(36,44,101,153),(37,45,102,154),(38,46,103,141),(39,47,104,142),(40,48,105,143),(41,49,106,144),(42,50,107,145),(71,210,140,114),(72,197,127,115),(73,198,128,116),(74,199,129,117),(75,200,130,118),(76,201,131,119),(77,202,132,120),(78,203,133,121),(79,204,134,122),(80,205,135,123),(81,206,136,124),(82,207,137,125),(83,208,138,126),(84,209,139,113)], [(1,111,190,39),(2,112,191,40),(3,99,192,41),(4,100,193,42),(5,101,194,29),(6,102,195,30),(7,103,196,31),(8,104,183,32),(9,105,184,33),(10,106,185,34),(11,107,186,35),(12,108,187,36),(13,109,188,37),(14,110,189,38),(15,127,85,79),(16,128,86,80),(17,129,87,81),(18,130,88,82),(19,131,89,83),(20,132,90,84),(21,133,91,71),(22,134,92,72),(23,135,93,73),(24,136,94,74),(25,137,95,75),(26,138,96,76),(27,139,97,77),(28,140,98,78),(43,214,145,181),(44,215,146,182),(45,216,147,169),(46,217,148,170),(47,218,149,171),(48,219,150,172),(49,220,151,173),(50,221,152,174),(51,222,153,175),(52,223,154,176),(53,224,141,177),(54,211,142,178),(55,212,143,179),(56,213,144,180),(57,115,164,204),(58,116,165,205),(59,117,166,206),(60,118,167,207),(61,119,168,208),(62,120,155,209),(63,121,156,210),(64,122,157,197),(65,123,158,198),(66,124,159,199),(67,125,160,200),(68,126,161,201),(69,113,162,202),(70,114,163,203)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | ··· | 4R | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 2 | ··· | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | D14 | D14 | C4○D28 | 2- 1+4 | D4⋊2D7 | Q8.10D14 |
| kernel | C14.52- 1+4 | Dic7⋊3Q8 | Dic7.Q8 | C28.3Q8 | C4⋊C4⋊7D7 | D14⋊2Q8 | C4⋊C4⋊D7 | C28.48D4 | C23.21D14 | C4×C7⋊D4 | C23.23D14 | C14×C4⋊C4 | C2×C4⋊C4 | C28 | C2×C14 | C4⋊C4 | C22×C4 | C4 | C14 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 3 | 4 | 4 | 12 | 9 | 24 | 1 | 6 | 6 |
Matrix representation of C14.52- 1+4 ►in GL6(𝔽29)
| 25 | 0 | 0 | 0 | 0 | 0 |
| 2 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 12 | 8 | 0 | 0 | 0 | 0 |
| 22 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 9 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 8 |
| 0 | 0 | 0 | 0 | 21 | 23 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 23 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 15 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 1 | 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 | 15 | 20 |
| 0 | 0 | 0 | 0 | 9 | 14 |
G:=sub<GL(6,GF(29))| [25,2,0,0,0,0,0,7,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[12,22,0,0,0,0,8,17,0,0,0,0,0,0,4,9,0,0,0,0,11,25,0,0,0,0,0,0,6,21,0,0,0,0,8,23],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,23,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,15,0,0,0,0,0,17,0,0,0,0,0,0,0,28,0,0,0,0,28,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,9,0,0,0,0,20,14] >;
C14.52- 1+4 in GAP, Magma, Sage, TeX
C_{14}._52_-^{1+4} % in TeX
G:=Group("C14.5ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,966);
// by ID
G=gap.SmallGroup(448,966);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,387,100,675,18822]);
// 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=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^7*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations