G = C14.222- 1+4 order 448 = 26·7
22nd non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.222- 1+4, C14.552+ 1+4, C22⋊Q8⋊17D7, C4⋊C4.195D14, (C2×Q8).77D14, D14⋊Q8⋊23C2, D14⋊D4.2C2, C22⋊C4.20D14, Dic7⋊Q8⋊17C2, Dic7⋊3Q8⋊28C2, D14.5D4⋊21C2, C28.23D4⋊15C2, (C2×C28).628C23, (C2×C14).184C24, D14⋊C4.26C22, Dic7.7(C4○D4), (C22×C4).246D14, C2.57(D4⋊6D14), Dic7.D4⋊26C2, (C2×D28).152C22, C23.D14⋊24C2, Dic7⋊C4.32C22, C4⋊Dic7.219C22, (Q8×C14).114C22, (C22×D7).75C23, C23.123(C22×D7), C22.205(C23×D7), C23.D7.35C22, C23.23D14⋊25C2, (C22×C28).383C22, (C22×C14).212C23, C7⋊5(C22.36C24), (C2×Dic7).238C23, (C4×Dic7).112C22, C2.23(Q8.10D14), (C2×Dic14).162C22, (C4×C7⋊D4)⋊59C2, C2.55(D7×C4○D4), C4⋊C4⋊7D7⋊28C2, C4⋊C4⋊D7⋊19C2, (C7×C22⋊Q8)⋊20C2, C14.167(C2×C4○D4), (C2×C4×D7).211C22, (C2×C4).54(C22×D7), (C7×C4⋊C4).165C22, (C2×C7⋊D4).131C22, (C7×C22⋊C4).39C22, SmallGroup(448,1093)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.222- 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, cd=dc, ce=ec, ede-1=a7b2d >
Subgroups: 988 in 216 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic7, Dic7, C28, D14, C2×C14, C2×C14, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4⋊Q8, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×Q8, C22×D7, C22×C14, C22.36C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C2×C7⋊D4, C22×C28, Q8×C14, C23.D14, D14⋊D4, Dic7.D4, Dic7⋊3Q8, C4⋊C4⋊7D7, D14.5D4, D14⋊Q8, C4⋊C4⋊D7, C4×C7⋊D4, C23.23D14, Dic7⋊Q8, C28.23D4, C7×C22⋊Q8, C14.222- 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, C23×D7, D4⋊6D14, Q8.10D14, D7×C4○D4, C14.222- 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 78 19 64)(2 79 20 65)(3 80 21 66)(4 81 22 67)(5 82 23 68)(6 83 24 69)(7 84 25 70)(8 71 26 57)(9 72 27 58)(10 73 28 59)(11 74 15 60)(12 75 16 61)(13 76 17 62)(14 77 18 63)(29 99 43 88)(30 100 44 89)(31 101 45 90)(32 102 46 91)(33 103 47 92)(34 104 48 93)(35 105 49 94)(36 106 50 95)(37 107 51 96)(38 108 52 97)(39 109 53 98)(40 110 54 85)(41 111 55 86)(42 112 56 87)(113 178 139 193)(114 179 140 194)(115 180 127 195)(116 181 128 196)(117 182 129 183)(118 169 130 184)(119 170 131 185)(120 171 132 186)(121 172 133 187)(122 173 134 188)(123 174 135 189)(124 175 136 190)(125 176 137 191)(126 177 138 192)(141 201 168 219)(142 202 155 220)(143 203 156 221)(144 204 157 222)(145 205 158 223)(146 206 159 224)(147 207 160 211)(148 208 161 212)(149 209 162 213)(150 210 163 214)(151 197 164 215)(152 198 165 216)(153 199 166 217)(154 200 167 218)
(1 192 8 185)(2 191 9 184)(3 190 10 183)(4 189 11 196)(5 188 12 195)(6 187 13 194)(7 186 14 193)(15 181 22 174)(16 180 23 173)(17 179 24 172)(18 178 25 171)(19 177 26 170)(20 176 27 169)(21 175 28 182)(29 223 36 216)(30 222 37 215)(31 221 38 214)(32 220 39 213)(33 219 40 212)(34 218 41 211)(35 217 42 224)(43 205 50 198)(44 204 51 197)(45 203 52 210)(46 202 53 209)(47 201 54 208)(48 200 55 207)(49 199 56 206)(57 126 64 119)(58 125 65 118)(59 124 66 117)(60 123 67 116)(61 122 68 115)(62 121 69 114)(63 120 70 113)(71 138 78 131)(72 137 79 130)(73 136 80 129)(74 135 81 128)(75 134 82 127)(76 133 83 140)(77 132 84 139)(85 141 92 148)(86 154 93 147)(87 153 94 146)(88 152 95 145)(89 151 96 144)(90 150 97 143)(91 149 98 142)(99 165 106 158)(100 164 107 157)(101 163 108 156)(102 162 109 155)(103 161 110 168)(104 160 111 167)(105 159 112 166)
(1 153 26 159)(2 152 27 158)(3 151 28 157)(4 150 15 156)(5 149 16 155)(6 148 17 168)(7 147 18 167)(8 146 19 166)(9 145 20 165)(10 144 21 164)(11 143 22 163)(12 142 23 162)(13 141 24 161)(14 154 25 160)(29 130 50 125)(30 129 51 124)(31 128 52 123)(32 127 53 122)(33 140 54 121)(34 139 55 120)(35 138 56 119)(36 137 43 118)(37 136 44 117)(38 135 45 116)(39 134 46 115)(40 133 47 114)(41 132 48 113)(42 131 49 126)(57 217 78 206)(58 216 79 205)(59 215 80 204)(60 214 81 203)(61 213 82 202)(62 212 83 201)(63 211 84 200)(64 224 71 199)(65 223 72 198)(66 222 73 197)(67 221 74 210)(68 220 75 209)(69 219 76 208)(70 218 77 207)(85 179 103 187)(86 178 104 186)(87 177 105 185)(88 176 106 184)(89 175 107 183)(90 174 108 196)(91 173 109 195)(92 172 110 194)(93 171 111 193)(94 170 112 192)(95 169 99 191)(96 182 100 190)(97 181 101 189)(98 180 102 188)
(1 71 19 57)(2 72 20 58)(3 73 21 59)(4 74 22 60)(5 75 23 61)(6 76 24 62)(7 77 25 63)(8 78 26 64)(9 79 27 65)(10 80 28 66)(11 81 15 67)(12 82 16 68)(13 83 17 69)(14 84 18 70)(29 99 43 88)(30 100 44 89)(31 101 45 90)(32 102 46 91)(33 103 47 92)(34 104 48 93)(35 105 49 94)(36 106 50 95)(37 107 51 96)(38 108 52 97)(39 109 53 98)(40 110 54 85)(41 111 55 86)(42 112 56 87)(113 193 139 178)(114 194 140 179)(115 195 127 180)(116 196 128 181)(117 183 129 182)(118 184 130 169)(119 185 131 170)(120 186 132 171)(121 187 133 172)(122 188 134 173)(123 189 135 174)(124 190 136 175)(125 191 137 176)(126 192 138 177)(141 212 168 208)(142 213 155 209)(143 214 156 210)(144 215 157 197)(145 216 158 198)(146 217 159 199)(147 218 160 200)(148 219 161 201)(149 220 162 202)(150 221 163 203)(151 222 164 204)(152 223 165 205)(153 224 166 206)(154 211 167 207)
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), (1,78,19,64)(2,79,20,65)(3,80,21,66)(4,81,22,67)(5,82,23,68)(6,83,24,69)(7,84,25,70)(8,71,26,57)(9,72,27,58)(10,73,28,59)(11,74,15,60)(12,75,16,61)(13,76,17,62)(14,77,18,63)(29,99,43,88)(30,100,44,89)(31,101,45,90)(32,102,46,91)(33,103,47,92)(34,104,48,93)(35,105,49,94)(36,106,50,95)(37,107,51,96)(38,108,52,97)(39,109,53,98)(40,110,54,85)(41,111,55,86)(42,112,56,87)(113,178,139,193)(114,179,140,194)(115,180,127,195)(116,181,128,196)(117,182,129,183)(118,169,130,184)(119,170,131,185)(120,171,132,186)(121,172,133,187)(122,173,134,188)(123,174,135,189)(124,175,136,190)(125,176,137,191)(126,177,138,192)(141,201,168,219)(142,202,155,220)(143,203,156,221)(144,204,157,222)(145,205,158,223)(146,206,159,224)(147,207,160,211)(148,208,161,212)(149,209,162,213)(150,210,163,214)(151,197,164,215)(152,198,165,216)(153,199,166,217)(154,200,167,218), (1,192,8,185)(2,191,9,184)(3,190,10,183)(4,189,11,196)(5,188,12,195)(6,187,13,194)(7,186,14,193)(15,181,22,174)(16,180,23,173)(17,179,24,172)(18,178,25,171)(19,177,26,170)(20,176,27,169)(21,175,28,182)(29,223,36,216)(30,222,37,215)(31,221,38,214)(32,220,39,213)(33,219,40,212)(34,218,41,211)(35,217,42,224)(43,205,50,198)(44,204,51,197)(45,203,52,210)(46,202,53,209)(47,201,54,208)(48,200,55,207)(49,199,56,206)(57,126,64,119)(58,125,65,118)(59,124,66,117)(60,123,67,116)(61,122,68,115)(62,121,69,114)(63,120,70,113)(71,138,78,131)(72,137,79,130)(73,136,80,129)(74,135,81,128)(75,134,82,127)(76,133,83,140)(77,132,84,139)(85,141,92,148)(86,154,93,147)(87,153,94,146)(88,152,95,145)(89,151,96,144)(90,150,97,143)(91,149,98,142)(99,165,106,158)(100,164,107,157)(101,163,108,156)(102,162,109,155)(103,161,110,168)(104,160,111,167)(105,159,112,166), (1,153,26,159)(2,152,27,158)(3,151,28,157)(4,150,15,156)(5,149,16,155)(6,148,17,168)(7,147,18,167)(8,146,19,166)(9,145,20,165)(10,144,21,164)(11,143,22,163)(12,142,23,162)(13,141,24,161)(14,154,25,160)(29,130,50,125)(30,129,51,124)(31,128,52,123)(32,127,53,122)(33,140,54,121)(34,139,55,120)(35,138,56,119)(36,137,43,118)(37,136,44,117)(38,135,45,116)(39,134,46,115)(40,133,47,114)(41,132,48,113)(42,131,49,126)(57,217,78,206)(58,216,79,205)(59,215,80,204)(60,214,81,203)(61,213,82,202)(62,212,83,201)(63,211,84,200)(64,224,71,199)(65,223,72,198)(66,222,73,197)(67,221,74,210)(68,220,75,209)(69,219,76,208)(70,218,77,207)(85,179,103,187)(86,178,104,186)(87,177,105,185)(88,176,106,184)(89,175,107,183)(90,174,108,196)(91,173,109,195)(92,172,110,194)(93,171,111,193)(94,170,112,192)(95,169,99,191)(96,182,100,190)(97,181,101,189)(98,180,102,188), (1,71,19,57)(2,72,20,58)(3,73,21,59)(4,74,22,60)(5,75,23,61)(6,76,24,62)(7,77,25,63)(8,78,26,64)(9,79,27,65)(10,80,28,66)(11,81,15,67)(12,82,16,68)(13,83,17,69)(14,84,18,70)(29,99,43,88)(30,100,44,89)(31,101,45,90)(32,102,46,91)(33,103,47,92)(34,104,48,93)(35,105,49,94)(36,106,50,95)(37,107,51,96)(38,108,52,97)(39,109,53,98)(40,110,54,85)(41,111,55,86)(42,112,56,87)(113,193,139,178)(114,194,140,179)(115,195,127,180)(116,196,128,181)(117,183,129,182)(118,184,130,169)(119,185,131,170)(120,186,132,171)(121,187,133,172)(122,188,134,173)(123,189,135,174)(124,190,136,175)(125,191,137,176)(126,192,138,177)(141,212,168,208)(142,213,155,209)(143,214,156,210)(144,215,157,197)(145,216,158,198)(146,217,159,199)(147,218,160,200)(148,219,161,201)(149,220,162,202)(150,221,163,203)(151,222,164,204)(152,223,165,205)(153,224,166,206)(154,211,167,207)>;
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), (1,78,19,64)(2,79,20,65)(3,80,21,66)(4,81,22,67)(5,82,23,68)(6,83,24,69)(7,84,25,70)(8,71,26,57)(9,72,27,58)(10,73,28,59)(11,74,15,60)(12,75,16,61)(13,76,17,62)(14,77,18,63)(29,99,43,88)(30,100,44,89)(31,101,45,90)(32,102,46,91)(33,103,47,92)(34,104,48,93)(35,105,49,94)(36,106,50,95)(37,107,51,96)(38,108,52,97)(39,109,53,98)(40,110,54,85)(41,111,55,86)(42,112,56,87)(113,178,139,193)(114,179,140,194)(115,180,127,195)(116,181,128,196)(117,182,129,183)(118,169,130,184)(119,170,131,185)(120,171,132,186)(121,172,133,187)(122,173,134,188)(123,174,135,189)(124,175,136,190)(125,176,137,191)(126,177,138,192)(141,201,168,219)(142,202,155,220)(143,203,156,221)(144,204,157,222)(145,205,158,223)(146,206,159,224)(147,207,160,211)(148,208,161,212)(149,209,162,213)(150,210,163,214)(151,197,164,215)(152,198,165,216)(153,199,166,217)(154,200,167,218), (1,192,8,185)(2,191,9,184)(3,190,10,183)(4,189,11,196)(5,188,12,195)(6,187,13,194)(7,186,14,193)(15,181,22,174)(16,180,23,173)(17,179,24,172)(18,178,25,171)(19,177,26,170)(20,176,27,169)(21,175,28,182)(29,223,36,216)(30,222,37,215)(31,221,38,214)(32,220,39,213)(33,219,40,212)(34,218,41,211)(35,217,42,224)(43,205,50,198)(44,204,51,197)(45,203,52,210)(46,202,53,209)(47,201,54,208)(48,200,55,207)(49,199,56,206)(57,126,64,119)(58,125,65,118)(59,124,66,117)(60,123,67,116)(61,122,68,115)(62,121,69,114)(63,120,70,113)(71,138,78,131)(72,137,79,130)(73,136,80,129)(74,135,81,128)(75,134,82,127)(76,133,83,140)(77,132,84,139)(85,141,92,148)(86,154,93,147)(87,153,94,146)(88,152,95,145)(89,151,96,144)(90,150,97,143)(91,149,98,142)(99,165,106,158)(100,164,107,157)(101,163,108,156)(102,162,109,155)(103,161,110,168)(104,160,111,167)(105,159,112,166), (1,153,26,159)(2,152,27,158)(3,151,28,157)(4,150,15,156)(5,149,16,155)(6,148,17,168)(7,147,18,167)(8,146,19,166)(9,145,20,165)(10,144,21,164)(11,143,22,163)(12,142,23,162)(13,141,24,161)(14,154,25,160)(29,130,50,125)(30,129,51,124)(31,128,52,123)(32,127,53,122)(33,140,54,121)(34,139,55,120)(35,138,56,119)(36,137,43,118)(37,136,44,117)(38,135,45,116)(39,134,46,115)(40,133,47,114)(41,132,48,113)(42,131,49,126)(57,217,78,206)(58,216,79,205)(59,215,80,204)(60,214,81,203)(61,213,82,202)(62,212,83,201)(63,211,84,200)(64,224,71,199)(65,223,72,198)(66,222,73,197)(67,221,74,210)(68,220,75,209)(69,219,76,208)(70,218,77,207)(85,179,103,187)(86,178,104,186)(87,177,105,185)(88,176,106,184)(89,175,107,183)(90,174,108,196)(91,173,109,195)(92,172,110,194)(93,171,111,193)(94,170,112,192)(95,169,99,191)(96,182,100,190)(97,181,101,189)(98,180,102,188), (1,71,19,57)(2,72,20,58)(3,73,21,59)(4,74,22,60)(5,75,23,61)(6,76,24,62)(7,77,25,63)(8,78,26,64)(9,79,27,65)(10,80,28,66)(11,81,15,67)(12,82,16,68)(13,83,17,69)(14,84,18,70)(29,99,43,88)(30,100,44,89)(31,101,45,90)(32,102,46,91)(33,103,47,92)(34,104,48,93)(35,105,49,94)(36,106,50,95)(37,107,51,96)(38,108,52,97)(39,109,53,98)(40,110,54,85)(41,111,55,86)(42,112,56,87)(113,193,139,178)(114,194,140,179)(115,195,127,180)(116,196,128,181)(117,183,129,182)(118,184,130,169)(119,185,131,170)(120,186,132,171)(121,187,133,172)(122,188,134,173)(123,189,135,174)(124,190,136,175)(125,191,137,176)(126,192,138,177)(141,212,168,208)(142,213,155,209)(143,214,156,210)(144,215,157,197)(145,216,158,198)(146,217,159,199)(147,218,160,200)(148,219,161,201)(149,220,162,202)(150,221,163,203)(151,222,164,204)(152,223,165,205)(153,224,166,206)(154,211,167,207) );
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)], [(1,78,19,64),(2,79,20,65),(3,80,21,66),(4,81,22,67),(5,82,23,68),(6,83,24,69),(7,84,25,70),(8,71,26,57),(9,72,27,58),(10,73,28,59),(11,74,15,60),(12,75,16,61),(13,76,17,62),(14,77,18,63),(29,99,43,88),(30,100,44,89),(31,101,45,90),(32,102,46,91),(33,103,47,92),(34,104,48,93),(35,105,49,94),(36,106,50,95),(37,107,51,96),(38,108,52,97),(39,109,53,98),(40,110,54,85),(41,111,55,86),(42,112,56,87),(113,178,139,193),(114,179,140,194),(115,180,127,195),(116,181,128,196),(117,182,129,183),(118,169,130,184),(119,170,131,185),(120,171,132,186),(121,172,133,187),(122,173,134,188),(123,174,135,189),(124,175,136,190),(125,176,137,191),(126,177,138,192),(141,201,168,219),(142,202,155,220),(143,203,156,221),(144,204,157,222),(145,205,158,223),(146,206,159,224),(147,207,160,211),(148,208,161,212),(149,209,162,213),(150,210,163,214),(151,197,164,215),(152,198,165,216),(153,199,166,217),(154,200,167,218)], [(1,192,8,185),(2,191,9,184),(3,190,10,183),(4,189,11,196),(5,188,12,195),(6,187,13,194),(7,186,14,193),(15,181,22,174),(16,180,23,173),(17,179,24,172),(18,178,25,171),(19,177,26,170),(20,176,27,169),(21,175,28,182),(29,223,36,216),(30,222,37,215),(31,221,38,214),(32,220,39,213),(33,219,40,212),(34,218,41,211),(35,217,42,224),(43,205,50,198),(44,204,51,197),(45,203,52,210),(46,202,53,209),(47,201,54,208),(48,200,55,207),(49,199,56,206),(57,126,64,119),(58,125,65,118),(59,124,66,117),(60,123,67,116),(61,122,68,115),(62,121,69,114),(63,120,70,113),(71,138,78,131),(72,137,79,130),(73,136,80,129),(74,135,81,128),(75,134,82,127),(76,133,83,140),(77,132,84,139),(85,141,92,148),(86,154,93,147),(87,153,94,146),(88,152,95,145),(89,151,96,144),(90,150,97,143),(91,149,98,142),(99,165,106,158),(100,164,107,157),(101,163,108,156),(102,162,109,155),(103,161,110,168),(104,160,111,167),(105,159,112,166)], [(1,153,26,159),(2,152,27,158),(3,151,28,157),(4,150,15,156),(5,149,16,155),(6,148,17,168),(7,147,18,167),(8,146,19,166),(9,145,20,165),(10,144,21,164),(11,143,22,163),(12,142,23,162),(13,141,24,161),(14,154,25,160),(29,130,50,125),(30,129,51,124),(31,128,52,123),(32,127,53,122),(33,140,54,121),(34,139,55,120),(35,138,56,119),(36,137,43,118),(37,136,44,117),(38,135,45,116),(39,134,46,115),(40,133,47,114),(41,132,48,113),(42,131,49,126),(57,217,78,206),(58,216,79,205),(59,215,80,204),(60,214,81,203),(61,213,82,202),(62,212,83,201),(63,211,84,200),(64,224,71,199),(65,223,72,198),(66,222,73,197),(67,221,74,210),(68,220,75,209),(69,219,76,208),(70,218,77,207),(85,179,103,187),(86,178,104,186),(87,177,105,185),(88,176,106,184),(89,175,107,183),(90,174,108,196),(91,173,109,195),(92,172,110,194),(93,171,111,193),(94,170,112,192),(95,169,99,191),(96,182,100,190),(97,181,101,189),(98,180,102,188)], [(1,71,19,57),(2,72,20,58),(3,73,21,59),(4,74,22,60),(5,75,23,61),(6,76,24,62),(7,77,25,63),(8,78,26,64),(9,79,27,65),(10,80,28,66),(11,81,15,67),(12,82,16,68),(13,83,17,69),(14,84,18,70),(29,99,43,88),(30,100,44,89),(31,101,45,90),(32,102,46,91),(33,103,47,92),(34,104,48,93),(35,105,49,94),(36,106,50,95),(37,107,51,96),(38,108,52,97),(39,109,53,98),(40,110,54,85),(41,111,55,86),(42,112,56,87),(113,193,139,178),(114,194,140,179),(115,195,127,180),(116,196,128,181),(117,183,129,182),(118,184,130,169),(119,185,131,170),(120,186,132,171),(121,187,133,172),(122,188,134,173),(123,189,135,174),(124,190,136,175),(125,191,137,176),(126,192,138,177),(141,212,168,208),(142,213,155,209),(143,214,156,210),(144,215,157,197),(145,216,158,198),(146,217,159,199),(147,218,160,200),(148,219,161,201),(149,220,162,202),(150,221,163,203),(151,222,164,204),(152,223,165,205),(153,224,166,206),(154,211,167,207)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 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 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 28 | 28 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | 2- 1+4 | D4⋊6D14 | Q8.10D14 | D7×C4○D4 |
| kernel | C14.222- 1+4 | C23.D14 | D14⋊D4 | Dic7.D4 | Dic7⋊3Q8 | C4⋊C4⋊7D7 | D14.5D4 | D14⋊Q8 | C4⋊C4⋊D7 | C4×C7⋊D4 | C23.23D14 | Dic7⋊Q8 | C28.23D4 | C7×C22⋊Q8 | C22⋊Q8 | Dic7 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C14 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 6 | 9 | 3 | 3 | 1 | 1 | 6 | 6 | 6 |
Matrix representation of C14.222- 1+4 ►in GL8(𝔽29)
| 18 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 25 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 2 |
| 12 | 0 | 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 26 | 22 |
| 19 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 | 6 | 6 |
| 0 | 0 | 0 | 0 | 7 | 26 | 14 | 23 |
| 0 | 0 | 0 | 0 | 26 | 26 | 26 | 26 |
| 0 | 0 | 0 | 0 | 22 | 3 | 22 | 3 |
| 25 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 18 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 8 | 4 | 25 | 0 | 0 | 0 | 0 |
| 7 | 8 | 11 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 2 | 17 | 2 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 12 |
| 17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 16 | 7 | 16 |
| 0 | 0 | 0 | 0 | 26 | 22 | 26 | 22 |
G:=sub<GL(8,GF(29))| [18,25,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,18,25,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,9,13,0,0,0,0,0,0,8,2,0,0,0,0,0,0,0,0,9,13,0,0,0,0,0,0,8,2],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,17,0,17,0,0,0,0,0,0,17,0,17,0,0,0,0,0,0,0,0,7,26,0,0,0,0,0,0,16,22,0,0,0,0,0,0,0,0,7,26,0,0,0,0,0,0,16,22],[19,16,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,19,16,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,3,7,26,22,0,0,0,0,3,26,26,3,0,0,0,0,6,14,26,22,0,0,0,0,6,23,26,3],[25,18,21,7,0,0,0,0,4,4,8,8,0,0,0,0,0,0,4,11,0,0,0,0,0,0,25,25,0,0,0,0,0,0,0,0,12,0,17,0,0,0,0,0,27,17,2,12,0,0,0,0,0,0,17,0,0,0,0,0,0,0,2,12],[17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,22,3,7,26,0,0,0,0,13,7,16,22,0,0,0,0,0,0,7,26,0,0,0,0,0,0,16,22] >;
C14.222- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{22}2_-^{1+4} % in TeX
G:=Group("C14.22ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1093);
// by ID
G=gap.SmallGroup(448,1093);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,758,555,100,1571,297,18822]);
// 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,d*b*d^-1=a^7*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations