metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.32- 1+4, C7⋊D4⋊9D4, C28⋊Q8⋊9C2, C28⋊8(C4○D4), C4⋊4(C4○D28), C4⋊D28⋊8C2, C7⋊1(D4⋊6D4), D28⋊C4⋊9C2, C28⋊7D4⋊27C2, C4⋊C4.262D14, D14⋊Q8⋊1C2, D14.13(C2×D4), C22.19(D4×D7), D14.5D4⋊1C2, (C2×C14).51C24, Dic7.15(C2×D4), C14.43(C22×D4), (C2×C28).136C23, D14⋊C4.61C22, (C22×C4).177D14, C22.85(C23×D7), (C2×D28).135C22, C4⋊Dic7.191C22, (C4×Dic7).64C22, C23.225(C22×D7), C23.D7.87C22, C23.23D14⋊11C2, Dic7⋊C4.148C22, (C22×C28).102C22, (C22×C14).400C23, C2.6(Q8.10D14), (C2×Dic7).189C23, (C22×D7).157C23, (C2×Dic14).227C22, (D7×C4⋊C4)⋊9C2, C2.16(C2×D4×D7), (C2×C4⋊C4)⋊16D7, (C14×C4⋊C4)⋊13C2, (C2×C4○D28)⋊4C2, (C4×C7⋊D4)⋊10C2, C2.22(C2×C4○D28), C14.20(C2×C4○D4), (C2×C14).389(C2×D4), (C2×C4×D7).190C22, (C7×C4⋊C4).295C22, (C2×C4).571(C22×D7), (C2×C7⋊D4).92C22, SmallGroup(448,960)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.2- 1+4
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=a7b2, e2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a7b-1, dbd-1=a7b, be=eb, dcd-1=a7c, ce=ec, ede-1=b2d >
Subgroups: 1380 in 292 conjugacy classes, 107 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, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, D4⋊6D4, C4×Dic7, Dic7⋊C4, Dic7⋊C4, C4⋊Dic7, D14⋊C4, D14⋊C4, C23.D7, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, C4○D28, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C22×C28, C28⋊Q8, D7×C4⋊C4, D28⋊C4, D14.5D4, C4⋊D28, D14⋊Q8, C4×C7⋊D4, C23.23D14, C28⋊7D4, C14×C4⋊C4, C2×C4○D28, C14.2- 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, 2- 1+4, C22×D7, D4⋊6D4, C4○D28, D4×D7, C23×D7, C2×C4○D28, C2×D4×D7, Q8.10D14, C14.2- 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)
(1 206 28 144)(2 207 15 145)(3 208 16 146)(4 209 17 147)(5 210 18 148)(6 197 19 149)(7 198 20 150)(8 199 21 151)(9 200 22 152)(10 201 23 153)(11 202 24 154)(12 203 25 141)(13 204 26 142)(14 205 27 143)(29 83 177 118)(30 84 178 119)(31 71 179 120)(32 72 180 121)(33 73 181 122)(34 74 182 123)(35 75 169 124)(36 76 170 125)(37 77 171 126)(38 78 172 113)(39 79 173 114)(40 80 174 115)(41 81 175 116)(42 82 176 117)(43 131 87 102)(44 132 88 103)(45 133 89 104)(46 134 90 105)(47 135 91 106)(48 136 92 107)(49 137 93 108)(50 138 94 109)(51 139 95 110)(52 140 96 111)(53 127 97 112)(54 128 98 99)(55 129 85 100)(56 130 86 101)(57 224 157 193)(58 211 158 194)(59 212 159 195)(60 213 160 196)(61 214 161 183)(62 215 162 184)(63 216 163 185)(64 217 164 186)(65 218 165 187)(66 219 166 188)(67 220 167 189)(68 221 168 190)(69 222 155 191)(70 223 156 192)
(1 206 8 199)(2 205 9 198)(3 204 10 197)(4 203 11 210)(5 202 12 209)(6 201 13 208)(7 200 14 207)(15 143 22 150)(16 142 23 149)(17 141 24 148)(18 154 25 147)(19 153 26 146)(20 152 27 145)(21 151 28 144)(29 81 36 74)(30 80 37 73)(31 79 38 72)(32 78 39 71)(33 77 40 84)(34 76 41 83)(35 75 42 82)(43 103 50 110)(44 102 51 109)(45 101 52 108)(46 100 53 107)(47 99 54 106)(48 112 55 105)(49 111 56 104)(57 190 64 183)(58 189 65 196)(59 188 66 195)(60 187 67 194)(61 186 68 193)(62 185 69 192)(63 184 70 191)(85 134 92 127)(86 133 93 140)(87 132 94 139)(88 131 95 138)(89 130 96 137)(90 129 97 136)(91 128 98 135)(113 173 120 180)(114 172 121 179)(115 171 122 178)(116 170 123 177)(117 169 124 176)(118 182 125 175)(119 181 126 174)(155 223 162 216)(156 222 163 215)(157 221 164 214)(158 220 165 213)(159 219 166 212)(160 218 167 211)(161 217 168 224)
(1 35 21 176)(2 34 22 175)(3 33 23 174)(4 32 24 173)(5 31 25 172)(6 30 26 171)(7 29 27 170)(8 42 28 169)(9 41 15 182)(10 40 16 181)(11 39 17 180)(12 38 18 179)(13 37 19 178)(14 36 20 177)(43 163 94 70)(44 162 95 69)(45 161 96 68)(46 160 97 67)(47 159 98 66)(48 158 85 65)(49 157 86 64)(50 156 87 63)(51 155 88 62)(52 168 89 61)(53 167 90 60)(54 166 91 59)(55 165 92 58)(56 164 93 57)(71 148 113 203)(72 147 114 202)(73 146 115 201)(74 145 116 200)(75 144 117 199)(76 143 118 198)(77 142 119 197)(78 141 120 210)(79 154 121 209)(80 153 122 208)(81 152 123 207)(82 151 124 206)(83 150 125 205)(84 149 126 204)(99 212 135 188)(100 211 136 187)(101 224 137 186)(102 223 138 185)(103 222 139 184)(104 221 140 183)(105 220 127 196)(106 219 128 195)(107 218 129 194)(108 217 130 193)(109 216 131 192)(110 215 132 191)(111 214 133 190)(112 213 134 189)
(1 219 28 188)(2 220 15 189)(3 221 16 190)(4 222 17 191)(5 223 18 192)(6 224 19 193)(7 211 20 194)(8 212 21 195)(9 213 22 196)(10 214 23 183)(11 215 24 184)(12 216 25 185)(13 217 26 186)(14 218 27 187)(29 107 177 136)(30 108 178 137)(31 109 179 138)(32 110 180 139)(33 111 181 140)(34 112 182 127)(35 99 169 128)(36 100 170 129)(37 101 171 130)(38 102 172 131)(39 103 173 132)(40 104 174 133)(41 105 175 134)(42 106 176 135)(43 113 87 78)(44 114 88 79)(45 115 89 80)(46 116 90 81)(47 117 91 82)(48 118 92 83)(49 119 93 84)(50 120 94 71)(51 121 95 72)(52 122 96 73)(53 123 97 74)(54 124 98 75)(55 125 85 76)(56 126 86 77)(57 197 157 149)(58 198 158 150)(59 199 159 151)(60 200 160 152)(61 201 161 153)(62 202 162 154)(63 203 163 141)(64 204 164 142)(65 205 165 143)(66 206 166 144)(67 207 167 145)(68 208 168 146)(69 209 155 147)(70 210 156 148)
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), (1,206,28,144)(2,207,15,145)(3,208,16,146)(4,209,17,147)(5,210,18,148)(6,197,19,149)(7,198,20,150)(8,199,21,151)(9,200,22,152)(10,201,23,153)(11,202,24,154)(12,203,25,141)(13,204,26,142)(14,205,27,143)(29,83,177,118)(30,84,178,119)(31,71,179,120)(32,72,180,121)(33,73,181,122)(34,74,182,123)(35,75,169,124)(36,76,170,125)(37,77,171,126)(38,78,172,113)(39,79,173,114)(40,80,174,115)(41,81,175,116)(42,82,176,117)(43,131,87,102)(44,132,88,103)(45,133,89,104)(46,134,90,105)(47,135,91,106)(48,136,92,107)(49,137,93,108)(50,138,94,109)(51,139,95,110)(52,140,96,111)(53,127,97,112)(54,128,98,99)(55,129,85,100)(56,130,86,101)(57,224,157,193)(58,211,158,194)(59,212,159,195)(60,213,160,196)(61,214,161,183)(62,215,162,184)(63,216,163,185)(64,217,164,186)(65,218,165,187)(66,219,166,188)(67,220,167,189)(68,221,168,190)(69,222,155,191)(70,223,156,192), (1,206,8,199)(2,205,9,198)(3,204,10,197)(4,203,11,210)(5,202,12,209)(6,201,13,208)(7,200,14,207)(15,143,22,150)(16,142,23,149)(17,141,24,148)(18,154,25,147)(19,153,26,146)(20,152,27,145)(21,151,28,144)(29,81,36,74)(30,80,37,73)(31,79,38,72)(32,78,39,71)(33,77,40,84)(34,76,41,83)(35,75,42,82)(43,103,50,110)(44,102,51,109)(45,101,52,108)(46,100,53,107)(47,99,54,106)(48,112,55,105)(49,111,56,104)(57,190,64,183)(58,189,65,196)(59,188,66,195)(60,187,67,194)(61,186,68,193)(62,185,69,192)(63,184,70,191)(85,134,92,127)(86,133,93,140)(87,132,94,139)(88,131,95,138)(89,130,96,137)(90,129,97,136)(91,128,98,135)(113,173,120,180)(114,172,121,179)(115,171,122,178)(116,170,123,177)(117,169,124,176)(118,182,125,175)(119,181,126,174)(155,223,162,216)(156,222,163,215)(157,221,164,214)(158,220,165,213)(159,219,166,212)(160,218,167,211)(161,217,168,224), (1,35,21,176)(2,34,22,175)(3,33,23,174)(4,32,24,173)(5,31,25,172)(6,30,26,171)(7,29,27,170)(8,42,28,169)(9,41,15,182)(10,40,16,181)(11,39,17,180)(12,38,18,179)(13,37,19,178)(14,36,20,177)(43,163,94,70)(44,162,95,69)(45,161,96,68)(46,160,97,67)(47,159,98,66)(48,158,85,65)(49,157,86,64)(50,156,87,63)(51,155,88,62)(52,168,89,61)(53,167,90,60)(54,166,91,59)(55,165,92,58)(56,164,93,57)(71,148,113,203)(72,147,114,202)(73,146,115,201)(74,145,116,200)(75,144,117,199)(76,143,118,198)(77,142,119,197)(78,141,120,210)(79,154,121,209)(80,153,122,208)(81,152,123,207)(82,151,124,206)(83,150,125,205)(84,149,126,204)(99,212,135,188)(100,211,136,187)(101,224,137,186)(102,223,138,185)(103,222,139,184)(104,221,140,183)(105,220,127,196)(106,219,128,195)(107,218,129,194)(108,217,130,193)(109,216,131,192)(110,215,132,191)(111,214,133,190)(112,213,134,189), (1,219,28,188)(2,220,15,189)(3,221,16,190)(4,222,17,191)(5,223,18,192)(6,224,19,193)(7,211,20,194)(8,212,21,195)(9,213,22,196)(10,214,23,183)(11,215,24,184)(12,216,25,185)(13,217,26,186)(14,218,27,187)(29,107,177,136)(30,108,178,137)(31,109,179,138)(32,110,180,139)(33,111,181,140)(34,112,182,127)(35,99,169,128)(36,100,170,129)(37,101,171,130)(38,102,172,131)(39,103,173,132)(40,104,174,133)(41,105,175,134)(42,106,176,135)(43,113,87,78)(44,114,88,79)(45,115,89,80)(46,116,90,81)(47,117,91,82)(48,118,92,83)(49,119,93,84)(50,120,94,71)(51,121,95,72)(52,122,96,73)(53,123,97,74)(54,124,98,75)(55,125,85,76)(56,126,86,77)(57,197,157,149)(58,198,158,150)(59,199,159,151)(60,200,160,152)(61,201,161,153)(62,202,162,154)(63,203,163,141)(64,204,164,142)(65,205,165,143)(66,206,166,144)(67,207,167,145)(68,208,168,146)(69,209,155,147)(70,210,156,148)>;
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), (1,206,28,144)(2,207,15,145)(3,208,16,146)(4,209,17,147)(5,210,18,148)(6,197,19,149)(7,198,20,150)(8,199,21,151)(9,200,22,152)(10,201,23,153)(11,202,24,154)(12,203,25,141)(13,204,26,142)(14,205,27,143)(29,83,177,118)(30,84,178,119)(31,71,179,120)(32,72,180,121)(33,73,181,122)(34,74,182,123)(35,75,169,124)(36,76,170,125)(37,77,171,126)(38,78,172,113)(39,79,173,114)(40,80,174,115)(41,81,175,116)(42,82,176,117)(43,131,87,102)(44,132,88,103)(45,133,89,104)(46,134,90,105)(47,135,91,106)(48,136,92,107)(49,137,93,108)(50,138,94,109)(51,139,95,110)(52,140,96,111)(53,127,97,112)(54,128,98,99)(55,129,85,100)(56,130,86,101)(57,224,157,193)(58,211,158,194)(59,212,159,195)(60,213,160,196)(61,214,161,183)(62,215,162,184)(63,216,163,185)(64,217,164,186)(65,218,165,187)(66,219,166,188)(67,220,167,189)(68,221,168,190)(69,222,155,191)(70,223,156,192), (1,206,8,199)(2,205,9,198)(3,204,10,197)(4,203,11,210)(5,202,12,209)(6,201,13,208)(7,200,14,207)(15,143,22,150)(16,142,23,149)(17,141,24,148)(18,154,25,147)(19,153,26,146)(20,152,27,145)(21,151,28,144)(29,81,36,74)(30,80,37,73)(31,79,38,72)(32,78,39,71)(33,77,40,84)(34,76,41,83)(35,75,42,82)(43,103,50,110)(44,102,51,109)(45,101,52,108)(46,100,53,107)(47,99,54,106)(48,112,55,105)(49,111,56,104)(57,190,64,183)(58,189,65,196)(59,188,66,195)(60,187,67,194)(61,186,68,193)(62,185,69,192)(63,184,70,191)(85,134,92,127)(86,133,93,140)(87,132,94,139)(88,131,95,138)(89,130,96,137)(90,129,97,136)(91,128,98,135)(113,173,120,180)(114,172,121,179)(115,171,122,178)(116,170,123,177)(117,169,124,176)(118,182,125,175)(119,181,126,174)(155,223,162,216)(156,222,163,215)(157,221,164,214)(158,220,165,213)(159,219,166,212)(160,218,167,211)(161,217,168,224), (1,35,21,176)(2,34,22,175)(3,33,23,174)(4,32,24,173)(5,31,25,172)(6,30,26,171)(7,29,27,170)(8,42,28,169)(9,41,15,182)(10,40,16,181)(11,39,17,180)(12,38,18,179)(13,37,19,178)(14,36,20,177)(43,163,94,70)(44,162,95,69)(45,161,96,68)(46,160,97,67)(47,159,98,66)(48,158,85,65)(49,157,86,64)(50,156,87,63)(51,155,88,62)(52,168,89,61)(53,167,90,60)(54,166,91,59)(55,165,92,58)(56,164,93,57)(71,148,113,203)(72,147,114,202)(73,146,115,201)(74,145,116,200)(75,144,117,199)(76,143,118,198)(77,142,119,197)(78,141,120,210)(79,154,121,209)(80,153,122,208)(81,152,123,207)(82,151,124,206)(83,150,125,205)(84,149,126,204)(99,212,135,188)(100,211,136,187)(101,224,137,186)(102,223,138,185)(103,222,139,184)(104,221,140,183)(105,220,127,196)(106,219,128,195)(107,218,129,194)(108,217,130,193)(109,216,131,192)(110,215,132,191)(111,214,133,190)(112,213,134,189), (1,219,28,188)(2,220,15,189)(3,221,16,190)(4,222,17,191)(5,223,18,192)(6,224,19,193)(7,211,20,194)(8,212,21,195)(9,213,22,196)(10,214,23,183)(11,215,24,184)(12,216,25,185)(13,217,26,186)(14,218,27,187)(29,107,177,136)(30,108,178,137)(31,109,179,138)(32,110,180,139)(33,111,181,140)(34,112,182,127)(35,99,169,128)(36,100,170,129)(37,101,171,130)(38,102,172,131)(39,103,173,132)(40,104,174,133)(41,105,175,134)(42,106,176,135)(43,113,87,78)(44,114,88,79)(45,115,89,80)(46,116,90,81)(47,117,91,82)(48,118,92,83)(49,119,93,84)(50,120,94,71)(51,121,95,72)(52,122,96,73)(53,123,97,74)(54,124,98,75)(55,125,85,76)(56,126,86,77)(57,197,157,149)(58,198,158,150)(59,199,159,151)(60,200,160,152)(61,201,161,153)(62,202,162,154)(63,203,163,141)(64,204,164,142)(65,205,165,143)(66,206,166,144)(67,207,167,145)(68,208,168,146)(69,209,155,147)(70,210,156,148) );
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)], [(1,206,28,144),(2,207,15,145),(3,208,16,146),(4,209,17,147),(5,210,18,148),(6,197,19,149),(7,198,20,150),(8,199,21,151),(9,200,22,152),(10,201,23,153),(11,202,24,154),(12,203,25,141),(13,204,26,142),(14,205,27,143),(29,83,177,118),(30,84,178,119),(31,71,179,120),(32,72,180,121),(33,73,181,122),(34,74,182,123),(35,75,169,124),(36,76,170,125),(37,77,171,126),(38,78,172,113),(39,79,173,114),(40,80,174,115),(41,81,175,116),(42,82,176,117),(43,131,87,102),(44,132,88,103),(45,133,89,104),(46,134,90,105),(47,135,91,106),(48,136,92,107),(49,137,93,108),(50,138,94,109),(51,139,95,110),(52,140,96,111),(53,127,97,112),(54,128,98,99),(55,129,85,100),(56,130,86,101),(57,224,157,193),(58,211,158,194),(59,212,159,195),(60,213,160,196),(61,214,161,183),(62,215,162,184),(63,216,163,185),(64,217,164,186),(65,218,165,187),(66,219,166,188),(67,220,167,189),(68,221,168,190),(69,222,155,191),(70,223,156,192)], [(1,206,8,199),(2,205,9,198),(3,204,10,197),(4,203,11,210),(5,202,12,209),(6,201,13,208),(7,200,14,207),(15,143,22,150),(16,142,23,149),(17,141,24,148),(18,154,25,147),(19,153,26,146),(20,152,27,145),(21,151,28,144),(29,81,36,74),(30,80,37,73),(31,79,38,72),(32,78,39,71),(33,77,40,84),(34,76,41,83),(35,75,42,82),(43,103,50,110),(44,102,51,109),(45,101,52,108),(46,100,53,107),(47,99,54,106),(48,112,55,105),(49,111,56,104),(57,190,64,183),(58,189,65,196),(59,188,66,195),(60,187,67,194),(61,186,68,193),(62,185,69,192),(63,184,70,191),(85,134,92,127),(86,133,93,140),(87,132,94,139),(88,131,95,138),(89,130,96,137),(90,129,97,136),(91,128,98,135),(113,173,120,180),(114,172,121,179),(115,171,122,178),(116,170,123,177),(117,169,124,176),(118,182,125,175),(119,181,126,174),(155,223,162,216),(156,222,163,215),(157,221,164,214),(158,220,165,213),(159,219,166,212),(160,218,167,211),(161,217,168,224)], [(1,35,21,176),(2,34,22,175),(3,33,23,174),(4,32,24,173),(5,31,25,172),(6,30,26,171),(7,29,27,170),(8,42,28,169),(9,41,15,182),(10,40,16,181),(11,39,17,180),(12,38,18,179),(13,37,19,178),(14,36,20,177),(43,163,94,70),(44,162,95,69),(45,161,96,68),(46,160,97,67),(47,159,98,66),(48,158,85,65),(49,157,86,64),(50,156,87,63),(51,155,88,62),(52,168,89,61),(53,167,90,60),(54,166,91,59),(55,165,92,58),(56,164,93,57),(71,148,113,203),(72,147,114,202),(73,146,115,201),(74,145,116,200),(75,144,117,199),(76,143,118,198),(77,142,119,197),(78,141,120,210),(79,154,121,209),(80,153,122,208),(81,152,123,207),(82,151,124,206),(83,150,125,205),(84,149,126,204),(99,212,135,188),(100,211,136,187),(101,224,137,186),(102,223,138,185),(103,222,139,184),(104,221,140,183),(105,220,127,196),(106,219,128,195),(107,218,129,194),(108,217,130,193),(109,216,131,192),(110,215,132,191),(111,214,133,190),(112,213,134,189)], [(1,219,28,188),(2,220,15,189),(3,221,16,190),(4,222,17,191),(5,223,18,192),(6,224,19,193),(7,211,20,194),(8,212,21,195),(9,213,22,196),(10,214,23,183),(11,215,24,184),(12,216,25,185),(13,217,26,186),(14,218,27,187),(29,107,177,136),(30,108,178,137),(31,109,179,138),(32,110,180,139),(33,111,181,140),(34,112,182,127),(35,99,169,128),(36,100,170,129),(37,101,171,130),(38,102,172,131),(39,103,173,132),(40,104,174,133),(41,105,175,134),(42,106,176,135),(43,113,87,78),(44,114,88,79),(45,115,89,80),(46,116,90,81),(47,117,91,82),(48,118,92,83),(49,119,93,84),(50,120,94,71),(51,121,95,72),(52,122,96,73),(53,123,97,74),(54,124,98,75),(55,125,85,76),(56,126,86,77),(57,197,157,149),(58,198,158,150),(59,199,159,151),(60,200,160,152),(61,201,161,153),(62,202,162,154),(63,203,163,141),(64,204,164,142),(65,205,165,143),(66,206,166,144),(67,207,167,145),(68,208,168,146),(69,209,155,147),(70,210,156,148)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 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 | 14 | 14 | 28 | 28 | 2 | ··· | 2 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 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 | D4 | D7 | C4○D4 | D14 | D14 | C4○D28 | 2- 1+4 | D4×D7 | Q8.10D14 |
| kernel | C14.2- 1+4 | C28⋊Q8 | D7×C4⋊C4 | D28⋊C4 | D14.5D4 | C4⋊D28 | D14⋊Q8 | C4×C7⋊D4 | C23.23D14 | C28⋊7D4 | C14×C4⋊C4 | C2×C4○D28 | C7⋊D4 | C2×C4⋊C4 | C28 | C4⋊C4 | C22×C4 | C4 | C14 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 | 3 | 4 | 12 | 9 | 24 | 1 | 6 | 6 |
Matrix representation of C14.2- 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 10 | 0 | 0 |
| 0 | 0 | 14 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 13 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 24 |
| 0 | 0 | 0 | 0 | 5 | 13 |
| 28 | 16 | 0 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 22 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 24 |
| 0 | 0 | 0 | 0 | 5 | 13 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 13 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 25 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 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 | 16 | 24 |
| 0 | 0 | 0 | 0 | 5 | 13 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,4,14,0,0,0,0,10,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,13,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,5,0,0,0,0,24,13],[28,18,0,0,0,0,16,1,0,0,0,0,0,0,0,4,0,0,0,0,22,0,0,0,0,0,0,0,16,5,0,0,0,0,24,13],[17,13,0,0,0,0,0,12,0,0,0,0,0,0,0,25,0,0,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,5,0,0,0,0,24,13] >;
C14.2- 1+4 in GAP, Magma, Sage, TeX
C_{14}.2_-^{1+4} % in TeX
G:=Group("C14.ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,960);
// by ID
G=gap.SmallGroup(448,960);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,100,675,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=a^7,d^2=a^7*b^2,e^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=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=b^2*d>;
// generators/relations