G = C14.722- 1+4 order 448 = 26·7
27th non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.722- 1+4, C14.392+ 1+4, (C4×D7)⋊2D4, C4⋊D4⋊11D7, D14.3(C2×D4), C4.184(D4×D7), C28⋊2D4⋊19C2, C4⋊C4.180D14, (C2×D4).92D14, C28.228(C2×D4), C22⋊C4.8D14, D14⋊D4⋊19C2, D14⋊2Q8⋊21C2, (C2×C28).39C23, Dic7.46(C2×D4), C14.67(C22×D4), Dic7⋊D4⋊13C2, C28.48D4⋊34C2, (C2×C14).152C24, D14⋊C4.15C22, (C22×C4).223D14, C2.41(D4⋊6D14), C23.16(C22×D7), C22⋊Dic14⋊19C2, (D4×C14).122C22, (C2×D28).220C22, Dic7⋊C4.18C22, C4⋊Dic7.207C22, (C2×Dic7).73C23, C22.173(C23×D7), C23.D7.25C22, (C22×C28).241C22, (C22×C14).187C23, C7⋊3(C22.31C24), (C22×D7).187C23, C2.30(D4.10D14), (C2×Dic14).154C22, (C22×Dic7).109C22, C2.40(C2×D4×D7), (D7×C4⋊C4)⋊21C2, (C2×C4○D28)⋊21C2, (C7×C4⋊D4)⋊14C2, (C2×D4⋊2D7)⋊13C2, (C2×C4×D7).83C22, (C7×C4⋊C4).144C22, (C2×C4).586(C22×D7), (C2×C7⋊D4).28C22, (C7×C22⋊C4).13C22, SmallGroup(448,1061)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.722- 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=e2=a7b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, dbd-1=ebe-1=a7b, dcd-1=ece-1=a7c, ede-1=b2d >
Subgroups: 1420 in 294 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D7, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C2×C4○D4, Dic14, C4×D7, C4×D7, D28, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C22×C14, C22.31C24, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×D28, C4○D28, D4⋊2D7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C22⋊Dic14, D14⋊D4, D7×C4⋊C4, D14⋊2Q8, C28.48D4, C28⋊2D4, C28⋊2D4, Dic7⋊D4, C7×C4⋊D4, C2×C4○D28, C2×D4⋊2D7, C14.722- 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, 2- 1+4, C22×D7, C22.31C24, D4×D7, C23×D7, C2×D4×D7, D4⋊6D14, D4.10D14, C14.722- 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 43 210 223)(2 56 197 222)(3 55 198 221)(4 54 199 220)(5 53 200 219)(6 52 201 218)(7 51 202 217)(8 50 203 216)(9 49 204 215)(10 48 205 214)(11 47 206 213)(12 46 207 212)(13 45 208 211)(14 44 209 224)(15 91 36 175)(16 90 37 174)(17 89 38 173)(18 88 39 172)(19 87 40 171)(20 86 41 170)(21 85 42 169)(22 98 29 182)(23 97 30 181)(24 96 31 180)(25 95 32 179)(26 94 33 178)(27 93 34 177)(28 92 35 176)(57 142 161 195)(58 141 162 194)(59 154 163 193)(60 153 164 192)(61 152 165 191)(62 151 166 190)(63 150 167 189)(64 149 168 188)(65 148 155 187)(66 147 156 186)(67 146 157 185)(68 145 158 184)(69 144 159 183)(70 143 160 196)(71 123 99 131)(72 122 100 130)(73 121 101 129)(74 120 102 128)(75 119 103 127)(76 118 104 140)(77 117 105 139)(78 116 106 138)(79 115 107 137)(80 114 108 136)(81 113 109 135)(82 126 110 134)(83 125 111 133)(84 124 112 132)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 176)(16 177)(17 178)(18 179)(19 180)(20 181)(21 182)(22 169)(23 170)(24 171)(25 172)(26 173)(27 174)(28 175)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(57 153)(58 154)(59 141)(60 142)(61 143)(62 144)(63 145)(64 146)(65 147)(66 148)(67 149)(68 150)(69 151)(70 152)(71 118)(72 119)(73 120)(74 121)(75 122)(76 123)(77 124)(78 125)(79 126)(80 113)(81 114)(82 115)(83 116)(84 117)(99 140)(100 127)(101 128)(102 129)(103 130)(104 131)(105 132)(106 133)(107 134)(108 135)(109 136)(110 137)(111 138)(112 139)(155 186)(156 187)(157 188)(158 189)(159 190)(160 191)(161 192)(162 193)(163 194)(164 195)(165 196)(166 183)(167 184)(168 185)(197 217)(198 218)(199 219)(200 220)(201 221)(202 222)(203 223)(204 224)(205 211)(206 212)(207 213)(208 214)(209 215)(210 216)
(1 172 203 95)(2 171 204 94)(3 170 205 93)(4 169 206 92)(5 182 207 91)(6 181 208 90)(7 180 209 89)(8 179 210 88)(9 178 197 87)(10 177 198 86)(11 176 199 85)(12 175 200 98)(13 174 201 97)(14 173 202 96)(15 212 29 53)(16 211 30 52)(17 224 31 51)(18 223 32 50)(19 222 33 49)(20 221 34 48)(21 220 35 47)(22 219 36 46)(23 218 37 45)(24 217 38 44)(25 216 39 43)(26 215 40 56)(27 214 41 55)(28 213 42 54)(57 103 168 82)(58 102 155 81)(59 101 156 80)(60 100 157 79)(61 99 158 78)(62 112 159 77)(63 111 160 76)(64 110 161 75)(65 109 162 74)(66 108 163 73)(67 107 164 72)(68 106 165 71)(69 105 166 84)(70 104 167 83)(113 148 128 194)(114 147 129 193)(115 146 130 192)(116 145 131 191)(117 144 132 190)(118 143 133 189)(119 142 134 188)(120 141 135 187)(121 154 136 186)(122 153 137 185)(123 152 138 184)(124 151 139 183)(125 150 140 196)(126 149 127 195)
(1 159 203 62)(2 160 204 63)(3 161 205 64)(4 162 206 65)(5 163 207 66)(6 164 208 67)(7 165 209 68)(8 166 210 69)(9 167 197 70)(10 168 198 57)(11 155 199 58)(12 156 200 59)(13 157 201 60)(14 158 202 61)(15 129 29 114)(16 130 30 115)(17 131 31 116)(18 132 32 117)(19 133 33 118)(20 134 34 119)(21 135 35 120)(22 136 36 121)(23 137 37 122)(24 138 38 123)(25 139 39 124)(26 140 40 125)(27 127 41 126)(28 128 42 113)(43 190 216 144)(44 191 217 145)(45 192 218 146)(46 193 219 147)(47 194 220 148)(48 195 221 149)(49 196 222 150)(50 183 223 151)(51 184 224 152)(52 185 211 153)(53 186 212 154)(54 187 213 141)(55 188 214 142)(56 189 215 143)(71 173 106 96)(72 174 107 97)(73 175 108 98)(74 176 109 85)(75 177 110 86)(76 178 111 87)(77 179 112 88)(78 180 99 89)(79 181 100 90)(80 182 101 91)(81 169 102 92)(82 170 103 93)(83 171 104 94)(84 172 105 95)
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), (1,43,210,223)(2,56,197,222)(3,55,198,221)(4,54,199,220)(5,53,200,219)(6,52,201,218)(7,51,202,217)(8,50,203,216)(9,49,204,215)(10,48,205,214)(11,47,206,213)(12,46,207,212)(13,45,208,211)(14,44,209,224)(15,91,36,175)(16,90,37,174)(17,89,38,173)(18,88,39,172)(19,87,40,171)(20,86,41,170)(21,85,42,169)(22,98,29,182)(23,97,30,181)(24,96,31,180)(25,95,32,179)(26,94,33,178)(27,93,34,177)(28,92,35,176)(57,142,161,195)(58,141,162,194)(59,154,163,193)(60,153,164,192)(61,152,165,191)(62,151,166,190)(63,150,167,189)(64,149,168,188)(65,148,155,187)(66,147,156,186)(67,146,157,185)(68,145,158,184)(69,144,159,183)(70,143,160,196)(71,123,99,131)(72,122,100,130)(73,121,101,129)(74,120,102,128)(75,119,103,127)(76,118,104,140)(77,117,105,139)(78,116,106,138)(79,115,107,137)(80,114,108,136)(81,113,109,135)(82,126,110,134)(83,125,111,133)(84,124,112,132), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,176)(16,177)(17,178)(18,179)(19,180)(20,181)(21,182)(22,169)(23,170)(24,171)(25,172)(26,173)(27,174)(28,175)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(57,153)(58,154)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,118)(72,119)(73,120)(74,121)(75,122)(76,123)(77,124)(78,125)(79,126)(80,113)(81,114)(82,115)(83,116)(84,117)(99,140)(100,127)(101,128)(102,129)(103,130)(104,131)(105,132)(106,133)(107,134)(108,135)(109,136)(110,137)(111,138)(112,139)(155,186)(156,187)(157,188)(158,189)(159,190)(160,191)(161,192)(162,193)(163,194)(164,195)(165,196)(166,183)(167,184)(168,185)(197,217)(198,218)(199,219)(200,220)(201,221)(202,222)(203,223)(204,224)(205,211)(206,212)(207,213)(208,214)(209,215)(210,216), (1,172,203,95)(2,171,204,94)(3,170,205,93)(4,169,206,92)(5,182,207,91)(6,181,208,90)(7,180,209,89)(8,179,210,88)(9,178,197,87)(10,177,198,86)(11,176,199,85)(12,175,200,98)(13,174,201,97)(14,173,202,96)(15,212,29,53)(16,211,30,52)(17,224,31,51)(18,223,32,50)(19,222,33,49)(20,221,34,48)(21,220,35,47)(22,219,36,46)(23,218,37,45)(24,217,38,44)(25,216,39,43)(26,215,40,56)(27,214,41,55)(28,213,42,54)(57,103,168,82)(58,102,155,81)(59,101,156,80)(60,100,157,79)(61,99,158,78)(62,112,159,77)(63,111,160,76)(64,110,161,75)(65,109,162,74)(66,108,163,73)(67,107,164,72)(68,106,165,71)(69,105,166,84)(70,104,167,83)(113,148,128,194)(114,147,129,193)(115,146,130,192)(116,145,131,191)(117,144,132,190)(118,143,133,189)(119,142,134,188)(120,141,135,187)(121,154,136,186)(122,153,137,185)(123,152,138,184)(124,151,139,183)(125,150,140,196)(126,149,127,195), (1,159,203,62)(2,160,204,63)(3,161,205,64)(4,162,206,65)(5,163,207,66)(6,164,208,67)(7,165,209,68)(8,166,210,69)(9,167,197,70)(10,168,198,57)(11,155,199,58)(12,156,200,59)(13,157,201,60)(14,158,202,61)(15,129,29,114)(16,130,30,115)(17,131,31,116)(18,132,32,117)(19,133,33,118)(20,134,34,119)(21,135,35,120)(22,136,36,121)(23,137,37,122)(24,138,38,123)(25,139,39,124)(26,140,40,125)(27,127,41,126)(28,128,42,113)(43,190,216,144)(44,191,217,145)(45,192,218,146)(46,193,219,147)(47,194,220,148)(48,195,221,149)(49,196,222,150)(50,183,223,151)(51,184,224,152)(52,185,211,153)(53,186,212,154)(54,187,213,141)(55,188,214,142)(56,189,215,143)(71,173,106,96)(72,174,107,97)(73,175,108,98)(74,176,109,85)(75,177,110,86)(76,178,111,87)(77,179,112,88)(78,180,99,89)(79,181,100,90)(80,182,101,91)(81,169,102,92)(82,170,103,93)(83,171,104,94)(84,172,105,95)>;
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), (1,43,210,223)(2,56,197,222)(3,55,198,221)(4,54,199,220)(5,53,200,219)(6,52,201,218)(7,51,202,217)(8,50,203,216)(9,49,204,215)(10,48,205,214)(11,47,206,213)(12,46,207,212)(13,45,208,211)(14,44,209,224)(15,91,36,175)(16,90,37,174)(17,89,38,173)(18,88,39,172)(19,87,40,171)(20,86,41,170)(21,85,42,169)(22,98,29,182)(23,97,30,181)(24,96,31,180)(25,95,32,179)(26,94,33,178)(27,93,34,177)(28,92,35,176)(57,142,161,195)(58,141,162,194)(59,154,163,193)(60,153,164,192)(61,152,165,191)(62,151,166,190)(63,150,167,189)(64,149,168,188)(65,148,155,187)(66,147,156,186)(67,146,157,185)(68,145,158,184)(69,144,159,183)(70,143,160,196)(71,123,99,131)(72,122,100,130)(73,121,101,129)(74,120,102,128)(75,119,103,127)(76,118,104,140)(77,117,105,139)(78,116,106,138)(79,115,107,137)(80,114,108,136)(81,113,109,135)(82,126,110,134)(83,125,111,133)(84,124,112,132), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,176)(16,177)(17,178)(18,179)(19,180)(20,181)(21,182)(22,169)(23,170)(24,171)(25,172)(26,173)(27,174)(28,175)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(57,153)(58,154)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,118)(72,119)(73,120)(74,121)(75,122)(76,123)(77,124)(78,125)(79,126)(80,113)(81,114)(82,115)(83,116)(84,117)(99,140)(100,127)(101,128)(102,129)(103,130)(104,131)(105,132)(106,133)(107,134)(108,135)(109,136)(110,137)(111,138)(112,139)(155,186)(156,187)(157,188)(158,189)(159,190)(160,191)(161,192)(162,193)(163,194)(164,195)(165,196)(166,183)(167,184)(168,185)(197,217)(198,218)(199,219)(200,220)(201,221)(202,222)(203,223)(204,224)(205,211)(206,212)(207,213)(208,214)(209,215)(210,216), (1,172,203,95)(2,171,204,94)(3,170,205,93)(4,169,206,92)(5,182,207,91)(6,181,208,90)(7,180,209,89)(8,179,210,88)(9,178,197,87)(10,177,198,86)(11,176,199,85)(12,175,200,98)(13,174,201,97)(14,173,202,96)(15,212,29,53)(16,211,30,52)(17,224,31,51)(18,223,32,50)(19,222,33,49)(20,221,34,48)(21,220,35,47)(22,219,36,46)(23,218,37,45)(24,217,38,44)(25,216,39,43)(26,215,40,56)(27,214,41,55)(28,213,42,54)(57,103,168,82)(58,102,155,81)(59,101,156,80)(60,100,157,79)(61,99,158,78)(62,112,159,77)(63,111,160,76)(64,110,161,75)(65,109,162,74)(66,108,163,73)(67,107,164,72)(68,106,165,71)(69,105,166,84)(70,104,167,83)(113,148,128,194)(114,147,129,193)(115,146,130,192)(116,145,131,191)(117,144,132,190)(118,143,133,189)(119,142,134,188)(120,141,135,187)(121,154,136,186)(122,153,137,185)(123,152,138,184)(124,151,139,183)(125,150,140,196)(126,149,127,195), (1,159,203,62)(2,160,204,63)(3,161,205,64)(4,162,206,65)(5,163,207,66)(6,164,208,67)(7,165,209,68)(8,166,210,69)(9,167,197,70)(10,168,198,57)(11,155,199,58)(12,156,200,59)(13,157,201,60)(14,158,202,61)(15,129,29,114)(16,130,30,115)(17,131,31,116)(18,132,32,117)(19,133,33,118)(20,134,34,119)(21,135,35,120)(22,136,36,121)(23,137,37,122)(24,138,38,123)(25,139,39,124)(26,140,40,125)(27,127,41,126)(28,128,42,113)(43,190,216,144)(44,191,217,145)(45,192,218,146)(46,193,219,147)(47,194,220,148)(48,195,221,149)(49,196,222,150)(50,183,223,151)(51,184,224,152)(52,185,211,153)(53,186,212,154)(54,187,213,141)(55,188,214,142)(56,189,215,143)(71,173,106,96)(72,174,107,97)(73,175,108,98)(74,176,109,85)(75,177,110,86)(76,178,111,87)(77,179,112,88)(78,180,99,89)(79,181,100,90)(80,182,101,91)(81,169,102,92)(82,170,103,93)(83,171,104,94)(84,172,105,95) );
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)], [(1,43,210,223),(2,56,197,222),(3,55,198,221),(4,54,199,220),(5,53,200,219),(6,52,201,218),(7,51,202,217),(8,50,203,216),(9,49,204,215),(10,48,205,214),(11,47,206,213),(12,46,207,212),(13,45,208,211),(14,44,209,224),(15,91,36,175),(16,90,37,174),(17,89,38,173),(18,88,39,172),(19,87,40,171),(20,86,41,170),(21,85,42,169),(22,98,29,182),(23,97,30,181),(24,96,31,180),(25,95,32,179),(26,94,33,178),(27,93,34,177),(28,92,35,176),(57,142,161,195),(58,141,162,194),(59,154,163,193),(60,153,164,192),(61,152,165,191),(62,151,166,190),(63,150,167,189),(64,149,168,188),(65,148,155,187),(66,147,156,186),(67,146,157,185),(68,145,158,184),(69,144,159,183),(70,143,160,196),(71,123,99,131),(72,122,100,130),(73,121,101,129),(74,120,102,128),(75,119,103,127),(76,118,104,140),(77,117,105,139),(78,116,106,138),(79,115,107,137),(80,114,108,136),(81,113,109,135),(82,126,110,134),(83,125,111,133),(84,124,112,132)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,176),(16,177),(17,178),(18,179),(19,180),(20,181),(21,182),(22,169),(23,170),(24,171),(25,172),(26,173),(27,174),(28,175),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(57,153),(58,154),(59,141),(60,142),(61,143),(62,144),(63,145),(64,146),(65,147),(66,148),(67,149),(68,150),(69,151),(70,152),(71,118),(72,119),(73,120),(74,121),(75,122),(76,123),(77,124),(78,125),(79,126),(80,113),(81,114),(82,115),(83,116),(84,117),(99,140),(100,127),(101,128),(102,129),(103,130),(104,131),(105,132),(106,133),(107,134),(108,135),(109,136),(110,137),(111,138),(112,139),(155,186),(156,187),(157,188),(158,189),(159,190),(160,191),(161,192),(162,193),(163,194),(164,195),(165,196),(166,183),(167,184),(168,185),(197,217),(198,218),(199,219),(200,220),(201,221),(202,222),(203,223),(204,224),(205,211),(206,212),(207,213),(208,214),(209,215),(210,216)], [(1,172,203,95),(2,171,204,94),(3,170,205,93),(4,169,206,92),(5,182,207,91),(6,181,208,90),(7,180,209,89),(8,179,210,88),(9,178,197,87),(10,177,198,86),(11,176,199,85),(12,175,200,98),(13,174,201,97),(14,173,202,96),(15,212,29,53),(16,211,30,52),(17,224,31,51),(18,223,32,50),(19,222,33,49),(20,221,34,48),(21,220,35,47),(22,219,36,46),(23,218,37,45),(24,217,38,44),(25,216,39,43),(26,215,40,56),(27,214,41,55),(28,213,42,54),(57,103,168,82),(58,102,155,81),(59,101,156,80),(60,100,157,79),(61,99,158,78),(62,112,159,77),(63,111,160,76),(64,110,161,75),(65,109,162,74),(66,108,163,73),(67,107,164,72),(68,106,165,71),(69,105,166,84),(70,104,167,83),(113,148,128,194),(114,147,129,193),(115,146,130,192),(116,145,131,191),(117,144,132,190),(118,143,133,189),(119,142,134,188),(120,141,135,187),(121,154,136,186),(122,153,137,185),(123,152,138,184),(124,151,139,183),(125,150,140,196),(126,149,127,195)], [(1,159,203,62),(2,160,204,63),(3,161,205,64),(4,162,206,65),(5,163,207,66),(6,164,208,67),(7,165,209,68),(8,166,210,69),(9,167,197,70),(10,168,198,57),(11,155,199,58),(12,156,200,59),(13,157,201,60),(14,158,202,61),(15,129,29,114),(16,130,30,115),(17,131,31,116),(18,132,32,117),(19,133,33,118),(20,134,34,119),(21,135,35,120),(22,136,36,121),(23,137,37,122),(24,138,38,123),(25,139,39,124),(26,140,40,125),(27,127,41,126),(28,128,42,113),(43,190,216,144),(44,191,217,145),(45,192,218,146),(46,193,219,147),(47,194,220,148),(48,195,221,149),(49,196,222,150),(50,183,223,151),(51,184,224,152),(52,185,211,153),(53,186,212,154),(54,187,213,141),(55,188,214,142),(56,189,215,143),(71,173,106,96),(72,174,107,97),(73,175,108,98),(74,176,109,85),(75,177,110,86),(76,178,111,87),(77,179,112,88),(78,180,99,89),(79,181,100,90),(80,182,101,91),(81,169,102,92),(82,170,103,93),(83,171,104,94),(84,172,105,95)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 14 | 14 | 28 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | D14 | 2+ 1+4 | 2- 1+4 | D4×D7 | D4⋊6D14 | D4.10D14 |
| kernel | C14.722- 1+4 | C22⋊Dic14 | D14⋊D4 | D7×C4⋊C4 | D14⋊2Q8 | C28.48D4 | C28⋊2D4 | Dic7⋊D4 | C7×C4⋊D4 | C2×C4○D28 | C2×D4⋊2D7 | C4×D7 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 1 | 4 | 3 | 6 | 3 | 3 | 9 | 1 | 1 | 6 | 6 | 6 |
Matrix representation of C14.722- 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 21 | 0 | 0 |
| 0 | 0 | 17 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 21 |
| 0 | 0 | 0 | 0 | 17 | 28 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 26 | 0 | 0 |
| 0 | 0 | 19 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 14 | 20 | 0 | 0 | 0 | 0 |
| 9 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 3 | 17 |
| 0 | 0 | 7 | 0 | 11 | 26 |
| 0 | 0 | 3 | 17 | 0 | 5 |
| 0 | 0 | 11 | 26 | 7 | 0 |
| 14 | 20 | 0 | 0 | 0 | 0 |
| 9 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 18 |
| 0 | 0 | 0 | 0 | 27 | 8 |
| 0 | 0 | 21 | 18 | 0 | 0 |
| 0 | 0 | 27 | 8 | 0 | 0 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,19,17,0,0,0,0,21,28,0,0,0,0,0,0,19,17,0,0,0,0,21,28],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,19,0,0,0,0,26,0,0,0,0,10,0,0,0,0,3,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[14,9,0,0,0,0,20,15,0,0,0,0,0,0,0,7,3,11,0,0,5,0,17,26,0,0,3,11,0,7,0,0,17,26,5,0],[14,9,0,0,0,0,20,15,0,0,0,0,0,0,0,0,21,27,0,0,0,0,18,8,0,0,21,27,0,0,0,0,18,8,0,0] >;
C14.722- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{72}2_-^{1+4} % in TeX
G:=Group("C14.72ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1061);
// by ID
G=gap.SmallGroup(448,1061);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,1123,570,185,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^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,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=b^2*d>;
// generators/relations