G = C14.742- 1+4 order 448 = 26·7
29th non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.742- 1+4, C14.492+ 1+4, C4⋊D4⋊24D7, C4⋊C4.93D14, C28⋊2D4⋊29C2, (C2×D4).96D14, D14⋊Q8⋊16C2, Dic7.Q8⋊14C2, (C2×C28).47C23, C22⋊C4.12D14, Dic7⋊D4⋊19C2, D14.D4⋊23C2, C28.48D4⋊45C2, C28.17D4⋊20C2, (C2×C14).165C24, D14⋊C4.18C22, (C22×C4).231D14, C4⋊Dic7.46C22, C2.51(D4⋊6D14), C22⋊Dic14⋊21C2, Dic7.D4⋊23C2, (D4×C14).129C22, Dic7⋊C4.79C22, (C2×Dic7).82C23, (C22×D7).72C23, C22.186(C23×D7), C23.115(C22×D7), C23.D7.29C22, C23.18D14⋊23C2, C23.23D14⋊13C2, (C22×C28).313C22, (C22×C14).193C23, C7⋊1(C22.56C24), (C4×Dic7).100C22, (C2×Dic14).34C22, C2.32(D4.10D14), (C22×Dic7).116C22, (C7×C4⋊D4)⋊27C2, (C2×C4×D7).90C22, (C2×C4).43(C22×D7), (C7×C4⋊C4).151C22, (C2×C7⋊D4).36C22, (C7×C22⋊C4).20C22, SmallGroup(448,1074)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.742- 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=e2=b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, bd=db, ebe-1=a7b, dcd-1=a7c, ce=ec, ede-1=a7b2d >
Subgroups: 1004 in 220 conjugacy classes, 91 normal (all characteristic)
C1, C2 [×3], C2 [×4], C4 [×11], C22, C22 [×12], C7, C2×C4 [×4], C2×C4 [×11], D4 [×6], Q8 [×2], C23 [×3], C23, D7, C14 [×3], C14 [×3], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×3], C2×D4 [×3], C2×D4 [×3], C2×Q8 [×2], Dic7 [×7], C28 [×4], D14 [×3], C2×C14, C2×C14 [×9], C4⋊D4, C4⋊D4 [×3], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C42.C2, Dic14 [×2], C4×D7, C2×Dic7 [×7], C2×Dic7 [×2], C7⋊D4 [×3], C2×C28 [×4], C2×C28, C7×D4 [×3], C22×D7, C22×C14 [×3], C22.56C24, C4×Dic7, Dic7⋊C4 [×7], C4⋊Dic7 [×2], D14⋊C4 [×3], C23.D7 [×7], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14 [×2], C2×C4×D7, C22×Dic7 [×2], C2×C7⋊D4 [×3], C22×C28, D4×C14 [×3], C22⋊Dic14 [×2], D14.D4, Dic7.D4, Dic7.Q8, D14⋊Q8, C28.48D4, C23.23D14, C23.18D14 [×2], C28.17D4, C28⋊2D4, Dic7⋊D4 [×2], C7×C4⋊D4, C14.742- 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ 1+4 [×2], 2- 1+4, C22×D7 [×7], C22.56C24, C23×D7, D4⋊6D14 [×2], D4.10D14, C14.742- 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)
(1 135 224 117)(2 134 211 116)(3 133 212 115)(4 132 213 114)(5 131 214 113)(6 130 215 126)(7 129 216 125)(8 128 217 124)(9 127 218 123)(10 140 219 122)(11 139 220 121)(12 138 221 120)(13 137 222 119)(14 136 223 118)(15 74 178 111)(16 73 179 110)(17 72 180 109)(18 71 181 108)(19 84 182 107)(20 83 169 106)(21 82 170 105)(22 81 171 104)(23 80 172 103)(24 79 173 102)(25 78 174 101)(26 77 175 100)(27 76 176 99)(28 75 177 112)(29 69 199 96)(30 68 200 95)(31 67 201 94)(32 66 202 93)(33 65 203 92)(34 64 204 91)(35 63 205 90)(36 62 206 89)(37 61 207 88)(38 60 208 87)(39 59 209 86)(40 58 210 85)(41 57 197 98)(42 70 198 97)(43 196 159 144)(44 195 160 143)(45 194 161 142)(46 193 162 141)(47 192 163 154)(48 191 164 153)(49 190 165 152)(50 189 166 151)(51 188 167 150)(52 187 168 149)(53 186 155 148)(54 185 156 147)(55 184 157 146)(56 183 158 145)
(1 141)(2 142)(3 143)(4 144)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 57)(25 58)(26 59)(27 60)(28 61)(29 81)(30 82)(31 83)(32 84)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 120)(50 121)(51 122)(52 123)(53 124)(54 125)(55 126)(56 113)(85 174)(86 175)(87 176)(88 177)(89 178)(90 179)(91 180)(92 181)(93 182)(94 169)(95 170)(96 171)(97 172)(98 173)(99 208)(100 209)(101 210)(102 197)(103 198)(104 199)(105 200)(106 201)(107 202)(108 203)(109 204)(110 205)(111 206)(112 207)(127 168)(128 155)(129 156)(130 157)(131 158)(132 159)(133 160)(134 161)(135 162)(136 163)(137 164)(138 165)(139 166)(140 167)(183 214)(184 215)(185 216)(186 217)(187 218)(188 219)(189 220)(190 221)(191 222)(192 223)(193 224)(194 211)(195 212)(196 213)
(1 135 224 117)(2 134 211 116)(3 133 212 115)(4 132 213 114)(5 131 214 113)(6 130 215 126)(7 129 216 125)(8 128 217 124)(9 127 218 123)(10 140 219 122)(11 139 220 121)(12 138 221 120)(13 137 222 119)(14 136 223 118)(15 111 178 74)(16 110 179 73)(17 109 180 72)(18 108 181 71)(19 107 182 84)(20 106 169 83)(21 105 170 82)(22 104 171 81)(23 103 172 80)(24 102 173 79)(25 101 174 78)(26 100 175 77)(27 99 176 76)(28 112 177 75)(29 62 199 89)(30 61 200 88)(31 60 201 87)(32 59 202 86)(33 58 203 85)(34 57 204 98)(35 70 205 97)(36 69 206 96)(37 68 207 95)(38 67 208 94)(39 66 209 93)(40 65 210 92)(41 64 197 91)(42 63 198 90)(43 151 159 189)(44 150 160 188)(45 149 161 187)(46 148 162 186)(47 147 163 185)(48 146 164 184)(49 145 165 183)(50 144 166 196)(51 143 167 195)(52 142 168 194)(53 141 155 193)(54 154 156 192)(55 153 157 191)(56 152 158 190)
(1 170 224 21)(2 171 211 22)(3 172 212 23)(4 173 213 24)(5 174 214 25)(6 175 215 26)(7 176 216 27)(8 177 217 28)(9 178 218 15)(10 179 219 16)(11 180 220 17)(12 181 221 18)(13 182 222 19)(14 169 223 20)(29 168 199 52)(30 155 200 53)(31 156 201 54)(32 157 202 55)(33 158 203 56)(34 159 204 43)(35 160 205 44)(36 161 206 45)(37 162 207 46)(38 163 208 47)(39 164 209 48)(40 165 210 49)(41 166 197 50)(42 167 198 51)(57 144 98 196)(58 145 85 183)(59 146 86 184)(60 147 87 185)(61 148 88 186)(62 149 89 187)(63 150 90 188)(64 151 91 189)(65 152 92 190)(66 153 93 191)(67 154 94 192)(68 141 95 193)(69 142 96 194)(70 143 97 195)(71 131 108 113)(72 132 109 114)(73 133 110 115)(74 134 111 116)(75 135 112 117)(76 136 99 118)(77 137 100 119)(78 138 101 120)(79 139 102 121)(80 140 103 122)(81 127 104 123)(82 128 105 124)(83 129 106 125)(84 130 107 126)
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), (1,135,224,117)(2,134,211,116)(3,133,212,115)(4,132,213,114)(5,131,214,113)(6,130,215,126)(7,129,216,125)(8,128,217,124)(9,127,218,123)(10,140,219,122)(11,139,220,121)(12,138,221,120)(13,137,222,119)(14,136,223,118)(15,74,178,111)(16,73,179,110)(17,72,180,109)(18,71,181,108)(19,84,182,107)(20,83,169,106)(21,82,170,105)(22,81,171,104)(23,80,172,103)(24,79,173,102)(25,78,174,101)(26,77,175,100)(27,76,176,99)(28,75,177,112)(29,69,199,96)(30,68,200,95)(31,67,201,94)(32,66,202,93)(33,65,203,92)(34,64,204,91)(35,63,205,90)(36,62,206,89)(37,61,207,88)(38,60,208,87)(39,59,209,86)(40,58,210,85)(41,57,197,98)(42,70,198,97)(43,196,159,144)(44,195,160,143)(45,194,161,142)(46,193,162,141)(47,192,163,154)(48,191,164,153)(49,190,165,152)(50,189,166,151)(51,188,167,150)(52,187,168,149)(53,186,155,148)(54,185,156,147)(55,184,157,146)(56,183,158,145), (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,57)(25,58)(26,59)(27,60)(28,61)(29,81)(30,82)(31,83)(32,84)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,120)(50,121)(51,122)(52,123)(53,124)(54,125)(55,126)(56,113)(85,174)(86,175)(87,176)(88,177)(89,178)(90,179)(91,180)(92,181)(93,182)(94,169)(95,170)(96,171)(97,172)(98,173)(99,208)(100,209)(101,210)(102,197)(103,198)(104,199)(105,200)(106,201)(107,202)(108,203)(109,204)(110,205)(111,206)(112,207)(127,168)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(136,163)(137,164)(138,165)(139,166)(140,167)(183,214)(184,215)(185,216)(186,217)(187,218)(188,219)(189,220)(190,221)(191,222)(192,223)(193,224)(194,211)(195,212)(196,213), (1,135,224,117)(2,134,211,116)(3,133,212,115)(4,132,213,114)(5,131,214,113)(6,130,215,126)(7,129,216,125)(8,128,217,124)(9,127,218,123)(10,140,219,122)(11,139,220,121)(12,138,221,120)(13,137,222,119)(14,136,223,118)(15,111,178,74)(16,110,179,73)(17,109,180,72)(18,108,181,71)(19,107,182,84)(20,106,169,83)(21,105,170,82)(22,104,171,81)(23,103,172,80)(24,102,173,79)(25,101,174,78)(26,100,175,77)(27,99,176,76)(28,112,177,75)(29,62,199,89)(30,61,200,88)(31,60,201,87)(32,59,202,86)(33,58,203,85)(34,57,204,98)(35,70,205,97)(36,69,206,96)(37,68,207,95)(38,67,208,94)(39,66,209,93)(40,65,210,92)(41,64,197,91)(42,63,198,90)(43,151,159,189)(44,150,160,188)(45,149,161,187)(46,148,162,186)(47,147,163,185)(48,146,164,184)(49,145,165,183)(50,144,166,196)(51,143,167,195)(52,142,168,194)(53,141,155,193)(54,154,156,192)(55,153,157,191)(56,152,158,190), (1,170,224,21)(2,171,211,22)(3,172,212,23)(4,173,213,24)(5,174,214,25)(6,175,215,26)(7,176,216,27)(8,177,217,28)(9,178,218,15)(10,179,219,16)(11,180,220,17)(12,181,221,18)(13,182,222,19)(14,169,223,20)(29,168,199,52)(30,155,200,53)(31,156,201,54)(32,157,202,55)(33,158,203,56)(34,159,204,43)(35,160,205,44)(36,161,206,45)(37,162,207,46)(38,163,208,47)(39,164,209,48)(40,165,210,49)(41,166,197,50)(42,167,198,51)(57,144,98,196)(58,145,85,183)(59,146,86,184)(60,147,87,185)(61,148,88,186)(62,149,89,187)(63,150,90,188)(64,151,91,189)(65,152,92,190)(66,153,93,191)(67,154,94,192)(68,141,95,193)(69,142,96,194)(70,143,97,195)(71,131,108,113)(72,132,109,114)(73,133,110,115)(74,134,111,116)(75,135,112,117)(76,136,99,118)(77,137,100,119)(78,138,101,120)(79,139,102,121)(80,140,103,122)(81,127,104,123)(82,128,105,124)(83,129,106,125)(84,130,107,126)>;
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), (1,135,224,117)(2,134,211,116)(3,133,212,115)(4,132,213,114)(5,131,214,113)(6,130,215,126)(7,129,216,125)(8,128,217,124)(9,127,218,123)(10,140,219,122)(11,139,220,121)(12,138,221,120)(13,137,222,119)(14,136,223,118)(15,74,178,111)(16,73,179,110)(17,72,180,109)(18,71,181,108)(19,84,182,107)(20,83,169,106)(21,82,170,105)(22,81,171,104)(23,80,172,103)(24,79,173,102)(25,78,174,101)(26,77,175,100)(27,76,176,99)(28,75,177,112)(29,69,199,96)(30,68,200,95)(31,67,201,94)(32,66,202,93)(33,65,203,92)(34,64,204,91)(35,63,205,90)(36,62,206,89)(37,61,207,88)(38,60,208,87)(39,59,209,86)(40,58,210,85)(41,57,197,98)(42,70,198,97)(43,196,159,144)(44,195,160,143)(45,194,161,142)(46,193,162,141)(47,192,163,154)(48,191,164,153)(49,190,165,152)(50,189,166,151)(51,188,167,150)(52,187,168,149)(53,186,155,148)(54,185,156,147)(55,184,157,146)(56,183,158,145), (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,68)(22,69)(23,70)(24,57)(25,58)(26,59)(27,60)(28,61)(29,81)(30,82)(31,83)(32,84)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,120)(50,121)(51,122)(52,123)(53,124)(54,125)(55,126)(56,113)(85,174)(86,175)(87,176)(88,177)(89,178)(90,179)(91,180)(92,181)(93,182)(94,169)(95,170)(96,171)(97,172)(98,173)(99,208)(100,209)(101,210)(102,197)(103,198)(104,199)(105,200)(106,201)(107,202)(108,203)(109,204)(110,205)(111,206)(112,207)(127,168)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(136,163)(137,164)(138,165)(139,166)(140,167)(183,214)(184,215)(185,216)(186,217)(187,218)(188,219)(189,220)(190,221)(191,222)(192,223)(193,224)(194,211)(195,212)(196,213), (1,135,224,117)(2,134,211,116)(3,133,212,115)(4,132,213,114)(5,131,214,113)(6,130,215,126)(7,129,216,125)(8,128,217,124)(9,127,218,123)(10,140,219,122)(11,139,220,121)(12,138,221,120)(13,137,222,119)(14,136,223,118)(15,111,178,74)(16,110,179,73)(17,109,180,72)(18,108,181,71)(19,107,182,84)(20,106,169,83)(21,105,170,82)(22,104,171,81)(23,103,172,80)(24,102,173,79)(25,101,174,78)(26,100,175,77)(27,99,176,76)(28,112,177,75)(29,62,199,89)(30,61,200,88)(31,60,201,87)(32,59,202,86)(33,58,203,85)(34,57,204,98)(35,70,205,97)(36,69,206,96)(37,68,207,95)(38,67,208,94)(39,66,209,93)(40,65,210,92)(41,64,197,91)(42,63,198,90)(43,151,159,189)(44,150,160,188)(45,149,161,187)(46,148,162,186)(47,147,163,185)(48,146,164,184)(49,145,165,183)(50,144,166,196)(51,143,167,195)(52,142,168,194)(53,141,155,193)(54,154,156,192)(55,153,157,191)(56,152,158,190), (1,170,224,21)(2,171,211,22)(3,172,212,23)(4,173,213,24)(5,174,214,25)(6,175,215,26)(7,176,216,27)(8,177,217,28)(9,178,218,15)(10,179,219,16)(11,180,220,17)(12,181,221,18)(13,182,222,19)(14,169,223,20)(29,168,199,52)(30,155,200,53)(31,156,201,54)(32,157,202,55)(33,158,203,56)(34,159,204,43)(35,160,205,44)(36,161,206,45)(37,162,207,46)(38,163,208,47)(39,164,209,48)(40,165,210,49)(41,166,197,50)(42,167,198,51)(57,144,98,196)(58,145,85,183)(59,146,86,184)(60,147,87,185)(61,148,88,186)(62,149,89,187)(63,150,90,188)(64,151,91,189)(65,152,92,190)(66,153,93,191)(67,154,94,192)(68,141,95,193)(69,142,96,194)(70,143,97,195)(71,131,108,113)(72,132,109,114)(73,133,110,115)(74,134,111,116)(75,135,112,117)(76,136,99,118)(77,137,100,119)(78,138,101,120)(79,139,102,121)(80,140,103,122)(81,127,104,123)(82,128,105,124)(83,129,106,125)(84,130,107,126) );
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)], [(1,135,224,117),(2,134,211,116),(3,133,212,115),(4,132,213,114),(5,131,214,113),(6,130,215,126),(7,129,216,125),(8,128,217,124),(9,127,218,123),(10,140,219,122),(11,139,220,121),(12,138,221,120),(13,137,222,119),(14,136,223,118),(15,74,178,111),(16,73,179,110),(17,72,180,109),(18,71,181,108),(19,84,182,107),(20,83,169,106),(21,82,170,105),(22,81,171,104),(23,80,172,103),(24,79,173,102),(25,78,174,101),(26,77,175,100),(27,76,176,99),(28,75,177,112),(29,69,199,96),(30,68,200,95),(31,67,201,94),(32,66,202,93),(33,65,203,92),(34,64,204,91),(35,63,205,90),(36,62,206,89),(37,61,207,88),(38,60,208,87),(39,59,209,86),(40,58,210,85),(41,57,197,98),(42,70,198,97),(43,196,159,144),(44,195,160,143),(45,194,161,142),(46,193,162,141),(47,192,163,154),(48,191,164,153),(49,190,165,152),(50,189,166,151),(51,188,167,150),(52,187,168,149),(53,186,155,148),(54,185,156,147),(55,184,157,146),(56,183,158,145)], [(1,141),(2,142),(3,143),(4,144),(5,145),(6,146),(7,147),(8,148),(9,149),(10,150),(11,151),(12,152),(13,153),(14,154),(15,62),(16,63),(17,64),(18,65),(19,66),(20,67),(21,68),(22,69),(23,70),(24,57),(25,58),(26,59),(27,60),(28,61),(29,81),(30,82),(31,83),(32,84),(33,71),(34,72),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78),(41,79),(42,80),(43,114),(44,115),(45,116),(46,117),(47,118),(48,119),(49,120),(50,121),(51,122),(52,123),(53,124),(54,125),(55,126),(56,113),(85,174),(86,175),(87,176),(88,177),(89,178),(90,179),(91,180),(92,181),(93,182),(94,169),(95,170),(96,171),(97,172),(98,173),(99,208),(100,209),(101,210),(102,197),(103,198),(104,199),(105,200),(106,201),(107,202),(108,203),(109,204),(110,205),(111,206),(112,207),(127,168),(128,155),(129,156),(130,157),(131,158),(132,159),(133,160),(134,161),(135,162),(136,163),(137,164),(138,165),(139,166),(140,167),(183,214),(184,215),(185,216),(186,217),(187,218),(188,219),(189,220),(190,221),(191,222),(192,223),(193,224),(194,211),(195,212),(196,213)], [(1,135,224,117),(2,134,211,116),(3,133,212,115),(4,132,213,114),(5,131,214,113),(6,130,215,126),(7,129,216,125),(8,128,217,124),(9,127,218,123),(10,140,219,122),(11,139,220,121),(12,138,221,120),(13,137,222,119),(14,136,223,118),(15,111,178,74),(16,110,179,73),(17,109,180,72),(18,108,181,71),(19,107,182,84),(20,106,169,83),(21,105,170,82),(22,104,171,81),(23,103,172,80),(24,102,173,79),(25,101,174,78),(26,100,175,77),(27,99,176,76),(28,112,177,75),(29,62,199,89),(30,61,200,88),(31,60,201,87),(32,59,202,86),(33,58,203,85),(34,57,204,98),(35,70,205,97),(36,69,206,96),(37,68,207,95),(38,67,208,94),(39,66,209,93),(40,65,210,92),(41,64,197,91),(42,63,198,90),(43,151,159,189),(44,150,160,188),(45,149,161,187),(46,148,162,186),(47,147,163,185),(48,146,164,184),(49,145,165,183),(50,144,166,196),(51,143,167,195),(52,142,168,194),(53,141,155,193),(54,154,156,192),(55,153,157,191),(56,152,158,190)], [(1,170,224,21),(2,171,211,22),(3,172,212,23),(4,173,213,24),(5,174,214,25),(6,175,215,26),(7,176,216,27),(8,177,217,28),(9,178,218,15),(10,179,219,16),(11,180,220,17),(12,181,221,18),(13,182,222,19),(14,169,223,20),(29,168,199,52),(30,155,200,53),(31,156,201,54),(32,157,202,55),(33,158,203,56),(34,159,204,43),(35,160,205,44),(36,161,206,45),(37,162,207,46),(38,163,208,47),(39,164,209,48),(40,165,210,49),(41,166,197,50),(42,167,198,51),(57,144,98,196),(58,145,85,183),(59,146,86,184),(60,147,87,185),(61,148,88,186),(62,149,89,187),(63,150,90,188),(64,151,91,189),(65,152,92,190),(66,153,93,191),(67,154,94,192),(68,141,95,193),(69,142,96,194),(70,143,97,195),(71,131,108,113),(72,132,109,114),(73,133,110,115),(74,134,111,116),(75,135,112,117),(76,136,99,118),(77,137,100,119),(78,138,101,120),(79,139,102,121),(80,140,103,122),(81,127,104,123),(82,128,105,124),(83,129,106,125),(84,130,107,126)])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4K | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 28 | 4 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | D14 | 2+ 1+4 | 2- 1+4 | D4⋊6D14 | D4.10D14 |
| kernel | C14.742- 1+4 | C22⋊Dic14 | D14.D4 | Dic7.D4 | Dic7.Q8 | D14⋊Q8 | C28.48D4 | C23.23D14 | C23.18D14 | C28.17D4 | C28⋊2D4 | Dic7⋊D4 | C7×C4⋊D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 3 | 6 | 3 | 3 | 9 | 2 | 1 | 12 | 6 |
Matrix representation of C14.742- 1+4 ►in GL8(𝔽29)
| 19 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 14 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 25 |
| 19 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 | 1 | 4 |
| 0 | 0 | 0 | 0 | 17 | 0 | 8 | 27 |
| 0 | 0 | 0 | 0 | 0 | 0 | 19 | 23 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 10 |
| 1 | 0 | 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 21 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 18 |
| 19 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 19 | 19 | 10 | 10 | 0 | 0 | 0 | 0 |
| 7 | 10 | 22 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 | 3 | 17 |
| 0 | 0 | 0 | 0 | 17 | 0 | 0 | 26 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 19 |
| 20 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 1 | 27 | 8 |
| 0 | 0 | 0 | 0 | 28 | 12 | 14 | 0 |
| 0 | 0 | 0 | 0 | 25 | 27 | 21 | 13 |
| 0 | 0 | 0 | 0 | 1 | 21 | 1 | 6 |
G:=sub<GL(8,GF(29))| [19,10,0,0,0,0,0,0,19,7,0,0,0,0,0,0,0,0,19,10,0,0,0,0,0,0,19,7,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,19,19,0,0,0,0,0,0,0,0,1,10,0,0,0,0,0,0,14,25],[19,7,0,0,0,0,0,0,19,10,0,0,0,0,0,0,0,0,19,7,0,0,0,0,0,0,19,10,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,0,1,8,19,12,0,0,0,0,4,27,23,10],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,27,0,28,0,0,0,0,0,0,27,0,28,0,0,0,0,0,0,0,0,20,15,0,0,0,0,0,0,14,9,0,0,0,0,0,0,0,0,11,15,0,0,0,0,0,0,21,18],[19,7,19,7,0,0,0,0,19,10,19,10,0,0,0,0,0,0,10,22,0,0,0,0,0,0,10,19,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,0,3,0,10,17,0,0,0,0,17,26,6,19],[20,14,0,0,0,0,0,0,15,9,0,0,0,0,0,0,0,0,20,14,0,0,0,0,0,0,15,9,0,0,0,0,0,0,0,0,19,28,25,1,0,0,0,0,1,12,27,21,0,0,0,0,27,14,21,1,0,0,0,0,8,0,13,6] >;
C14.742- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{74}2_-^{1+4} % in TeX
G:=Group("C14.74ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1074);
// by ID
G=gap.SmallGroup(448,1074);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,1571,570,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=e^2=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,b*d=d*b,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