G = C14.712- 1+4 order 448 = 26·7
26th non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.712- 1+4, C14.362+ 1+4, C28⋊Q8⋊19C2, C4⋊D4.9D7, C4⋊C4.179D14, (D4×Dic7)⋊19C2, (C2×D4).91D14, C22⋊C4.7D14, (C2×C28).37C23, Dic7⋊3Q8⋊22C2, C28.202(C4○D4), C28.48D4⋊32C2, C4.68(D4⋊2D7), C28.17D4⋊16C2, (C2×C14).147C24, (C22×C4).222D14, C2.38(D4⋊6D14), C23.13(C22×D7), C22⋊Dic14⋊18C2, (D4×C14).121C22, C23.18D14⋊8C2, C23.D14⋊17C2, Dic7⋊C4.17C22, C4⋊Dic7.310C22, (C4×Dic7).94C22, (C2×Dic7).68C23, C22.168(C23×D7), C23.D7.24C22, C23.21D14⋊25C2, (C22×C14).185C23, (C22×C28).239C22, C7⋊4(C22.36C24), C2.29(D4.10D14), (C2×Dic14).153C22, (C22×Dic7).108C22, C14.83(C2×C4○D4), (C7×C4⋊D4).9C2, C2.35(C2×D4⋊2D7), (C2×C4).36(C22×D7), (C7×C4⋊C4).143C22, (C7×C22⋊C4).12C22, SmallGroup(448,1056)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.712- 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=a7b-1, dbd-1=a7b, be=eb, dcd-1=a7c, ce=ec, ede-1=a7b2d >
Subgroups: 844 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4⋊Q8, Dic14, C2×Dic7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22.36C24, C4×Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C23.D7, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×Dic14, C22×Dic7, C22×C28, D4×C14, D4×C14, C22⋊Dic14, C23.D14, Dic7⋊3Q8, C28⋊Q8, C28.48D4, C23.21D14, D4×Dic7, C23.18D14, C28.17D4, C28.17D4, C7×C4⋊D4, C14.712- 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D7, C22.36C24, D4⋊2D7, C23×D7, C2×D4⋊2D7, D4⋊6D14, D4.10D14, C14.712- 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)
(1 175 212 82)(2 174 213 81)(3 173 214 80)(4 172 215 79)(5 171 216 78)(6 170 217 77)(7 169 218 76)(8 182 219 75)(9 181 220 74)(10 180 221 73)(11 179 222 72)(12 178 223 71)(13 177 224 84)(14 176 211 83)(15 37 117 145)(16 36 118 144)(17 35 119 143)(18 34 120 142)(19 33 121 141)(20 32 122 154)(21 31 123 153)(22 30 124 152)(23 29 125 151)(24 42 126 150)(25 41 113 149)(26 40 114 148)(27 39 115 147)(28 38 116 146)(43 191 210 107)(44 190 197 106)(45 189 198 105)(46 188 199 104)(47 187 200 103)(48 186 201 102)(49 185 202 101)(50 184 203 100)(51 183 204 99)(52 196 205 112)(53 195 206 111)(54 194 207 110)(55 193 208 109)(56 192 209 108)(57 158 91 131)(58 157 92 130)(59 156 93 129)(60 155 94 128)(61 168 95 127)(62 167 96 140)(63 166 97 139)(64 165 98 138)(65 164 85 137)(66 163 86 136)(67 162 87 135)(68 161 88 134)(69 160 89 133)(70 159 90 132)
(15 117)(16 118)(17 119)(18 120)(19 121)(20 122)(21 123)(22 124)(23 125)(24 126)(25 113)(26 114)(27 115)(28 116)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 210)(44 197)(45 198)(46 199)(47 200)(48 201)(49 202)(50 203)(51 204)(52 205)(53 206)(54 207)(55 208)(56 209)(71 171)(72 172)(73 173)(74 174)(75 175)(76 176)(77 177)(78 178)(79 179)(80 180)(81 181)(82 182)(83 169)(84 170)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(127 161)(128 162)(129 163)(130 164)(131 165)(132 166)(133 167)(134 168)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)
(1 112 219 189)(2 111 220 188)(3 110 221 187)(4 109 222 186)(5 108 223 185)(6 107 224 184)(7 106 211 183)(8 105 212 196)(9 104 213 195)(10 103 214 194)(11 102 215 193)(12 101 216 192)(13 100 217 191)(14 99 218 190)(15 167 124 133)(16 166 125 132)(17 165 126 131)(18 164 113 130)(19 163 114 129)(20 162 115 128)(21 161 116 127)(22 160 117 140)(23 159 118 139)(24 158 119 138)(25 157 120 137)(26 156 121 136)(27 155 122 135)(28 168 123 134)(29 97 144 70)(30 96 145 69)(31 95 146 68)(32 94 147 67)(33 93 148 66)(34 92 149 65)(35 91 150 64)(36 90 151 63)(37 89 152 62)(38 88 153 61)(39 87 154 60)(40 86 141 59)(41 85 142 58)(42 98 143 57)(43 77 203 177)(44 76 204 176)(45 75 205 175)(46 74 206 174)(47 73 207 173)(48 72 208 172)(49 71 209 171)(50 84 210 170)(51 83 197 169)(52 82 198 182)(53 81 199 181)(54 80 200 180)(55 79 201 179)(56 78 202 178)
(1 89 219 62)(2 90 220 63)(3 91 221 64)(4 92 222 65)(5 93 223 66)(6 94 224 67)(7 95 211 68)(8 96 212 69)(9 97 213 70)(10 98 214 57)(11 85 215 58)(12 86 216 59)(13 87 217 60)(14 88 218 61)(15 45 124 205)(16 46 125 206)(17 47 126 207)(18 48 113 208)(19 49 114 209)(20 50 115 210)(21 51 116 197)(22 52 117 198)(23 53 118 199)(24 54 119 200)(25 55 120 201)(26 56 121 202)(27 43 122 203)(28 44 123 204)(29 195 144 104)(30 196 145 105)(31 183 146 106)(32 184 147 107)(33 185 148 108)(34 186 149 109)(35 187 150 110)(36 188 151 111)(37 189 152 112)(38 190 153 99)(39 191 154 100)(40 192 141 101)(41 193 142 102)(42 194 143 103)(71 163 171 129)(72 164 172 130)(73 165 173 131)(74 166 174 132)(75 167 175 133)(76 168 176 134)(77 155 177 135)(78 156 178 136)(79 157 179 137)(80 158 180 138)(81 159 181 139)(82 160 182 140)(83 161 169 127)(84 162 170 128)
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), (1,175,212,82)(2,174,213,81)(3,173,214,80)(4,172,215,79)(5,171,216,78)(6,170,217,77)(7,169,218,76)(8,182,219,75)(9,181,220,74)(10,180,221,73)(11,179,222,72)(12,178,223,71)(13,177,224,84)(14,176,211,83)(15,37,117,145)(16,36,118,144)(17,35,119,143)(18,34,120,142)(19,33,121,141)(20,32,122,154)(21,31,123,153)(22,30,124,152)(23,29,125,151)(24,42,126,150)(25,41,113,149)(26,40,114,148)(27,39,115,147)(28,38,116,146)(43,191,210,107)(44,190,197,106)(45,189,198,105)(46,188,199,104)(47,187,200,103)(48,186,201,102)(49,185,202,101)(50,184,203,100)(51,183,204,99)(52,196,205,112)(53,195,206,111)(54,194,207,110)(55,193,208,109)(56,192,209,108)(57,158,91,131)(58,157,92,130)(59,156,93,129)(60,155,94,128)(61,168,95,127)(62,167,96,140)(63,166,97,139)(64,165,98,138)(65,164,85,137)(66,163,86,136)(67,162,87,135)(68,161,88,134)(69,160,89,133)(70,159,90,132), (15,117)(16,118)(17,119)(18,120)(19,121)(20,122)(21,123)(22,124)(23,125)(24,126)(25,113)(26,114)(27,115)(28,116)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,210)(44,197)(45,198)(46,199)(47,200)(48,201)(49,202)(50,203)(51,204)(52,205)(53,206)(54,207)(55,208)(56,209)(71,171)(72,172)(73,173)(74,174)(75,175)(76,176)(77,177)(78,178)(79,179)(80,180)(81,181)(82,182)(83,169)(84,170)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(127,161)(128,162)(129,163)(130,164)(131,165)(132,166)(133,167)(134,168)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196), (1,112,219,189)(2,111,220,188)(3,110,221,187)(4,109,222,186)(5,108,223,185)(6,107,224,184)(7,106,211,183)(8,105,212,196)(9,104,213,195)(10,103,214,194)(11,102,215,193)(12,101,216,192)(13,100,217,191)(14,99,218,190)(15,167,124,133)(16,166,125,132)(17,165,126,131)(18,164,113,130)(19,163,114,129)(20,162,115,128)(21,161,116,127)(22,160,117,140)(23,159,118,139)(24,158,119,138)(25,157,120,137)(26,156,121,136)(27,155,122,135)(28,168,123,134)(29,97,144,70)(30,96,145,69)(31,95,146,68)(32,94,147,67)(33,93,148,66)(34,92,149,65)(35,91,150,64)(36,90,151,63)(37,89,152,62)(38,88,153,61)(39,87,154,60)(40,86,141,59)(41,85,142,58)(42,98,143,57)(43,77,203,177)(44,76,204,176)(45,75,205,175)(46,74,206,174)(47,73,207,173)(48,72,208,172)(49,71,209,171)(50,84,210,170)(51,83,197,169)(52,82,198,182)(53,81,199,181)(54,80,200,180)(55,79,201,179)(56,78,202,178), (1,89,219,62)(2,90,220,63)(3,91,221,64)(4,92,222,65)(5,93,223,66)(6,94,224,67)(7,95,211,68)(8,96,212,69)(9,97,213,70)(10,98,214,57)(11,85,215,58)(12,86,216,59)(13,87,217,60)(14,88,218,61)(15,45,124,205)(16,46,125,206)(17,47,126,207)(18,48,113,208)(19,49,114,209)(20,50,115,210)(21,51,116,197)(22,52,117,198)(23,53,118,199)(24,54,119,200)(25,55,120,201)(26,56,121,202)(27,43,122,203)(28,44,123,204)(29,195,144,104)(30,196,145,105)(31,183,146,106)(32,184,147,107)(33,185,148,108)(34,186,149,109)(35,187,150,110)(36,188,151,111)(37,189,152,112)(38,190,153,99)(39,191,154,100)(40,192,141,101)(41,193,142,102)(42,194,143,103)(71,163,171,129)(72,164,172,130)(73,165,173,131)(74,166,174,132)(75,167,175,133)(76,168,176,134)(77,155,177,135)(78,156,178,136)(79,157,179,137)(80,158,180,138)(81,159,181,139)(82,160,182,140)(83,161,169,127)(84,162,170,128)>;
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), (1,175,212,82)(2,174,213,81)(3,173,214,80)(4,172,215,79)(5,171,216,78)(6,170,217,77)(7,169,218,76)(8,182,219,75)(9,181,220,74)(10,180,221,73)(11,179,222,72)(12,178,223,71)(13,177,224,84)(14,176,211,83)(15,37,117,145)(16,36,118,144)(17,35,119,143)(18,34,120,142)(19,33,121,141)(20,32,122,154)(21,31,123,153)(22,30,124,152)(23,29,125,151)(24,42,126,150)(25,41,113,149)(26,40,114,148)(27,39,115,147)(28,38,116,146)(43,191,210,107)(44,190,197,106)(45,189,198,105)(46,188,199,104)(47,187,200,103)(48,186,201,102)(49,185,202,101)(50,184,203,100)(51,183,204,99)(52,196,205,112)(53,195,206,111)(54,194,207,110)(55,193,208,109)(56,192,209,108)(57,158,91,131)(58,157,92,130)(59,156,93,129)(60,155,94,128)(61,168,95,127)(62,167,96,140)(63,166,97,139)(64,165,98,138)(65,164,85,137)(66,163,86,136)(67,162,87,135)(68,161,88,134)(69,160,89,133)(70,159,90,132), (15,117)(16,118)(17,119)(18,120)(19,121)(20,122)(21,123)(22,124)(23,125)(24,126)(25,113)(26,114)(27,115)(28,116)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,210)(44,197)(45,198)(46,199)(47,200)(48,201)(49,202)(50,203)(51,204)(52,205)(53,206)(54,207)(55,208)(56,209)(71,171)(72,172)(73,173)(74,174)(75,175)(76,176)(77,177)(78,178)(79,179)(80,180)(81,181)(82,182)(83,169)(84,170)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(127,161)(128,162)(129,163)(130,164)(131,165)(132,166)(133,167)(134,168)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196), (1,112,219,189)(2,111,220,188)(3,110,221,187)(4,109,222,186)(5,108,223,185)(6,107,224,184)(7,106,211,183)(8,105,212,196)(9,104,213,195)(10,103,214,194)(11,102,215,193)(12,101,216,192)(13,100,217,191)(14,99,218,190)(15,167,124,133)(16,166,125,132)(17,165,126,131)(18,164,113,130)(19,163,114,129)(20,162,115,128)(21,161,116,127)(22,160,117,140)(23,159,118,139)(24,158,119,138)(25,157,120,137)(26,156,121,136)(27,155,122,135)(28,168,123,134)(29,97,144,70)(30,96,145,69)(31,95,146,68)(32,94,147,67)(33,93,148,66)(34,92,149,65)(35,91,150,64)(36,90,151,63)(37,89,152,62)(38,88,153,61)(39,87,154,60)(40,86,141,59)(41,85,142,58)(42,98,143,57)(43,77,203,177)(44,76,204,176)(45,75,205,175)(46,74,206,174)(47,73,207,173)(48,72,208,172)(49,71,209,171)(50,84,210,170)(51,83,197,169)(52,82,198,182)(53,81,199,181)(54,80,200,180)(55,79,201,179)(56,78,202,178), (1,89,219,62)(2,90,220,63)(3,91,221,64)(4,92,222,65)(5,93,223,66)(6,94,224,67)(7,95,211,68)(8,96,212,69)(9,97,213,70)(10,98,214,57)(11,85,215,58)(12,86,216,59)(13,87,217,60)(14,88,218,61)(15,45,124,205)(16,46,125,206)(17,47,126,207)(18,48,113,208)(19,49,114,209)(20,50,115,210)(21,51,116,197)(22,52,117,198)(23,53,118,199)(24,54,119,200)(25,55,120,201)(26,56,121,202)(27,43,122,203)(28,44,123,204)(29,195,144,104)(30,196,145,105)(31,183,146,106)(32,184,147,107)(33,185,148,108)(34,186,149,109)(35,187,150,110)(36,188,151,111)(37,189,152,112)(38,190,153,99)(39,191,154,100)(40,192,141,101)(41,193,142,102)(42,194,143,103)(71,163,171,129)(72,164,172,130)(73,165,173,131)(74,166,174,132)(75,167,175,133)(76,168,176,134)(77,155,177,135)(78,156,178,136)(79,157,179,137)(80,158,180,138)(81,159,181,139)(82,160,182,140)(83,161,169,127)(84,162,170,128) );
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)], [(1,175,212,82),(2,174,213,81),(3,173,214,80),(4,172,215,79),(5,171,216,78),(6,170,217,77),(7,169,218,76),(8,182,219,75),(9,181,220,74),(10,180,221,73),(11,179,222,72),(12,178,223,71),(13,177,224,84),(14,176,211,83),(15,37,117,145),(16,36,118,144),(17,35,119,143),(18,34,120,142),(19,33,121,141),(20,32,122,154),(21,31,123,153),(22,30,124,152),(23,29,125,151),(24,42,126,150),(25,41,113,149),(26,40,114,148),(27,39,115,147),(28,38,116,146),(43,191,210,107),(44,190,197,106),(45,189,198,105),(46,188,199,104),(47,187,200,103),(48,186,201,102),(49,185,202,101),(50,184,203,100),(51,183,204,99),(52,196,205,112),(53,195,206,111),(54,194,207,110),(55,193,208,109),(56,192,209,108),(57,158,91,131),(58,157,92,130),(59,156,93,129),(60,155,94,128),(61,168,95,127),(62,167,96,140),(63,166,97,139),(64,165,98,138),(65,164,85,137),(66,163,86,136),(67,162,87,135),(68,161,88,134),(69,160,89,133),(70,159,90,132)], [(15,117),(16,118),(17,119),(18,120),(19,121),(20,122),(21,123),(22,124),(23,125),(24,126),(25,113),(26,114),(27,115),(28,116),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,210),(44,197),(45,198),(46,199),(47,200),(48,201),(49,202),(50,203),(51,204),(52,205),(53,206),(54,207),(55,208),(56,209),(71,171),(72,172),(73,173),(74,174),(75,175),(76,176),(77,177),(78,178),(79,179),(80,180),(81,181),(82,182),(83,169),(84,170),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(127,161),(128,162),(129,163),(130,164),(131,165),(132,166),(133,167),(134,168),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160),(141,148),(142,149),(143,150),(144,151),(145,152),(146,153),(147,154),(183,190),(184,191),(185,192),(186,193),(187,194),(188,195),(189,196)], [(1,112,219,189),(2,111,220,188),(3,110,221,187),(4,109,222,186),(5,108,223,185),(6,107,224,184),(7,106,211,183),(8,105,212,196),(9,104,213,195),(10,103,214,194),(11,102,215,193),(12,101,216,192),(13,100,217,191),(14,99,218,190),(15,167,124,133),(16,166,125,132),(17,165,126,131),(18,164,113,130),(19,163,114,129),(20,162,115,128),(21,161,116,127),(22,160,117,140),(23,159,118,139),(24,158,119,138),(25,157,120,137),(26,156,121,136),(27,155,122,135),(28,168,123,134),(29,97,144,70),(30,96,145,69),(31,95,146,68),(32,94,147,67),(33,93,148,66),(34,92,149,65),(35,91,150,64),(36,90,151,63),(37,89,152,62),(38,88,153,61),(39,87,154,60),(40,86,141,59),(41,85,142,58),(42,98,143,57),(43,77,203,177),(44,76,204,176),(45,75,205,175),(46,74,206,174),(47,73,207,173),(48,72,208,172),(49,71,209,171),(50,84,210,170),(51,83,197,169),(52,82,198,182),(53,81,199,181),(54,80,200,180),(55,79,201,179),(56,78,202,178)], [(1,89,219,62),(2,90,220,63),(3,91,221,64),(4,92,222,65),(5,93,223,66),(6,94,224,67),(7,95,211,68),(8,96,212,69),(9,97,213,70),(10,98,214,57),(11,85,215,58),(12,86,216,59),(13,87,217,60),(14,88,218,61),(15,45,124,205),(16,46,125,206),(17,47,126,207),(18,48,113,208),(19,49,114,209),(20,50,115,210),(21,51,116,197),(22,52,117,198),(23,53,118,199),(24,54,119,200),(25,55,120,201),(26,56,121,202),(27,43,122,203),(28,44,123,204),(29,195,144,104),(30,196,145,105),(31,183,146,106),(32,184,147,107),(33,185,148,108),(34,186,149,109),(35,187,150,110),(36,188,151,111),(37,189,152,112),(38,190,153,99),(39,191,154,100),(40,192,141,101),(41,193,142,102),(42,194,143,103),(71,163,171,129),(72,164,172,130),(73,165,173,131),(74,166,174,132),(75,167,175,133),(76,168,176,134),(77,155,177,135),(78,156,178,136),(79,157,179,137),(80,158,180,138),(81,159,181,139),(82,160,182,140),(83,161,169,127),(84,162,170,128)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | ··· | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 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 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | 2- 1+4 | D4⋊2D7 | D4⋊6D14 | D4.10D14 |
| kernel | C14.712- 1+4 | C22⋊Dic14 | C23.D14 | Dic7⋊3Q8 | C28⋊Q8 | C28.48D4 | C23.21D14 | D4×Dic7 | C23.18D14 | C28.17D4 | C7×C4⋊D4 | C4⋊D4 | C28 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 1 | 3 | 4 | 6 | 3 | 3 | 9 | 1 | 1 | 6 | 6 | 6 |
Matrix representation of C14.712- 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 0 |
| 0 | 0 | 0 | 0 | 0 | 22 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 24 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 9 |
| 0 | 0 | 0 | 0 | 20 | 14 |
| 0 | 0 | 14 | 20 | 0 | 0 |
| 0 | 0 | 9 | 15 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 27 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 17 | 12 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 15 |
| 0 | 0 | 0 | 0 | 14 | 20 |
| 0 | 0 | 20 | 14 | 0 | 0 |
| 0 | 0 | 15 | 9 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 9 | 0 | 0 |
| 0 | 0 | 20 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 9 |
| 0 | 0 | 0 | 0 | 20 | 14 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,22,0,0,0,0,0,0,22],[12,24,0,0,0,0,0,17,0,0,0,0,0,0,0,0,14,9,0,0,0,0,20,15,0,0,15,20,0,0,0,0,9,14,0,0],[28,27,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[17,5,0,0,0,0,12,12,0,0,0,0,0,0,0,0,20,15,0,0,0,0,14,9,0,0,9,14,0,0,0,0,15,20,0,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,15,20,0,0,0,0,9,14,0,0,0,0,0,0,15,20,0,0,0,0,9,14] >;
C14.712- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{71}2_-^{1+4} % in TeX
G:=Group("C14.71ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1056);
// by ID
G=gap.SmallGroup(448,1056);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,219,675,570,297,18822]);
// 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=a^7*b^-1,d*b*d^-1=a^7*b,b*e=e*b,d*c*d^-1=a^7*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations