G = C14.212- 1+4 order 448 = 26·7
21st non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.212- 1+4, C22⋊Q8⋊16D7, C4⋊C4.194D14, (Q8×Dic7)⋊14C2, D14⋊3Q8⋊21C2, Dic7.Q8⋊21C2, (C2×C28).61C23, (C2×Q8).130D14, C22⋊C4.19D14, C28.3Q8⋊24C2, D14.20(C4○D4), C28.211(C4○D4), (C2×C14).183C24, D14.D4.2C2, (C22×C4).245D14, C4.100(D4⋊2D7), D14⋊C4.129C22, C23.D14⋊23C2, Dic7⋊C4.31C22, C4⋊Dic7.375C22, (Q8×C14).113C22, (C2×Dic7).93C23, C22.204(C23×D7), C23.122(C22×D7), C23.21D14⋊29C2, (C22×C14).211C23, (C22×C28).259C22, C7⋊6(C22.46C24), (C4×Dic7).111C22, (C22×D7).204C23, C23.D7.123C22, C2.22(Q8.10D14), (D7×C4⋊C4)⋊30C2, C2.54(D7×C4○D4), (C4×C7⋊D4).9C2, C4⋊C4⋊7D7⋊27C2, (C7×C22⋊Q8)⋊19C2, C14.166(C2×C4○D4), C2.47(C2×D4⋊2D7), (C2×C4×D7).101C22, (C2×C4).53(C22×D7), (C7×C4⋊C4).164C22, (C2×C7⋊D4).130C22, (C7×C22⋊C4).38C22, SmallGroup(448,1092)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.212- 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, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 828 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×Q8, C22×D7, C22×C14, C22.46C24, C4×Dic7, C4×Dic7, Dic7⋊C4, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, D14⋊C4, D14⋊C4, C23.D7, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C4×D7, C2×C4×D7, C2×C7⋊D4, C22×C28, Q8×C14, C23.D14, D14.D4, Dic7.Q8, C28.3Q8, D7×C4⋊C4, C4⋊C4⋊7D7, C23.21D14, C4×C7⋊D4, Q8×Dic7, D14⋊3Q8, C7×C22⋊Q8, C14.212- 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2- 1+4, C22×D7, C22.46C24, D4⋊2D7, C23×D7, C2×D4⋊2D7, Q8.10D14, D7×C4○D4, C14.212- 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)
(1 162 27 149)(2 163 28 150)(3 164 15 151)(4 165 16 152)(5 166 17 153)(6 167 18 154)(7 168 19 141)(8 155 20 142)(9 156 21 143)(10 157 22 144)(11 158 23 145)(12 159 24 146)(13 160 25 147)(14 161 26 148)(29 129 44 123)(30 130 45 124)(31 131 46 125)(32 132 47 126)(33 133 48 113)(34 134 49 114)(35 135 50 115)(36 136 51 116)(37 137 52 117)(38 138 53 118)(39 139 54 119)(40 140 55 120)(41 127 56 121)(42 128 43 122)(57 213 71 199)(58 214 72 200)(59 215 73 201)(60 216 74 202)(61 217 75 203)(62 218 76 204)(63 219 77 205)(64 220 78 206)(65 221 79 207)(66 222 80 208)(67 223 81 209)(68 224 82 210)(69 211 83 197)(70 212 84 198)(85 191 107 172)(86 192 108 173)(87 193 109 174)(88 194 110 175)(89 195 111 176)(90 196 112 177)(91 183 99 178)(92 184 100 179)(93 185 101 180)(94 186 102 181)(95 187 103 182)(96 188 104 169)(97 189 105 170)(98 190 106 171)
(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)(15 160 22 167)(16 159 23 166)(17 158 24 165)(18 157 25 164)(19 156 26 163)(20 155 27 162)(21 168 28 161)(29 128 36 135)(30 127 37 134)(31 140 38 133)(32 139 39 132)(33 138 40 131)(34 137 41 130)(35 136 42 129)(43 123 50 116)(44 122 51 115)(45 121 52 114)(46 120 53 113)(47 119 54 126)(48 118 55 125)(49 117 56 124)(57 207 64 200)(58 206 65 199)(59 205 66 198)(60 204 67 197)(61 203 68 210)(62 202 69 209)(63 201 70 208)(71 221 78 214)(72 220 79 213)(73 219 80 212)(74 218 81 211)(75 217 82 224)(76 216 83 223)(77 215 84 222)(85 186 92 193)(86 185 93 192)(87 184 94 191)(88 183 95 190)(89 196 96 189)(90 195 97 188)(91 194 98 187)(99 175 106 182)(100 174 107 181)(101 173 108 180)(102 172 109 179)(103 171 110 178)(104 170 111 177)(105 169 112 176)
(1 68 20 75)(2 67 21 74)(3 66 22 73)(4 65 23 72)(5 64 24 71)(6 63 25 84)(7 62 26 83)(8 61 27 82)(9 60 28 81)(10 59 15 80)(11 58 16 79)(12 57 17 78)(13 70 18 77)(14 69 19 76)(29 111 51 96)(30 110 52 95)(31 109 53 94)(32 108 54 93)(33 107 55 92)(34 106 56 91)(35 105 43 90)(36 104 44 89)(37 103 45 88)(38 102 46 87)(39 101 47 86)(40 100 48 85)(41 99 49 98)(42 112 50 97)(113 191 140 179)(114 190 127 178)(115 189 128 177)(116 188 129 176)(117 187 130 175)(118 186 131 174)(119 185 132 173)(120 184 133 172)(121 183 134 171)(122 196 135 170)(123 195 136 169)(124 194 137 182)(125 193 138 181)(126 192 139 180)(141 204 161 211)(142 203 162 224)(143 202 163 223)(144 201 164 222)(145 200 165 221)(146 199 166 220)(147 198 167 219)(148 197 168 218)(149 210 155 217)(150 209 156 216)(151 208 157 215)(152 207 158 214)(153 206 159 213)(154 205 160 212)
(1 139 27 119)(2 140 28 120)(3 127 15 121)(4 128 16 122)(5 129 17 123)(6 130 18 124)(7 131 19 125)(8 132 20 126)(9 133 21 113)(10 134 22 114)(11 135 23 115)(12 136 24 116)(13 137 25 117)(14 138 26 118)(29 166 44 153)(30 167 45 154)(31 168 46 141)(32 155 47 142)(33 156 48 143)(34 157 49 144)(35 158 50 145)(36 159 51 146)(37 160 52 147)(38 161 53 148)(39 162 54 149)(40 163 55 150)(41 164 56 151)(42 165 43 152)(57 188 71 169)(58 189 72 170)(59 190 73 171)(60 191 74 172)(61 192 75 173)(62 193 76 174)(63 194 77 175)(64 195 78 176)(65 196 79 177)(66 183 80 178)(67 184 81 179)(68 185 82 180)(69 186 83 181)(70 187 84 182)(85 216 107 202)(86 217 108 203)(87 218 109 204)(88 219 110 205)(89 220 111 206)(90 221 112 207)(91 222 99 208)(92 223 100 209)(93 224 101 210)(94 211 102 197)(95 212 103 198)(96 213 104 199)(97 214 105 200)(98 215 106 201)
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), (1,162,27,149)(2,163,28,150)(3,164,15,151)(4,165,16,152)(5,166,17,153)(6,167,18,154)(7,168,19,141)(8,155,20,142)(9,156,21,143)(10,157,22,144)(11,158,23,145)(12,159,24,146)(13,160,25,147)(14,161,26,148)(29,129,44,123)(30,130,45,124)(31,131,46,125)(32,132,47,126)(33,133,48,113)(34,134,49,114)(35,135,50,115)(36,136,51,116)(37,137,52,117)(38,138,53,118)(39,139,54,119)(40,140,55,120)(41,127,56,121)(42,128,43,122)(57,213,71,199)(58,214,72,200)(59,215,73,201)(60,216,74,202)(61,217,75,203)(62,218,76,204)(63,219,77,205)(64,220,78,206)(65,221,79,207)(66,222,80,208)(67,223,81,209)(68,224,82,210)(69,211,83,197)(70,212,84,198)(85,191,107,172)(86,192,108,173)(87,193,109,174)(88,194,110,175)(89,195,111,176)(90,196,112,177)(91,183,99,178)(92,184,100,179)(93,185,101,180)(94,186,102,181)(95,187,103,182)(96,188,104,169)(97,189,105,170)(98,190,106,171), (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)(15,160,22,167)(16,159,23,166)(17,158,24,165)(18,157,25,164)(19,156,26,163)(20,155,27,162)(21,168,28,161)(29,128,36,135)(30,127,37,134)(31,140,38,133)(32,139,39,132)(33,138,40,131)(34,137,41,130)(35,136,42,129)(43,123,50,116)(44,122,51,115)(45,121,52,114)(46,120,53,113)(47,119,54,126)(48,118,55,125)(49,117,56,124)(57,207,64,200)(58,206,65,199)(59,205,66,198)(60,204,67,197)(61,203,68,210)(62,202,69,209)(63,201,70,208)(71,221,78,214)(72,220,79,213)(73,219,80,212)(74,218,81,211)(75,217,82,224)(76,216,83,223)(77,215,84,222)(85,186,92,193)(86,185,93,192)(87,184,94,191)(88,183,95,190)(89,196,96,189)(90,195,97,188)(91,194,98,187)(99,175,106,182)(100,174,107,181)(101,173,108,180)(102,172,109,179)(103,171,110,178)(104,170,111,177)(105,169,112,176), (1,68,20,75)(2,67,21,74)(3,66,22,73)(4,65,23,72)(5,64,24,71)(6,63,25,84)(7,62,26,83)(8,61,27,82)(9,60,28,81)(10,59,15,80)(11,58,16,79)(12,57,17,78)(13,70,18,77)(14,69,19,76)(29,111,51,96)(30,110,52,95)(31,109,53,94)(32,108,54,93)(33,107,55,92)(34,106,56,91)(35,105,43,90)(36,104,44,89)(37,103,45,88)(38,102,46,87)(39,101,47,86)(40,100,48,85)(41,99,49,98)(42,112,50,97)(113,191,140,179)(114,190,127,178)(115,189,128,177)(116,188,129,176)(117,187,130,175)(118,186,131,174)(119,185,132,173)(120,184,133,172)(121,183,134,171)(122,196,135,170)(123,195,136,169)(124,194,137,182)(125,193,138,181)(126,192,139,180)(141,204,161,211)(142,203,162,224)(143,202,163,223)(144,201,164,222)(145,200,165,221)(146,199,166,220)(147,198,167,219)(148,197,168,218)(149,210,155,217)(150,209,156,216)(151,208,157,215)(152,207,158,214)(153,206,159,213)(154,205,160,212), (1,139,27,119)(2,140,28,120)(3,127,15,121)(4,128,16,122)(5,129,17,123)(6,130,18,124)(7,131,19,125)(8,132,20,126)(9,133,21,113)(10,134,22,114)(11,135,23,115)(12,136,24,116)(13,137,25,117)(14,138,26,118)(29,166,44,153)(30,167,45,154)(31,168,46,141)(32,155,47,142)(33,156,48,143)(34,157,49,144)(35,158,50,145)(36,159,51,146)(37,160,52,147)(38,161,53,148)(39,162,54,149)(40,163,55,150)(41,164,56,151)(42,165,43,152)(57,188,71,169)(58,189,72,170)(59,190,73,171)(60,191,74,172)(61,192,75,173)(62,193,76,174)(63,194,77,175)(64,195,78,176)(65,196,79,177)(66,183,80,178)(67,184,81,179)(68,185,82,180)(69,186,83,181)(70,187,84,182)(85,216,107,202)(86,217,108,203)(87,218,109,204)(88,219,110,205)(89,220,111,206)(90,221,112,207)(91,222,99,208)(92,223,100,209)(93,224,101,210)(94,211,102,197)(95,212,103,198)(96,213,104,199)(97,214,105,200)(98,215,106,201)>;
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), (1,162,27,149)(2,163,28,150)(3,164,15,151)(4,165,16,152)(5,166,17,153)(6,167,18,154)(7,168,19,141)(8,155,20,142)(9,156,21,143)(10,157,22,144)(11,158,23,145)(12,159,24,146)(13,160,25,147)(14,161,26,148)(29,129,44,123)(30,130,45,124)(31,131,46,125)(32,132,47,126)(33,133,48,113)(34,134,49,114)(35,135,50,115)(36,136,51,116)(37,137,52,117)(38,138,53,118)(39,139,54,119)(40,140,55,120)(41,127,56,121)(42,128,43,122)(57,213,71,199)(58,214,72,200)(59,215,73,201)(60,216,74,202)(61,217,75,203)(62,218,76,204)(63,219,77,205)(64,220,78,206)(65,221,79,207)(66,222,80,208)(67,223,81,209)(68,224,82,210)(69,211,83,197)(70,212,84,198)(85,191,107,172)(86,192,108,173)(87,193,109,174)(88,194,110,175)(89,195,111,176)(90,196,112,177)(91,183,99,178)(92,184,100,179)(93,185,101,180)(94,186,102,181)(95,187,103,182)(96,188,104,169)(97,189,105,170)(98,190,106,171), (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)(15,160,22,167)(16,159,23,166)(17,158,24,165)(18,157,25,164)(19,156,26,163)(20,155,27,162)(21,168,28,161)(29,128,36,135)(30,127,37,134)(31,140,38,133)(32,139,39,132)(33,138,40,131)(34,137,41,130)(35,136,42,129)(43,123,50,116)(44,122,51,115)(45,121,52,114)(46,120,53,113)(47,119,54,126)(48,118,55,125)(49,117,56,124)(57,207,64,200)(58,206,65,199)(59,205,66,198)(60,204,67,197)(61,203,68,210)(62,202,69,209)(63,201,70,208)(71,221,78,214)(72,220,79,213)(73,219,80,212)(74,218,81,211)(75,217,82,224)(76,216,83,223)(77,215,84,222)(85,186,92,193)(86,185,93,192)(87,184,94,191)(88,183,95,190)(89,196,96,189)(90,195,97,188)(91,194,98,187)(99,175,106,182)(100,174,107,181)(101,173,108,180)(102,172,109,179)(103,171,110,178)(104,170,111,177)(105,169,112,176), (1,68,20,75)(2,67,21,74)(3,66,22,73)(4,65,23,72)(5,64,24,71)(6,63,25,84)(7,62,26,83)(8,61,27,82)(9,60,28,81)(10,59,15,80)(11,58,16,79)(12,57,17,78)(13,70,18,77)(14,69,19,76)(29,111,51,96)(30,110,52,95)(31,109,53,94)(32,108,54,93)(33,107,55,92)(34,106,56,91)(35,105,43,90)(36,104,44,89)(37,103,45,88)(38,102,46,87)(39,101,47,86)(40,100,48,85)(41,99,49,98)(42,112,50,97)(113,191,140,179)(114,190,127,178)(115,189,128,177)(116,188,129,176)(117,187,130,175)(118,186,131,174)(119,185,132,173)(120,184,133,172)(121,183,134,171)(122,196,135,170)(123,195,136,169)(124,194,137,182)(125,193,138,181)(126,192,139,180)(141,204,161,211)(142,203,162,224)(143,202,163,223)(144,201,164,222)(145,200,165,221)(146,199,166,220)(147,198,167,219)(148,197,168,218)(149,210,155,217)(150,209,156,216)(151,208,157,215)(152,207,158,214)(153,206,159,213)(154,205,160,212), (1,139,27,119)(2,140,28,120)(3,127,15,121)(4,128,16,122)(5,129,17,123)(6,130,18,124)(7,131,19,125)(8,132,20,126)(9,133,21,113)(10,134,22,114)(11,135,23,115)(12,136,24,116)(13,137,25,117)(14,138,26,118)(29,166,44,153)(30,167,45,154)(31,168,46,141)(32,155,47,142)(33,156,48,143)(34,157,49,144)(35,158,50,145)(36,159,51,146)(37,160,52,147)(38,161,53,148)(39,162,54,149)(40,163,55,150)(41,164,56,151)(42,165,43,152)(57,188,71,169)(58,189,72,170)(59,190,73,171)(60,191,74,172)(61,192,75,173)(62,193,76,174)(63,194,77,175)(64,195,78,176)(65,196,79,177)(66,183,80,178)(67,184,81,179)(68,185,82,180)(69,186,83,181)(70,187,84,182)(85,216,107,202)(86,217,108,203)(87,218,109,204)(88,219,110,205)(89,220,111,206)(90,221,112,207)(91,222,99,208)(92,223,100,209)(93,224,101,210)(94,211,102,197)(95,212,103,198)(96,213,104,199)(97,214,105,200)(98,215,106,201) );
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)], [(1,162,27,149),(2,163,28,150),(3,164,15,151),(4,165,16,152),(5,166,17,153),(6,167,18,154),(7,168,19,141),(8,155,20,142),(9,156,21,143),(10,157,22,144),(11,158,23,145),(12,159,24,146),(13,160,25,147),(14,161,26,148),(29,129,44,123),(30,130,45,124),(31,131,46,125),(32,132,47,126),(33,133,48,113),(34,134,49,114),(35,135,50,115),(36,136,51,116),(37,137,52,117),(38,138,53,118),(39,139,54,119),(40,140,55,120),(41,127,56,121),(42,128,43,122),(57,213,71,199),(58,214,72,200),(59,215,73,201),(60,216,74,202),(61,217,75,203),(62,218,76,204),(63,219,77,205),(64,220,78,206),(65,221,79,207),(66,222,80,208),(67,223,81,209),(68,224,82,210),(69,211,83,197),(70,212,84,198),(85,191,107,172),(86,192,108,173),(87,193,109,174),(88,194,110,175),(89,195,111,176),(90,196,112,177),(91,183,99,178),(92,184,100,179),(93,185,101,180),(94,186,102,181),(95,187,103,182),(96,188,104,169),(97,189,105,170),(98,190,106,171)], [(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),(15,160,22,167),(16,159,23,166),(17,158,24,165),(18,157,25,164),(19,156,26,163),(20,155,27,162),(21,168,28,161),(29,128,36,135),(30,127,37,134),(31,140,38,133),(32,139,39,132),(33,138,40,131),(34,137,41,130),(35,136,42,129),(43,123,50,116),(44,122,51,115),(45,121,52,114),(46,120,53,113),(47,119,54,126),(48,118,55,125),(49,117,56,124),(57,207,64,200),(58,206,65,199),(59,205,66,198),(60,204,67,197),(61,203,68,210),(62,202,69,209),(63,201,70,208),(71,221,78,214),(72,220,79,213),(73,219,80,212),(74,218,81,211),(75,217,82,224),(76,216,83,223),(77,215,84,222),(85,186,92,193),(86,185,93,192),(87,184,94,191),(88,183,95,190),(89,196,96,189),(90,195,97,188),(91,194,98,187),(99,175,106,182),(100,174,107,181),(101,173,108,180),(102,172,109,179),(103,171,110,178),(104,170,111,177),(105,169,112,176)], [(1,68,20,75),(2,67,21,74),(3,66,22,73),(4,65,23,72),(5,64,24,71),(6,63,25,84),(7,62,26,83),(8,61,27,82),(9,60,28,81),(10,59,15,80),(11,58,16,79),(12,57,17,78),(13,70,18,77),(14,69,19,76),(29,111,51,96),(30,110,52,95),(31,109,53,94),(32,108,54,93),(33,107,55,92),(34,106,56,91),(35,105,43,90),(36,104,44,89),(37,103,45,88),(38,102,46,87),(39,101,47,86),(40,100,48,85),(41,99,49,98),(42,112,50,97),(113,191,140,179),(114,190,127,178),(115,189,128,177),(116,188,129,176),(117,187,130,175),(118,186,131,174),(119,185,132,173),(120,184,133,172),(121,183,134,171),(122,196,135,170),(123,195,136,169),(124,194,137,182),(125,193,138,181),(126,192,139,180),(141,204,161,211),(142,203,162,224),(143,202,163,223),(144,201,164,222),(145,200,165,221),(146,199,166,220),(147,198,167,219),(148,197,168,218),(149,210,155,217),(150,209,156,216),(151,208,157,215),(152,207,158,214),(153,206,159,213),(154,205,160,212)], [(1,139,27,119),(2,140,28,120),(3,127,15,121),(4,128,16,122),(5,129,17,123),(6,130,18,124),(7,131,19,125),(8,132,20,126),(9,133,21,113),(10,134,22,114),(11,135,23,115),(12,136,24,116),(13,137,25,117),(14,138,26,118),(29,166,44,153),(30,167,45,154),(31,168,46,141),(32,155,47,142),(33,156,48,143),(34,157,49,144),(35,158,50,145),(36,159,51,146),(37,160,52,147),(38,161,53,148),(39,162,54,149),(40,163,55,150),(41,164,56,151),(42,165,43,152),(57,188,71,169),(58,189,72,170),(59,190,73,171),(60,191,74,172),(61,192,75,173),(62,193,76,174),(63,194,77,175),(64,195,78,176),(65,196,79,177),(66,183,80,178),(67,184,81,179),(68,185,82,180),(69,186,83,181),(70,187,84,182),(85,216,107,202),(86,217,108,203),(87,218,109,204),(88,219,110,205),(89,220,111,206),(90,221,112,207),(91,222,99,208),(92,223,100,209),(93,224,101,210),(94,211,102,197),(95,212,103,198),(96,213,104,199),(97,214,105,200),(98,215,106,201)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 4Q | 4R | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 14 | 14 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | 2- 1+4 | D4⋊2D7 | Q8.10D14 | D7×C4○D4 |
| kernel | C14.212- 1+4 | C23.D14 | D14.D4 | Dic7.Q8 | C28.3Q8 | D7×C4⋊C4 | C4⋊C4⋊7D7 | C23.21D14 | C4×C7⋊D4 | Q8×Dic7 | D14⋊3Q8 | C7×C22⋊Q8 | C22⋊Q8 | C28 | D14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 6 | 9 | 3 | 3 | 1 | 6 | 6 | 6 |
Matrix representation of C14.212- 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 19 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 3 |
| 0 | 0 | 0 | 0 | 8 | 21 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,10,19,0,0,0,0,10,22,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,1,7,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,1,7,0,0,0,0,0,28,0,0,0,0,0,0,8,8,0,0,0,0,3,21],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,5,17] >;
C14.212- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{21}2_-^{1+4} % in TeX
G:=Group("C14.21ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1092);
// by ID
G=gap.SmallGroup(448,1092);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,268,1571,297,18822]);
// 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,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations