G = C14.842- 1+4 order 448 = 26·7
39th non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.842- 1+4, C14.652+ 1+4, C28⋊Q8⋊33C2, C4⋊C4.199D14, D14⋊Q8⋊32C2, (C2×D4).102D14, C22⋊C4.29D14, Dic7⋊3Q8⋊32C2, D14.D4⋊35C2, C28.17D4⋊22C2, C28.48D4⋊23C2, (C2×C14).206C24, (C2×C28).181C23, D14⋊C4.34C22, Dic7.9(C4○D4), Dic7⋊D4.2C2, (C22×C4).260D14, C22.D4⋊11D7, C4⋊Dic7.49C22, C2.67(D4⋊6D14), C23.30(C22×D7), C22⋊Dic14⋊34C2, Dic7.D4⋊34C2, (D4×C14).144C22, C23.D14⋊34C2, Dic7⋊C4.44C22, (C22×C14).38C23, (C22×D7).87C23, C22.227(C23×D7), C23.D7.45C22, C23.11D14⋊15C2, (C22×C28).116C22, C7⋊7(C22.36C24), (C2×Dic7).107C23, (C4×Dic7).125C22, C2.45(D4.10D14), (C2×Dic14).169C22, (C22×Dic7).132C22, (C4×C7⋊D4)⋊8C2, C2.68(D7×C4○D4), C4⋊C4⋊D7⋊30C2, C14.180(C2×C4○D4), (C2×C4×D7).212C22, (C2×C4).68(C22×D7), (C7×C4⋊C4).179C22, (C2×C7⋊D4).50C22, (C7×C22.D4)⋊14C2, (C7×C22⋊C4).54C22, SmallGroup(448,1115)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.842- 1+4
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=a7b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=a7b-1, bd=db, ebe=a7b, cd=dc, ce=ec, ede=a7b2d >
Subgroups: 940 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×D4, C2×Q8, Dic7, Dic7, C28, D14, C2×C14, C2×C14, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C42⋊2C2, C4⋊Q8, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, 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, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23.11D14, C22⋊Dic14, C23.D14, D14.D4, Dic7.D4, Dic7⋊3Q8, C28⋊Q8, D14⋊Q8, C4⋊C4⋊D7, C28.48D4, C4×C7⋊D4, C28.17D4, Dic7⋊D4, C7×C22.D4, C14.842- 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, D7×C4○D4, D4.10D14, C14.842- 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)
(1 106 25 94)(2 107 26 95)(3 108 27 96)(4 109 28 97)(5 110 15 98)(6 111 16 85)(7 112 17 86)(8 99 18 87)(9 100 19 88)(10 101 20 89)(11 102 21 90)(12 103 22 91)(13 104 23 92)(14 105 24 93)(29 82 56 61)(30 83 43 62)(31 84 44 63)(32 71 45 64)(33 72 46 65)(34 73 47 66)(35 74 48 67)(36 75 49 68)(37 76 50 69)(38 77 51 70)(39 78 52 57)(40 79 53 58)(41 80 54 59)(42 81 55 60)(113 212 133 205)(114 213 134 206)(115 214 135 207)(116 215 136 208)(117 216 137 209)(118 217 138 210)(119 218 139 197)(120 219 140 198)(121 220 127 199)(122 221 128 200)(123 222 129 201)(124 223 130 202)(125 224 131 203)(126 211 132 204)(141 183 165 181)(142 184 166 182)(143 185 167 169)(144 186 168 170)(145 187 155 171)(146 188 156 172)(147 189 157 173)(148 190 158 174)(149 191 159 175)(150 192 160 176)(151 193 161 177)(152 194 162 178)(153 195 163 179)(154 196 164 180)
(1 68 8 61)(2 69 9 62)(3 70 10 63)(4 57 11 64)(5 58 12 65)(6 59 13 66)(7 60 14 67)(15 79 22 72)(16 80 23 73)(17 81 24 74)(18 82 25 75)(19 83 26 76)(20 84 27 77)(21 71 28 78)(29 87 36 94)(30 88 37 95)(31 89 38 96)(32 90 39 97)(33 91 40 98)(34 92 41 85)(35 93 42 86)(43 100 50 107)(44 101 51 108)(45 102 52 109)(46 103 53 110)(47 104 54 111)(48 105 55 112)(49 106 56 99)(113 180 120 173)(114 181 121 174)(115 182 122 175)(116 169 123 176)(117 170 124 177)(118 171 125 178)(119 172 126 179)(127 190 134 183)(128 191 135 184)(129 192 136 185)(130 193 137 186)(131 194 138 187)(132 195 139 188)(133 196 140 189)(141 206 148 199)(142 207 149 200)(143 208 150 201)(144 209 151 202)(145 210 152 203)(146 197 153 204)(147 198 154 205)(155 217 162 224)(156 218 163 211)(157 219 164 212)(158 220 165 213)(159 221 166 214)(160 222 167 215)(161 223 168 216)
(1 122 18 135)(2 121 19 134)(3 120 20 133)(4 119 21 132)(5 118 22 131)(6 117 23 130)(7 116 24 129)(8 115 25 128)(9 114 26 127)(10 113 27 140)(11 126 28 139)(12 125 15 138)(13 124 16 137)(14 123 17 136)(29 142 49 159)(30 141 50 158)(31 154 51 157)(32 153 52 156)(33 152 53 155)(34 151 54 168)(35 150 55 167)(36 149 56 166)(37 148 43 165)(38 147 44 164)(39 146 45 163)(40 145 46 162)(41 144 47 161)(42 143 48 160)(57 172 71 195)(58 171 72 194)(59 170 73 193)(60 169 74 192)(61 182 75 191)(62 181 76 190)(63 180 77 189)(64 179 78 188)(65 178 79 187)(66 177 80 186)(67 176 81 185)(68 175 82 184)(69 174 83 183)(70 173 84 196)(85 209 104 223)(86 208 105 222)(87 207 106 221)(88 206 107 220)(89 205 108 219)(90 204 109 218)(91 203 110 217)(92 202 111 216)(93 201 112 215)(94 200 99 214)(95 199 100 213)(96 198 101 212)(97 197 102 211)(98 210 103 224)
(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(113 140)(114 127)(115 128)(116 129)(117 130)(118 131)(119 132)(120 133)(121 134)(122 135)(123 136)(124 137)(125 138)(126 139)(141 165)(142 166)(143 167)(144 168)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)(151 161)(152 162)(153 163)(154 164)(169 192)(170 193)(171 194)(172 195)(173 196)(174 183)(175 184)(176 185)(177 186)(178 187)(179 188)(180 189)(181 190)(182 191)(197 218)(198 219)(199 220)(200 221)(201 222)(202 223)(203 224)(204 211)(205 212)(206 213)(207 214)(208 215)(209 216)(210 217)
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), (1,106,25,94)(2,107,26,95)(3,108,27,96)(4,109,28,97)(5,110,15,98)(6,111,16,85)(7,112,17,86)(8,99,18,87)(9,100,19,88)(10,101,20,89)(11,102,21,90)(12,103,22,91)(13,104,23,92)(14,105,24,93)(29,82,56,61)(30,83,43,62)(31,84,44,63)(32,71,45,64)(33,72,46,65)(34,73,47,66)(35,74,48,67)(36,75,49,68)(37,76,50,69)(38,77,51,70)(39,78,52,57)(40,79,53,58)(41,80,54,59)(42,81,55,60)(113,212,133,205)(114,213,134,206)(115,214,135,207)(116,215,136,208)(117,216,137,209)(118,217,138,210)(119,218,139,197)(120,219,140,198)(121,220,127,199)(122,221,128,200)(123,222,129,201)(124,223,130,202)(125,224,131,203)(126,211,132,204)(141,183,165,181)(142,184,166,182)(143,185,167,169)(144,186,168,170)(145,187,155,171)(146,188,156,172)(147,189,157,173)(148,190,158,174)(149,191,159,175)(150,192,160,176)(151,193,161,177)(152,194,162,178)(153,195,163,179)(154,196,164,180), (1,68,8,61)(2,69,9,62)(3,70,10,63)(4,57,11,64)(5,58,12,65)(6,59,13,66)(7,60,14,67)(15,79,22,72)(16,80,23,73)(17,81,24,74)(18,82,25,75)(19,83,26,76)(20,84,27,77)(21,71,28,78)(29,87,36,94)(30,88,37,95)(31,89,38,96)(32,90,39,97)(33,91,40,98)(34,92,41,85)(35,93,42,86)(43,100,50,107)(44,101,51,108)(45,102,52,109)(46,103,53,110)(47,104,54,111)(48,105,55,112)(49,106,56,99)(113,180,120,173)(114,181,121,174)(115,182,122,175)(116,169,123,176)(117,170,124,177)(118,171,125,178)(119,172,126,179)(127,190,134,183)(128,191,135,184)(129,192,136,185)(130,193,137,186)(131,194,138,187)(132,195,139,188)(133,196,140,189)(141,206,148,199)(142,207,149,200)(143,208,150,201)(144,209,151,202)(145,210,152,203)(146,197,153,204)(147,198,154,205)(155,217,162,224)(156,218,163,211)(157,219,164,212)(158,220,165,213)(159,221,166,214)(160,222,167,215)(161,223,168,216), (1,122,18,135)(2,121,19,134)(3,120,20,133)(4,119,21,132)(5,118,22,131)(6,117,23,130)(7,116,24,129)(8,115,25,128)(9,114,26,127)(10,113,27,140)(11,126,28,139)(12,125,15,138)(13,124,16,137)(14,123,17,136)(29,142,49,159)(30,141,50,158)(31,154,51,157)(32,153,52,156)(33,152,53,155)(34,151,54,168)(35,150,55,167)(36,149,56,166)(37,148,43,165)(38,147,44,164)(39,146,45,163)(40,145,46,162)(41,144,47,161)(42,143,48,160)(57,172,71,195)(58,171,72,194)(59,170,73,193)(60,169,74,192)(61,182,75,191)(62,181,76,190)(63,180,77,189)(64,179,78,188)(65,178,79,187)(66,177,80,186)(67,176,81,185)(68,175,82,184)(69,174,83,183)(70,173,84,196)(85,209,104,223)(86,208,105,222)(87,207,106,221)(88,206,107,220)(89,205,108,219)(90,204,109,218)(91,203,110,217)(92,202,111,216)(93,201,112,215)(94,200,99,214)(95,199,100,213)(96,198,101,212)(97,197,102,211)(98,210,103,224), (29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,140)(114,127)(115,128)(116,129)(117,130)(118,131)(119,132)(120,133)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(141,165)(142,166)(143,167)(144,168)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160)(151,161)(152,162)(153,163)(154,164)(169,192)(170,193)(171,194)(172,195)(173,196)(174,183)(175,184)(176,185)(177,186)(178,187)(179,188)(180,189)(181,190)(182,191)(197,218)(198,219)(199,220)(200,221)(201,222)(202,223)(203,224)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217)>;
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), (1,106,25,94)(2,107,26,95)(3,108,27,96)(4,109,28,97)(5,110,15,98)(6,111,16,85)(7,112,17,86)(8,99,18,87)(9,100,19,88)(10,101,20,89)(11,102,21,90)(12,103,22,91)(13,104,23,92)(14,105,24,93)(29,82,56,61)(30,83,43,62)(31,84,44,63)(32,71,45,64)(33,72,46,65)(34,73,47,66)(35,74,48,67)(36,75,49,68)(37,76,50,69)(38,77,51,70)(39,78,52,57)(40,79,53,58)(41,80,54,59)(42,81,55,60)(113,212,133,205)(114,213,134,206)(115,214,135,207)(116,215,136,208)(117,216,137,209)(118,217,138,210)(119,218,139,197)(120,219,140,198)(121,220,127,199)(122,221,128,200)(123,222,129,201)(124,223,130,202)(125,224,131,203)(126,211,132,204)(141,183,165,181)(142,184,166,182)(143,185,167,169)(144,186,168,170)(145,187,155,171)(146,188,156,172)(147,189,157,173)(148,190,158,174)(149,191,159,175)(150,192,160,176)(151,193,161,177)(152,194,162,178)(153,195,163,179)(154,196,164,180), (1,68,8,61)(2,69,9,62)(3,70,10,63)(4,57,11,64)(5,58,12,65)(6,59,13,66)(7,60,14,67)(15,79,22,72)(16,80,23,73)(17,81,24,74)(18,82,25,75)(19,83,26,76)(20,84,27,77)(21,71,28,78)(29,87,36,94)(30,88,37,95)(31,89,38,96)(32,90,39,97)(33,91,40,98)(34,92,41,85)(35,93,42,86)(43,100,50,107)(44,101,51,108)(45,102,52,109)(46,103,53,110)(47,104,54,111)(48,105,55,112)(49,106,56,99)(113,180,120,173)(114,181,121,174)(115,182,122,175)(116,169,123,176)(117,170,124,177)(118,171,125,178)(119,172,126,179)(127,190,134,183)(128,191,135,184)(129,192,136,185)(130,193,137,186)(131,194,138,187)(132,195,139,188)(133,196,140,189)(141,206,148,199)(142,207,149,200)(143,208,150,201)(144,209,151,202)(145,210,152,203)(146,197,153,204)(147,198,154,205)(155,217,162,224)(156,218,163,211)(157,219,164,212)(158,220,165,213)(159,221,166,214)(160,222,167,215)(161,223,168,216), (1,122,18,135)(2,121,19,134)(3,120,20,133)(4,119,21,132)(5,118,22,131)(6,117,23,130)(7,116,24,129)(8,115,25,128)(9,114,26,127)(10,113,27,140)(11,126,28,139)(12,125,15,138)(13,124,16,137)(14,123,17,136)(29,142,49,159)(30,141,50,158)(31,154,51,157)(32,153,52,156)(33,152,53,155)(34,151,54,168)(35,150,55,167)(36,149,56,166)(37,148,43,165)(38,147,44,164)(39,146,45,163)(40,145,46,162)(41,144,47,161)(42,143,48,160)(57,172,71,195)(58,171,72,194)(59,170,73,193)(60,169,74,192)(61,182,75,191)(62,181,76,190)(63,180,77,189)(64,179,78,188)(65,178,79,187)(66,177,80,186)(67,176,81,185)(68,175,82,184)(69,174,83,183)(70,173,84,196)(85,209,104,223)(86,208,105,222)(87,207,106,221)(88,206,107,220)(89,205,108,219)(90,204,109,218)(91,203,110,217)(92,202,111,216)(93,201,112,215)(94,200,99,214)(95,199,100,213)(96,198,101,212)(97,197,102,211)(98,210,103,224), (29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,140)(114,127)(115,128)(116,129)(117,130)(118,131)(119,132)(120,133)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(141,165)(142,166)(143,167)(144,168)(145,155)(146,156)(147,157)(148,158)(149,159)(150,160)(151,161)(152,162)(153,163)(154,164)(169,192)(170,193)(171,194)(172,195)(173,196)(174,183)(175,184)(176,185)(177,186)(178,187)(179,188)(180,189)(181,190)(182,191)(197,218)(198,219)(199,220)(200,221)(201,222)(202,223)(203,224)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217) );
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)], [(1,106,25,94),(2,107,26,95),(3,108,27,96),(4,109,28,97),(5,110,15,98),(6,111,16,85),(7,112,17,86),(8,99,18,87),(9,100,19,88),(10,101,20,89),(11,102,21,90),(12,103,22,91),(13,104,23,92),(14,105,24,93),(29,82,56,61),(30,83,43,62),(31,84,44,63),(32,71,45,64),(33,72,46,65),(34,73,47,66),(35,74,48,67),(36,75,49,68),(37,76,50,69),(38,77,51,70),(39,78,52,57),(40,79,53,58),(41,80,54,59),(42,81,55,60),(113,212,133,205),(114,213,134,206),(115,214,135,207),(116,215,136,208),(117,216,137,209),(118,217,138,210),(119,218,139,197),(120,219,140,198),(121,220,127,199),(122,221,128,200),(123,222,129,201),(124,223,130,202),(125,224,131,203),(126,211,132,204),(141,183,165,181),(142,184,166,182),(143,185,167,169),(144,186,168,170),(145,187,155,171),(146,188,156,172),(147,189,157,173),(148,190,158,174),(149,191,159,175),(150,192,160,176),(151,193,161,177),(152,194,162,178),(153,195,163,179),(154,196,164,180)], [(1,68,8,61),(2,69,9,62),(3,70,10,63),(4,57,11,64),(5,58,12,65),(6,59,13,66),(7,60,14,67),(15,79,22,72),(16,80,23,73),(17,81,24,74),(18,82,25,75),(19,83,26,76),(20,84,27,77),(21,71,28,78),(29,87,36,94),(30,88,37,95),(31,89,38,96),(32,90,39,97),(33,91,40,98),(34,92,41,85),(35,93,42,86),(43,100,50,107),(44,101,51,108),(45,102,52,109),(46,103,53,110),(47,104,54,111),(48,105,55,112),(49,106,56,99),(113,180,120,173),(114,181,121,174),(115,182,122,175),(116,169,123,176),(117,170,124,177),(118,171,125,178),(119,172,126,179),(127,190,134,183),(128,191,135,184),(129,192,136,185),(130,193,137,186),(131,194,138,187),(132,195,139,188),(133,196,140,189),(141,206,148,199),(142,207,149,200),(143,208,150,201),(144,209,151,202),(145,210,152,203),(146,197,153,204),(147,198,154,205),(155,217,162,224),(156,218,163,211),(157,219,164,212),(158,220,165,213),(159,221,166,214),(160,222,167,215),(161,223,168,216)], [(1,122,18,135),(2,121,19,134),(3,120,20,133),(4,119,21,132),(5,118,22,131),(6,117,23,130),(7,116,24,129),(8,115,25,128),(9,114,26,127),(10,113,27,140),(11,126,28,139),(12,125,15,138),(13,124,16,137),(14,123,17,136),(29,142,49,159),(30,141,50,158),(31,154,51,157),(32,153,52,156),(33,152,53,155),(34,151,54,168),(35,150,55,167),(36,149,56,166),(37,148,43,165),(38,147,44,164),(39,146,45,163),(40,145,46,162),(41,144,47,161),(42,143,48,160),(57,172,71,195),(58,171,72,194),(59,170,73,193),(60,169,74,192),(61,182,75,191),(62,181,76,190),(63,180,77,189),(64,179,78,188),(65,178,79,187),(66,177,80,186),(67,176,81,185),(68,175,82,184),(69,174,83,183),(70,173,84,196),(85,209,104,223),(86,208,105,222),(87,207,106,221),(88,206,107,220),(89,205,108,219),(90,204,109,218),(91,203,110,217),(92,202,111,216),(93,201,112,215),(94,200,99,214),(95,199,100,213),(96,198,101,212),(97,197,102,211),(98,210,103,224)], [(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(113,140),(114,127),(115,128),(116,129),(117,130),(118,131),(119,132),(120,133),(121,134),(122,135),(123,136),(124,137),(125,138),(126,139),(141,165),(142,166),(143,167),(144,168),(145,155),(146,156),(147,157),(148,158),(149,159),(150,160),(151,161),(152,162),(153,163),(154,164),(169,192),(170,193),(171,194),(172,195),(173,196),(174,183),(175,184),(176,185),(177,186),(178,187),(179,188),(180,189),(181,190),(182,191),(197,218),(198,219),(199,220),(200,221),(201,222),(202,223),(203,224),(204,211),(205,212),(206,213),(207,214),(208,215),(209,216),(210,217)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| 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 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 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 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | 2- 1+4 | D4⋊6D14 | D7×C4○D4 | D4.10D14 |
| kernel | C14.842- 1+4 | C23.11D14 | C22⋊Dic14 | C23.D14 | D14.D4 | Dic7.D4 | Dic7⋊3Q8 | C28⋊Q8 | D14⋊Q8 | C4⋊C4⋊D7 | C28.48D4 | C4×C7⋊D4 | C28.17D4 | Dic7⋊D4 | C7×C22.D4 | C22.D4 | Dic7 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 9 | 6 | 3 | 3 | 1 | 1 | 6 | 6 | 6 |
Matrix representation of C14.842- 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 0 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 9 | 0 | 0 |
| 0 | 0 | 20 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 9 |
| 0 | 0 | 0 | 0 | 20 | 15 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 15 | 0 | 0 |
| 0 | 0 | 14 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 15 |
| 0 | 0 | 0 | 0 | 14 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 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 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,25,0,0,0,0,0,0,25,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,14,20,0,0,0,0,9,15,0,0,0,0,0,0,14,20,0,0,0,0,9,15],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,20,14,0,0,0,0,15,9,0,0,0,0,0,0,20,14,0,0,0,0,15,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,0,28,0,0],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28] >;
C14.842- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{84}2_-^{1+4} % in TeX
G:=Group("C14.84ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1115);
// by ID
G=gap.SmallGroup(448,1115);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,675,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=a^7*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^7*b^-1,b*d=d*b,e*b*e=a^7*b,c*d=d*c,c*e=e*c,e*d*e=a^7*b^2*d>;
// generators/relations