G = C14.72+ 1+4 order 448 = 26·7
7th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.72+ 1+4, C14.12- 1+4, C28⋊Q8⋊8C2, (C2×C28)⋊3Q8, (C2×C4)⋊3Dic14, C28.67(C2×Q8), C4⋊C4.261D14, C28.3Q8⋊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, C22.11(C2×Dic14), C2.11(C22×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 C14.72+ 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, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C42.C2, C4⋊Q8, Dic14, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.41C23, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C23.D7, C7×C4⋊C4, C2×Dic14, C22×C28, C22×C28, C28⋊Q8, C28.3Q8, C28.48D4, C23.21D14, C14×C4⋊C4, C14.72+ 1+4
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C24, D14, C22×Q8, 2+ 1+4, 2- 1+4, Dic14, C22×D7, C23.41C23, C2×Dic14, C23×D7, C22×Dic14, D4⋊6D14, Q8.10D14, C14.72+ 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 64 79 142)(2 63 80 141)(3 62 81 154)(4 61 82 153)(5 60 83 152)(6 59 84 151)(7 58 71 150)(8 57 72 149)(9 70 73 148)(10 69 74 147)(11 68 75 146)(12 67 76 145)(13 66 77 144)(14 65 78 143)(15 170 211 159)(16 169 212 158)(17 182 213 157)(18 181 214 156)(19 180 215 155)(20 179 216 168)(21 178 217 167)(22 177 218 166)(23 176 219 165)(24 175 220 164)(25 174 221 163)(26 173 222 162)(27 172 223 161)(28 171 224 160)(29 194 110 96)(30 193 111 95)(31 192 112 94)(32 191 99 93)(33 190 100 92)(34 189 101 91)(35 188 102 90)(36 187 103 89)(37 186 104 88)(38 185 105 87)(39 184 106 86)(40 183 107 85)(41 196 108 98)(42 195 109 97)(43 209 135 121)(44 208 136 120)(45 207 137 119)(46 206 138 118)(47 205 139 117)(48 204 140 116)(49 203 127 115)(50 202 128 114)(51 201 129 113)(52 200 130 126)(53 199 131 125)(54 198 132 124)(55 197 133 123)(56 210 134 122)
(1 149 8 142)(2 148 9 141)(3 147 10 154)(4 146 11 153)(5 145 12 152)(6 144 13 151)(7 143 14 150)(15 170 22 177)(16 169 23 176)(17 182 24 175)(18 181 25 174)(19 180 26 173)(20 179 27 172)(21 178 28 171)(29 89 36 96)(30 88 37 95)(31 87 38 94)(32 86 39 93)(33 85 40 92)(34 98 41 91)(35 97 42 90)(43 209 50 202)(44 208 51 201)(45 207 52 200)(46 206 53 199)(47 205 54 198)(48 204 55 197)(49 203 56 210)(57 72 64 79)(58 71 65 78)(59 84 66 77)(60 83 67 76)(61 82 68 75)(62 81 69 74)(63 80 70 73)(99 184 106 191)(100 183 107 190)(101 196 108 189)(102 195 109 188)(103 194 110 187)(104 193 111 186)(105 192 112 185)(113 136 120 129)(114 135 121 128)(115 134 122 127)(116 133 123 140)(117 132 124 139)(118 131 125 138)(119 130 126 137)(155 222 162 215)(156 221 163 214)(157 220 164 213)(158 219 165 212)(159 218 166 211)(160 217 167 224)(161 216 168 223)
(1 203 79 115)(2 204 80 116)(3 205 81 117)(4 206 82 118)(5 207 83 119)(6 208 84 120)(7 209 71 121)(8 210 72 122)(9 197 73 123)(10 198 74 124)(11 199 75 125)(12 200 76 126)(13 201 77 113)(14 202 78 114)(15 85 211 183)(16 86 212 184)(17 87 213 185)(18 88 214 186)(19 89 215 187)(20 90 216 188)(21 91 217 189)(22 92 218 190)(23 93 219 191)(24 94 220 192)(25 95 221 193)(26 96 222 194)(27 97 223 195)(28 98 224 196)(29 155 110 180)(30 156 111 181)(31 157 112 182)(32 158 99 169)(33 159 100 170)(34 160 101 171)(35 161 102 172)(36 162 103 173)(37 163 104 174)(38 164 105 175)(39 165 106 176)(40 166 107 177)(41 167 108 178)(42 168 109 179)(43 65 135 143)(44 66 136 144)(45 67 137 145)(46 68 138 146)(47 69 139 147)(48 70 140 148)(49 57 127 149)(50 58 128 150)(51 59 129 151)(52 60 130 152)(53 61 131 153)(54 62 132 154)(55 63 133 141)(56 64 134 142)
(1 41 79 108)(2 42 80 109)(3 29 81 110)(4 30 82 111)(5 31 83 112)(6 32 84 99)(7 33 71 100)(8 34 72 101)(9 35 73 102)(10 36 74 103)(11 37 75 104)(12 38 76 105)(13 39 77 106)(14 40 78 107)(15 135 211 43)(16 136 212 44)(17 137 213 45)(18 138 214 46)(19 139 215 47)(20 140 216 48)(21 127 217 49)(22 128 218 50)(23 129 219 51)(24 130 220 52)(25 131 221 53)(26 132 222 54)(27 133 223 55)(28 134 224 56)(57 196 149 98)(58 183 150 85)(59 184 151 86)(60 185 152 87)(61 186 153 88)(62 187 154 89)(63 188 141 90)(64 189 142 91)(65 190 143 92)(66 191 144 93)(67 192 145 94)(68 193 146 95)(69 194 147 96)(70 195 148 97)(113 158 201 169)(114 159 202 170)(115 160 203 171)(116 161 204 172)(117 162 205 173)(118 163 206 174)(119 164 207 175)(120 165 208 176)(121 166 209 177)(122 167 210 178)(123 168 197 179)(124 155 198 180)(125 156 199 181)(126 157 200 182)
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), (1,64,79,142)(2,63,80,141)(3,62,81,154)(4,61,82,153)(5,60,83,152)(6,59,84,151)(7,58,71,150)(8,57,72,149)(9,70,73,148)(10,69,74,147)(11,68,75,146)(12,67,76,145)(13,66,77,144)(14,65,78,143)(15,170,211,159)(16,169,212,158)(17,182,213,157)(18,181,214,156)(19,180,215,155)(20,179,216,168)(21,178,217,167)(22,177,218,166)(23,176,219,165)(24,175,220,164)(25,174,221,163)(26,173,222,162)(27,172,223,161)(28,171,224,160)(29,194,110,96)(30,193,111,95)(31,192,112,94)(32,191,99,93)(33,190,100,92)(34,189,101,91)(35,188,102,90)(36,187,103,89)(37,186,104,88)(38,185,105,87)(39,184,106,86)(40,183,107,85)(41,196,108,98)(42,195,109,97)(43,209,135,121)(44,208,136,120)(45,207,137,119)(46,206,138,118)(47,205,139,117)(48,204,140,116)(49,203,127,115)(50,202,128,114)(51,201,129,113)(52,200,130,126)(53,199,131,125)(54,198,132,124)(55,197,133,123)(56,210,134,122), (1,149,8,142)(2,148,9,141)(3,147,10,154)(4,146,11,153)(5,145,12,152)(6,144,13,151)(7,143,14,150)(15,170,22,177)(16,169,23,176)(17,182,24,175)(18,181,25,174)(19,180,26,173)(20,179,27,172)(21,178,28,171)(29,89,36,96)(30,88,37,95)(31,87,38,94)(32,86,39,93)(33,85,40,92)(34,98,41,91)(35,97,42,90)(43,209,50,202)(44,208,51,201)(45,207,52,200)(46,206,53,199)(47,205,54,198)(48,204,55,197)(49,203,56,210)(57,72,64,79)(58,71,65,78)(59,84,66,77)(60,83,67,76)(61,82,68,75)(62,81,69,74)(63,80,70,73)(99,184,106,191)(100,183,107,190)(101,196,108,189)(102,195,109,188)(103,194,110,187)(104,193,111,186)(105,192,112,185)(113,136,120,129)(114,135,121,128)(115,134,122,127)(116,133,123,140)(117,132,124,139)(118,131,125,138)(119,130,126,137)(155,222,162,215)(156,221,163,214)(157,220,164,213)(158,219,165,212)(159,218,166,211)(160,217,167,224)(161,216,168,223), (1,203,79,115)(2,204,80,116)(3,205,81,117)(4,206,82,118)(5,207,83,119)(6,208,84,120)(7,209,71,121)(8,210,72,122)(9,197,73,123)(10,198,74,124)(11,199,75,125)(12,200,76,126)(13,201,77,113)(14,202,78,114)(15,85,211,183)(16,86,212,184)(17,87,213,185)(18,88,214,186)(19,89,215,187)(20,90,216,188)(21,91,217,189)(22,92,218,190)(23,93,219,191)(24,94,220,192)(25,95,221,193)(26,96,222,194)(27,97,223,195)(28,98,224,196)(29,155,110,180)(30,156,111,181)(31,157,112,182)(32,158,99,169)(33,159,100,170)(34,160,101,171)(35,161,102,172)(36,162,103,173)(37,163,104,174)(38,164,105,175)(39,165,106,176)(40,166,107,177)(41,167,108,178)(42,168,109,179)(43,65,135,143)(44,66,136,144)(45,67,137,145)(46,68,138,146)(47,69,139,147)(48,70,140,148)(49,57,127,149)(50,58,128,150)(51,59,129,151)(52,60,130,152)(53,61,131,153)(54,62,132,154)(55,63,133,141)(56,64,134,142), (1,41,79,108)(2,42,80,109)(3,29,81,110)(4,30,82,111)(5,31,83,112)(6,32,84,99)(7,33,71,100)(8,34,72,101)(9,35,73,102)(10,36,74,103)(11,37,75,104)(12,38,76,105)(13,39,77,106)(14,40,78,107)(15,135,211,43)(16,136,212,44)(17,137,213,45)(18,138,214,46)(19,139,215,47)(20,140,216,48)(21,127,217,49)(22,128,218,50)(23,129,219,51)(24,130,220,52)(25,131,221,53)(26,132,222,54)(27,133,223,55)(28,134,224,56)(57,196,149,98)(58,183,150,85)(59,184,151,86)(60,185,152,87)(61,186,153,88)(62,187,154,89)(63,188,141,90)(64,189,142,91)(65,190,143,92)(66,191,144,93)(67,192,145,94)(68,193,146,95)(69,194,147,96)(70,195,148,97)(113,158,201,169)(114,159,202,170)(115,160,203,171)(116,161,204,172)(117,162,205,173)(118,163,206,174)(119,164,207,175)(120,165,208,176)(121,166,209,177)(122,167,210,178)(123,168,197,179)(124,155,198,180)(125,156,199,181)(126,157,200,182)>;
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), (1,64,79,142)(2,63,80,141)(3,62,81,154)(4,61,82,153)(5,60,83,152)(6,59,84,151)(7,58,71,150)(8,57,72,149)(9,70,73,148)(10,69,74,147)(11,68,75,146)(12,67,76,145)(13,66,77,144)(14,65,78,143)(15,170,211,159)(16,169,212,158)(17,182,213,157)(18,181,214,156)(19,180,215,155)(20,179,216,168)(21,178,217,167)(22,177,218,166)(23,176,219,165)(24,175,220,164)(25,174,221,163)(26,173,222,162)(27,172,223,161)(28,171,224,160)(29,194,110,96)(30,193,111,95)(31,192,112,94)(32,191,99,93)(33,190,100,92)(34,189,101,91)(35,188,102,90)(36,187,103,89)(37,186,104,88)(38,185,105,87)(39,184,106,86)(40,183,107,85)(41,196,108,98)(42,195,109,97)(43,209,135,121)(44,208,136,120)(45,207,137,119)(46,206,138,118)(47,205,139,117)(48,204,140,116)(49,203,127,115)(50,202,128,114)(51,201,129,113)(52,200,130,126)(53,199,131,125)(54,198,132,124)(55,197,133,123)(56,210,134,122), (1,149,8,142)(2,148,9,141)(3,147,10,154)(4,146,11,153)(5,145,12,152)(6,144,13,151)(7,143,14,150)(15,170,22,177)(16,169,23,176)(17,182,24,175)(18,181,25,174)(19,180,26,173)(20,179,27,172)(21,178,28,171)(29,89,36,96)(30,88,37,95)(31,87,38,94)(32,86,39,93)(33,85,40,92)(34,98,41,91)(35,97,42,90)(43,209,50,202)(44,208,51,201)(45,207,52,200)(46,206,53,199)(47,205,54,198)(48,204,55,197)(49,203,56,210)(57,72,64,79)(58,71,65,78)(59,84,66,77)(60,83,67,76)(61,82,68,75)(62,81,69,74)(63,80,70,73)(99,184,106,191)(100,183,107,190)(101,196,108,189)(102,195,109,188)(103,194,110,187)(104,193,111,186)(105,192,112,185)(113,136,120,129)(114,135,121,128)(115,134,122,127)(116,133,123,140)(117,132,124,139)(118,131,125,138)(119,130,126,137)(155,222,162,215)(156,221,163,214)(157,220,164,213)(158,219,165,212)(159,218,166,211)(160,217,167,224)(161,216,168,223), (1,203,79,115)(2,204,80,116)(3,205,81,117)(4,206,82,118)(5,207,83,119)(6,208,84,120)(7,209,71,121)(8,210,72,122)(9,197,73,123)(10,198,74,124)(11,199,75,125)(12,200,76,126)(13,201,77,113)(14,202,78,114)(15,85,211,183)(16,86,212,184)(17,87,213,185)(18,88,214,186)(19,89,215,187)(20,90,216,188)(21,91,217,189)(22,92,218,190)(23,93,219,191)(24,94,220,192)(25,95,221,193)(26,96,222,194)(27,97,223,195)(28,98,224,196)(29,155,110,180)(30,156,111,181)(31,157,112,182)(32,158,99,169)(33,159,100,170)(34,160,101,171)(35,161,102,172)(36,162,103,173)(37,163,104,174)(38,164,105,175)(39,165,106,176)(40,166,107,177)(41,167,108,178)(42,168,109,179)(43,65,135,143)(44,66,136,144)(45,67,137,145)(46,68,138,146)(47,69,139,147)(48,70,140,148)(49,57,127,149)(50,58,128,150)(51,59,129,151)(52,60,130,152)(53,61,131,153)(54,62,132,154)(55,63,133,141)(56,64,134,142), (1,41,79,108)(2,42,80,109)(3,29,81,110)(4,30,82,111)(5,31,83,112)(6,32,84,99)(7,33,71,100)(8,34,72,101)(9,35,73,102)(10,36,74,103)(11,37,75,104)(12,38,76,105)(13,39,77,106)(14,40,78,107)(15,135,211,43)(16,136,212,44)(17,137,213,45)(18,138,214,46)(19,139,215,47)(20,140,216,48)(21,127,217,49)(22,128,218,50)(23,129,219,51)(24,130,220,52)(25,131,221,53)(26,132,222,54)(27,133,223,55)(28,134,224,56)(57,196,149,98)(58,183,150,85)(59,184,151,86)(60,185,152,87)(61,186,153,88)(62,187,154,89)(63,188,141,90)(64,189,142,91)(65,190,143,92)(66,191,144,93)(67,192,145,94)(68,193,146,95)(69,194,147,96)(70,195,148,97)(113,158,201,169)(114,159,202,170)(115,160,203,171)(116,161,204,172)(117,162,205,173)(118,163,206,174)(119,164,207,175)(120,165,208,176)(121,166,209,177)(122,167,210,178)(123,168,197,179)(124,155,198,180)(125,156,199,181)(126,157,200,182) );
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)], [(1,64,79,142),(2,63,80,141),(3,62,81,154),(4,61,82,153),(5,60,83,152),(6,59,84,151),(7,58,71,150),(8,57,72,149),(9,70,73,148),(10,69,74,147),(11,68,75,146),(12,67,76,145),(13,66,77,144),(14,65,78,143),(15,170,211,159),(16,169,212,158),(17,182,213,157),(18,181,214,156),(19,180,215,155),(20,179,216,168),(21,178,217,167),(22,177,218,166),(23,176,219,165),(24,175,220,164),(25,174,221,163),(26,173,222,162),(27,172,223,161),(28,171,224,160),(29,194,110,96),(30,193,111,95),(31,192,112,94),(32,191,99,93),(33,190,100,92),(34,189,101,91),(35,188,102,90),(36,187,103,89),(37,186,104,88),(38,185,105,87),(39,184,106,86),(40,183,107,85),(41,196,108,98),(42,195,109,97),(43,209,135,121),(44,208,136,120),(45,207,137,119),(46,206,138,118),(47,205,139,117),(48,204,140,116),(49,203,127,115),(50,202,128,114),(51,201,129,113),(52,200,130,126),(53,199,131,125),(54,198,132,124),(55,197,133,123),(56,210,134,122)], [(1,149,8,142),(2,148,9,141),(3,147,10,154),(4,146,11,153),(5,145,12,152),(6,144,13,151),(7,143,14,150),(15,170,22,177),(16,169,23,176),(17,182,24,175),(18,181,25,174),(19,180,26,173),(20,179,27,172),(21,178,28,171),(29,89,36,96),(30,88,37,95),(31,87,38,94),(32,86,39,93),(33,85,40,92),(34,98,41,91),(35,97,42,90),(43,209,50,202),(44,208,51,201),(45,207,52,200),(46,206,53,199),(47,205,54,198),(48,204,55,197),(49,203,56,210),(57,72,64,79),(58,71,65,78),(59,84,66,77),(60,83,67,76),(61,82,68,75),(62,81,69,74),(63,80,70,73),(99,184,106,191),(100,183,107,190),(101,196,108,189),(102,195,109,188),(103,194,110,187),(104,193,111,186),(105,192,112,185),(113,136,120,129),(114,135,121,128),(115,134,122,127),(116,133,123,140),(117,132,124,139),(118,131,125,138),(119,130,126,137),(155,222,162,215),(156,221,163,214),(157,220,164,213),(158,219,165,212),(159,218,166,211),(160,217,167,224),(161,216,168,223)], [(1,203,79,115),(2,204,80,116),(3,205,81,117),(4,206,82,118),(5,207,83,119),(6,208,84,120),(7,209,71,121),(8,210,72,122),(9,197,73,123),(10,198,74,124),(11,199,75,125),(12,200,76,126),(13,201,77,113),(14,202,78,114),(15,85,211,183),(16,86,212,184),(17,87,213,185),(18,88,214,186),(19,89,215,187),(20,90,216,188),(21,91,217,189),(22,92,218,190),(23,93,219,191),(24,94,220,192),(25,95,221,193),(26,96,222,194),(27,97,223,195),(28,98,224,196),(29,155,110,180),(30,156,111,181),(31,157,112,182),(32,158,99,169),(33,159,100,170),(34,160,101,171),(35,161,102,172),(36,162,103,173),(37,163,104,174),(38,164,105,175),(39,165,106,176),(40,166,107,177),(41,167,108,178),(42,168,109,179),(43,65,135,143),(44,66,136,144),(45,67,137,145),(46,68,138,146),(47,69,139,147),(48,70,140,148),(49,57,127,149),(50,58,128,150),(51,59,129,151),(52,60,130,152),(53,61,131,153),(54,62,132,154),(55,63,133,141),(56,64,134,142)], [(1,41,79,108),(2,42,80,109),(3,29,81,110),(4,30,82,111),(5,31,83,112),(6,32,84,99),(7,33,71,100),(8,34,72,101),(9,35,73,102),(10,36,74,103),(11,37,75,104),(12,38,76,105),(13,39,77,106),(14,40,78,107),(15,135,211,43),(16,136,212,44),(17,137,213,45),(18,138,214,46),(19,139,215,47),(20,140,216,48),(21,127,217,49),(22,128,218,50),(23,129,219,51),(24,130,220,52),(25,131,221,53),(26,132,222,54),(27,133,223,55),(28,134,224,56),(57,196,149,98),(58,183,150,85),(59,184,151,86),(60,185,152,87),(61,186,153,88),(62,187,154,89),(63,188,141,90),(64,189,142,91),(65,190,143,92),(66,191,144,93),(67,192,145,94),(68,193,146,95),(69,194,147,96),(70,195,148,97),(113,158,201,169),(114,159,202,170),(115,160,203,171),(116,161,204,172),(117,162,205,173),(118,163,206,174),(119,164,207,175),(120,165,208,176),(121,166,209,177),(122,167,210,178),(123,168,197,179),(124,155,198,180),(125,156,199,181),(126,157,200,182)]])
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 | C14.72+ 1+4 | C28⋊Q8 | C28.3Q8 | 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 C14.72+ 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] >;
C14.72+ 1+4 in GAP, Magma, Sage, TeX
C_{14}._72_+^{1+4} % in TeX
G:=Group("C14.7ES+(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