G = C14.822- 1+4 order 448 = 26·7
37th non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.822- 1+4, C7⋊D4⋊7D4, C28⋊Q8⋊32C2, C7⋊6(D4⋊6D4), C4⋊C4.106D14, D14.23(C2×D4), D14⋊D4⋊30C2, C22.13(D4×D7), D14⋊Q8⋊28C2, Dic7⋊6(C4○D4), (C2×D4).164D14, (C2×C28).72C23, C22⋊C4.27D14, Dic7.27(C2×D4), C14.84(C22×D4), C22.D4⋊5D7, Dic7⋊4D4⋊19C2, Dic7⋊D4⋊21C2, D14.5D4⋊28C2, D14.D4⋊30C2, (C2×C14).199C24, D14⋊C4.32C22, (C22×C4).257D14, C22⋊Dic14⋊31C2, (D4×C14).137C22, (C2×D28).222C22, C22.D28⋊20C2, Dic7⋊C4.41C22, C4⋊Dic7.227C22, (C22×C14).34C23, C23.201(C22×D7), C22.220(C23×D7), C23.D7.42C22, (C22×C28).113C22, (C2×Dic7).103C23, (C4×Dic7).123C22, (C22×D7).207C23, C2.43(D4.10D14), (C2×Dic14).168C22, (C22×Dic7).129C22, C2.57(C2×D4×D7), (D7×C4⋊C4)⋊32C2, C2.61(D7×C4○D4), (C2×C4○D28)⋊12C2, (C2×C14).60(C2×D4), (C2×D4⋊2D7)⋊16C2, (C2×Dic7⋊C4)⋊27C2, C14.173(C2×C4○D4), (C2×C4×D7).110C22, (C2×C4).62(C22×D7), (C7×C4⋊C4).176C22, (C7×C22.D4)⋊7C2, (C2×C7⋊D4).134C22, (C7×C22⋊C4).51C22, SmallGroup(448,1108)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.822- 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=e2=b2, bab-1=cac=dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=a7b2d >
Subgroups: 1356 in 292 conjugacy classes, 105 normal (91 characteristic)
C1, C2, C2, C4, C22, 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, C4○D4, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C22.D4, C4⋊Q8, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, D4⋊6D4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, D4⋊2D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C22⋊Dic14, Dic7⋊4D4, D14.D4, D14⋊D4, C22.D28, C28⋊Q8, D7×C4⋊C4, D14.5D4, D14⋊Q8, C2×Dic7⋊C4, Dic7⋊D4, C7×C22.D4, C2×C4○D28, C2×D4⋊2D7, C14.822- 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, D4×D7, C23×D7, C2×D4×D7, D7×C4○D4, D4.10D14, C14.822- 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)
(1 211 116 44)(2 224 117 43)(3 223 118 56)(4 222 119 55)(5 221 120 54)(6 220 121 53)(7 219 122 52)(8 218 123 51)(9 217 124 50)(10 216 125 49)(11 215 126 48)(12 214 113 47)(13 213 114 46)(14 212 115 45)(15 162 195 101)(16 161 196 100)(17 160 183 99)(18 159 184 112)(19 158 185 111)(20 157 186 110)(21 156 187 109)(22 155 188 108)(23 168 189 107)(24 167 190 106)(25 166 191 105)(26 165 192 104)(27 164 193 103)(28 163 194 102)(29 135 72 148)(30 134 73 147)(31 133 74 146)(32 132 75 145)(33 131 76 144)(34 130 77 143)(35 129 78 142)(36 128 79 141)(37 127 80 154)(38 140 81 153)(39 139 82 152)(40 138 83 151)(41 137 84 150)(42 136 71 149)(57 173 201 94)(58 172 202 93)(59 171 203 92)(60 170 204 91)(61 169 205 90)(62 182 206 89)(63 181 207 88)(64 180 208 87)(65 179 209 86)(66 178 210 85)(67 177 197 98)(68 176 198 97)(69 175 199 96)(70 174 200 95)
(1 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 196)(16 195)(17 194)(18 193)(19 192)(20 191)(21 190)(22 189)(23 188)(24 187)(25 186)(26 185)(27 184)(28 183)(29 36)(30 35)(31 34)(32 33)(37 42)(38 41)(39 40)(43 219)(44 218)(45 217)(46 216)(47 215)(48 214)(49 213)(50 212)(51 211)(52 224)(53 223)(54 222)(55 221)(56 220)(57 62)(58 61)(59 60)(63 70)(64 69)(65 68)(66 67)(71 80)(72 79)(73 78)(74 77)(75 76)(81 84)(82 83)(85 177)(86 176)(87 175)(88 174)(89 173)(90 172)(91 171)(92 170)(93 169)(94 182)(95 181)(96 180)(97 179)(98 178)(99 102)(100 101)(103 112)(104 111)(105 110)(106 109)(107 108)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 120)(127 149)(128 148)(129 147)(130 146)(131 145)(132 144)(133 143)(134 142)(135 141)(136 154)(137 153)(138 152)(139 151)(140 150)(155 168)(156 167)(157 166)(158 165)(159 164)(160 163)(161 162)(197 210)(198 209)(199 208)(200 207)(201 206)(202 205)(203 204)
(1 36 116 79)(2 35 117 78)(3 34 118 77)(4 33 119 76)(5 32 120 75)(6 31 121 74)(7 30 122 73)(8 29 123 72)(9 42 124 71)(10 41 125 84)(11 40 126 83)(12 39 113 82)(13 38 114 81)(14 37 115 80)(15 92 195 171)(16 91 196 170)(17 90 183 169)(18 89 184 182)(19 88 185 181)(20 87 186 180)(21 86 187 179)(22 85 188 178)(23 98 189 177)(24 97 190 176)(25 96 191 175)(26 95 192 174)(27 94 193 173)(28 93 194 172)(43 142 224 129)(44 141 211 128)(45 154 212 127)(46 153 213 140)(47 152 214 139)(48 151 215 138)(49 150 216 137)(50 149 217 136)(51 148 218 135)(52 147 219 134)(53 146 220 133)(54 145 221 132)(55 144 222 131)(56 143 223 130)(57 103 201 164)(58 102 202 163)(59 101 203 162)(60 100 204 161)(61 99 205 160)(62 112 206 159)(63 111 207 158)(64 110 208 157)(65 109 209 156)(66 108 210 155)(67 107 197 168)(68 106 198 167)(69 105 199 166)(70 104 200 165)
(1 165 116 104)(2 166 117 105)(3 167 118 106)(4 168 119 107)(5 155 120 108)(6 156 121 109)(7 157 122 110)(8 158 123 111)(9 159 124 112)(10 160 125 99)(11 161 126 100)(12 162 113 101)(13 163 114 102)(14 164 115 103)(15 214 195 47)(16 215 196 48)(17 216 183 49)(18 217 184 50)(19 218 185 51)(20 219 186 52)(21 220 187 53)(22 221 188 54)(23 222 189 55)(24 223 190 56)(25 224 191 43)(26 211 192 44)(27 212 193 45)(28 213 194 46)(29 200 72 70)(30 201 73 57)(31 202 74 58)(32 203 75 59)(33 204 76 60)(34 205 77 61)(35 206 78 62)(36 207 79 63)(37 208 80 64)(38 209 81 65)(39 210 82 66)(40 197 83 67)(41 198 84 68)(42 199 71 69)(85 152 178 139)(86 153 179 140)(87 154 180 127)(88 141 181 128)(89 142 182 129)(90 143 169 130)(91 144 170 131)(92 145 171 132)(93 146 172 133)(94 147 173 134)(95 148 174 135)(96 149 175 136)(97 150 176 137)(98 151 177 138)
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), (1,211,116,44)(2,224,117,43)(3,223,118,56)(4,222,119,55)(5,221,120,54)(6,220,121,53)(7,219,122,52)(8,218,123,51)(9,217,124,50)(10,216,125,49)(11,215,126,48)(12,214,113,47)(13,213,114,46)(14,212,115,45)(15,162,195,101)(16,161,196,100)(17,160,183,99)(18,159,184,112)(19,158,185,111)(20,157,186,110)(21,156,187,109)(22,155,188,108)(23,168,189,107)(24,167,190,106)(25,166,191,105)(26,165,192,104)(27,164,193,103)(28,163,194,102)(29,135,72,148)(30,134,73,147)(31,133,74,146)(32,132,75,145)(33,131,76,144)(34,130,77,143)(35,129,78,142)(36,128,79,141)(37,127,80,154)(38,140,81,153)(39,139,82,152)(40,138,83,151)(41,137,84,150)(42,136,71,149)(57,173,201,94)(58,172,202,93)(59,171,203,92)(60,170,204,91)(61,169,205,90)(62,182,206,89)(63,181,207,88)(64,180,208,87)(65,179,209,86)(66,178,210,85)(67,177,197,98)(68,176,198,97)(69,175,199,96)(70,174,200,95), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,196)(16,195)(17,194)(18,193)(19,192)(20,191)(21,190)(22,189)(23,188)(24,187)(25,186)(26,185)(27,184)(28,183)(29,36)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(43,219)(44,218)(45,217)(46,216)(47,215)(48,214)(49,213)(50,212)(51,211)(52,224)(53,223)(54,222)(55,221)(56,220)(57,62)(58,61)(59,60)(63,70)(64,69)(65,68)(66,67)(71,80)(72,79)(73,78)(74,77)(75,76)(81,84)(82,83)(85,177)(86,176)(87,175)(88,174)(89,173)(90,172)(91,171)(92,170)(93,169)(94,182)(95,181)(96,180)(97,179)(98,178)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,120)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,154)(137,153)(138,152)(139,151)(140,150)(155,168)(156,167)(157,166)(158,165)(159,164)(160,163)(161,162)(197,210)(198,209)(199,208)(200,207)(201,206)(202,205)(203,204), (1,36,116,79)(2,35,117,78)(3,34,118,77)(4,33,119,76)(5,32,120,75)(6,31,121,74)(7,30,122,73)(8,29,123,72)(9,42,124,71)(10,41,125,84)(11,40,126,83)(12,39,113,82)(13,38,114,81)(14,37,115,80)(15,92,195,171)(16,91,196,170)(17,90,183,169)(18,89,184,182)(19,88,185,181)(20,87,186,180)(21,86,187,179)(22,85,188,178)(23,98,189,177)(24,97,190,176)(25,96,191,175)(26,95,192,174)(27,94,193,173)(28,93,194,172)(43,142,224,129)(44,141,211,128)(45,154,212,127)(46,153,213,140)(47,152,214,139)(48,151,215,138)(49,150,216,137)(50,149,217,136)(51,148,218,135)(52,147,219,134)(53,146,220,133)(54,145,221,132)(55,144,222,131)(56,143,223,130)(57,103,201,164)(58,102,202,163)(59,101,203,162)(60,100,204,161)(61,99,205,160)(62,112,206,159)(63,111,207,158)(64,110,208,157)(65,109,209,156)(66,108,210,155)(67,107,197,168)(68,106,198,167)(69,105,199,166)(70,104,200,165), (1,165,116,104)(2,166,117,105)(3,167,118,106)(4,168,119,107)(5,155,120,108)(6,156,121,109)(7,157,122,110)(8,158,123,111)(9,159,124,112)(10,160,125,99)(11,161,126,100)(12,162,113,101)(13,163,114,102)(14,164,115,103)(15,214,195,47)(16,215,196,48)(17,216,183,49)(18,217,184,50)(19,218,185,51)(20,219,186,52)(21,220,187,53)(22,221,188,54)(23,222,189,55)(24,223,190,56)(25,224,191,43)(26,211,192,44)(27,212,193,45)(28,213,194,46)(29,200,72,70)(30,201,73,57)(31,202,74,58)(32,203,75,59)(33,204,76,60)(34,205,77,61)(35,206,78,62)(36,207,79,63)(37,208,80,64)(38,209,81,65)(39,210,82,66)(40,197,83,67)(41,198,84,68)(42,199,71,69)(85,152,178,139)(86,153,179,140)(87,154,180,127)(88,141,181,128)(89,142,182,129)(90,143,169,130)(91,144,170,131)(92,145,171,132)(93,146,172,133)(94,147,173,134)(95,148,174,135)(96,149,175,136)(97,150,176,137)(98,151,177,138)>;
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), (1,211,116,44)(2,224,117,43)(3,223,118,56)(4,222,119,55)(5,221,120,54)(6,220,121,53)(7,219,122,52)(8,218,123,51)(9,217,124,50)(10,216,125,49)(11,215,126,48)(12,214,113,47)(13,213,114,46)(14,212,115,45)(15,162,195,101)(16,161,196,100)(17,160,183,99)(18,159,184,112)(19,158,185,111)(20,157,186,110)(21,156,187,109)(22,155,188,108)(23,168,189,107)(24,167,190,106)(25,166,191,105)(26,165,192,104)(27,164,193,103)(28,163,194,102)(29,135,72,148)(30,134,73,147)(31,133,74,146)(32,132,75,145)(33,131,76,144)(34,130,77,143)(35,129,78,142)(36,128,79,141)(37,127,80,154)(38,140,81,153)(39,139,82,152)(40,138,83,151)(41,137,84,150)(42,136,71,149)(57,173,201,94)(58,172,202,93)(59,171,203,92)(60,170,204,91)(61,169,205,90)(62,182,206,89)(63,181,207,88)(64,180,208,87)(65,179,209,86)(66,178,210,85)(67,177,197,98)(68,176,198,97)(69,175,199,96)(70,174,200,95), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,12)(15,196)(16,195)(17,194)(18,193)(19,192)(20,191)(21,190)(22,189)(23,188)(24,187)(25,186)(26,185)(27,184)(28,183)(29,36)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(43,219)(44,218)(45,217)(46,216)(47,215)(48,214)(49,213)(50,212)(51,211)(52,224)(53,223)(54,222)(55,221)(56,220)(57,62)(58,61)(59,60)(63,70)(64,69)(65,68)(66,67)(71,80)(72,79)(73,78)(74,77)(75,76)(81,84)(82,83)(85,177)(86,176)(87,175)(88,174)(89,173)(90,172)(91,171)(92,170)(93,169)(94,182)(95,181)(96,180)(97,179)(98,178)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,120)(127,149)(128,148)(129,147)(130,146)(131,145)(132,144)(133,143)(134,142)(135,141)(136,154)(137,153)(138,152)(139,151)(140,150)(155,168)(156,167)(157,166)(158,165)(159,164)(160,163)(161,162)(197,210)(198,209)(199,208)(200,207)(201,206)(202,205)(203,204), (1,36,116,79)(2,35,117,78)(3,34,118,77)(4,33,119,76)(5,32,120,75)(6,31,121,74)(7,30,122,73)(8,29,123,72)(9,42,124,71)(10,41,125,84)(11,40,126,83)(12,39,113,82)(13,38,114,81)(14,37,115,80)(15,92,195,171)(16,91,196,170)(17,90,183,169)(18,89,184,182)(19,88,185,181)(20,87,186,180)(21,86,187,179)(22,85,188,178)(23,98,189,177)(24,97,190,176)(25,96,191,175)(26,95,192,174)(27,94,193,173)(28,93,194,172)(43,142,224,129)(44,141,211,128)(45,154,212,127)(46,153,213,140)(47,152,214,139)(48,151,215,138)(49,150,216,137)(50,149,217,136)(51,148,218,135)(52,147,219,134)(53,146,220,133)(54,145,221,132)(55,144,222,131)(56,143,223,130)(57,103,201,164)(58,102,202,163)(59,101,203,162)(60,100,204,161)(61,99,205,160)(62,112,206,159)(63,111,207,158)(64,110,208,157)(65,109,209,156)(66,108,210,155)(67,107,197,168)(68,106,198,167)(69,105,199,166)(70,104,200,165), (1,165,116,104)(2,166,117,105)(3,167,118,106)(4,168,119,107)(5,155,120,108)(6,156,121,109)(7,157,122,110)(8,158,123,111)(9,159,124,112)(10,160,125,99)(11,161,126,100)(12,162,113,101)(13,163,114,102)(14,164,115,103)(15,214,195,47)(16,215,196,48)(17,216,183,49)(18,217,184,50)(19,218,185,51)(20,219,186,52)(21,220,187,53)(22,221,188,54)(23,222,189,55)(24,223,190,56)(25,224,191,43)(26,211,192,44)(27,212,193,45)(28,213,194,46)(29,200,72,70)(30,201,73,57)(31,202,74,58)(32,203,75,59)(33,204,76,60)(34,205,77,61)(35,206,78,62)(36,207,79,63)(37,208,80,64)(38,209,81,65)(39,210,82,66)(40,197,83,67)(41,198,84,68)(42,199,71,69)(85,152,178,139)(86,153,179,140)(87,154,180,127)(88,141,181,128)(89,142,182,129)(90,143,169,130)(91,144,170,131)(92,145,171,132)(93,146,172,133)(94,147,173,134)(95,148,174,135)(96,149,175,136)(97,150,176,137)(98,151,177,138) );
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)], [(1,211,116,44),(2,224,117,43),(3,223,118,56),(4,222,119,55),(5,221,120,54),(6,220,121,53),(7,219,122,52),(8,218,123,51),(9,217,124,50),(10,216,125,49),(11,215,126,48),(12,214,113,47),(13,213,114,46),(14,212,115,45),(15,162,195,101),(16,161,196,100),(17,160,183,99),(18,159,184,112),(19,158,185,111),(20,157,186,110),(21,156,187,109),(22,155,188,108),(23,168,189,107),(24,167,190,106),(25,166,191,105),(26,165,192,104),(27,164,193,103),(28,163,194,102),(29,135,72,148),(30,134,73,147),(31,133,74,146),(32,132,75,145),(33,131,76,144),(34,130,77,143),(35,129,78,142),(36,128,79,141),(37,127,80,154),(38,140,81,153),(39,139,82,152),(40,138,83,151),(41,137,84,150),(42,136,71,149),(57,173,201,94),(58,172,202,93),(59,171,203,92),(60,170,204,91),(61,169,205,90),(62,182,206,89),(63,181,207,88),(64,180,208,87),(65,179,209,86),(66,178,210,85),(67,177,197,98),(68,176,198,97),(69,175,199,96),(70,174,200,95)], [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,12),(15,196),(16,195),(17,194),(18,193),(19,192),(20,191),(21,190),(22,189),(23,188),(24,187),(25,186),(26,185),(27,184),(28,183),(29,36),(30,35),(31,34),(32,33),(37,42),(38,41),(39,40),(43,219),(44,218),(45,217),(46,216),(47,215),(48,214),(49,213),(50,212),(51,211),(52,224),(53,223),(54,222),(55,221),(56,220),(57,62),(58,61),(59,60),(63,70),(64,69),(65,68),(66,67),(71,80),(72,79),(73,78),(74,77),(75,76),(81,84),(82,83),(85,177),(86,176),(87,175),(88,174),(89,173),(90,172),(91,171),(92,170),(93,169),(94,182),(95,181),(96,180),(97,179),(98,178),(99,102),(100,101),(103,112),(104,111),(105,110),(106,109),(107,108),(113,126),(114,125),(115,124),(116,123),(117,122),(118,121),(119,120),(127,149),(128,148),(129,147),(130,146),(131,145),(132,144),(133,143),(134,142),(135,141),(136,154),(137,153),(138,152),(139,151),(140,150),(155,168),(156,167),(157,166),(158,165),(159,164),(160,163),(161,162),(197,210),(198,209),(199,208),(200,207),(201,206),(202,205),(203,204)], [(1,36,116,79),(2,35,117,78),(3,34,118,77),(4,33,119,76),(5,32,120,75),(6,31,121,74),(7,30,122,73),(8,29,123,72),(9,42,124,71),(10,41,125,84),(11,40,126,83),(12,39,113,82),(13,38,114,81),(14,37,115,80),(15,92,195,171),(16,91,196,170),(17,90,183,169),(18,89,184,182),(19,88,185,181),(20,87,186,180),(21,86,187,179),(22,85,188,178),(23,98,189,177),(24,97,190,176),(25,96,191,175),(26,95,192,174),(27,94,193,173),(28,93,194,172),(43,142,224,129),(44,141,211,128),(45,154,212,127),(46,153,213,140),(47,152,214,139),(48,151,215,138),(49,150,216,137),(50,149,217,136),(51,148,218,135),(52,147,219,134),(53,146,220,133),(54,145,221,132),(55,144,222,131),(56,143,223,130),(57,103,201,164),(58,102,202,163),(59,101,203,162),(60,100,204,161),(61,99,205,160),(62,112,206,159),(63,111,207,158),(64,110,208,157),(65,109,209,156),(66,108,210,155),(67,107,197,168),(68,106,198,167),(69,105,199,166),(70,104,200,165)], [(1,165,116,104),(2,166,117,105),(3,167,118,106),(4,168,119,107),(5,155,120,108),(6,156,121,109),(7,157,122,110),(8,158,123,111),(9,159,124,112),(10,160,125,99),(11,161,126,100),(12,162,113,101),(13,163,114,102),(14,164,115,103),(15,214,195,47),(16,215,196,48),(17,216,183,49),(18,217,184,50),(19,218,185,51),(20,219,186,52),(21,220,187,53),(22,221,188,54),(23,222,189,55),(24,223,190,56),(25,224,191,43),(26,211,192,44),(27,212,193,45),(28,213,194,46),(29,200,72,70),(30,201,73,57),(31,202,74,58),(32,203,75,59),(33,204,76,60),(34,205,77,61),(35,206,78,62),(36,207,79,63),(37,208,80,64),(38,209,81,65),(39,210,82,66),(40,197,83,67),(41,198,84,68),(42,199,71,69),(85,152,178,139),(86,153,179,140),(87,154,180,127),(88,141,181,128),(89,142,182,129),(90,143,169,130),(91,144,170,131),(92,145,171,132),(93,146,172,133),(94,147,173,134),(95,148,174,135),(96,149,175,136),(97,150,176,137),(98,151,177,138)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 4M | 4N | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| 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 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 14 | 14 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2- 1+4 | D4×D7 | D7×C4○D4 | D4.10D14 |
| kernel | C14.822- 1+4 | C22⋊Dic14 | Dic7⋊4D4 | D14.D4 | D14⋊D4 | C22.D28 | C28⋊Q8 | D7×C4⋊C4 | D14.5D4 | D14⋊Q8 | C2×Dic7⋊C4 | Dic7⋊D4 | C7×C22.D4 | C2×C4○D28 | C2×D4⋊2D7 | C7⋊D4 | C22.D4 | Dic7 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 9 | 6 | 3 | 3 | 1 | 6 | 6 | 6 |
Matrix representation of C14.822- 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 21 | 0 | 0 |
| 0 | 0 | 11 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 24 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 1 | 0 | 0 |
| 0 | 0 | 14 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 3 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 1 | 0 | 0 |
| 0 | 0 | 14 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 1 | 0 | 0 |
| 0 | 0 | 14 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 18 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 22 |
| 0 | 0 | 0 | 0 | 28 | 20 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,11,0,0,0,0,21,18,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[3,24,0,0,0,0,2,26,0,0,0,0,0,0,25,14,0,0,0,0,1,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,3,0,0,0,0,0,1,0,0,0,0,0,0,25,14,0,0,0,0,1,4,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,25,14,0,0,0,0,1,4,0,0,0,0,0,0,1,0,0,0,0,0,18,28],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,9,28,0,0,0,0,22,20] >;
C14.822- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{82}2_-^{1+4} % in TeX
G:=Group("C14.82ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1108);
// by ID
G=gap.SmallGroup(448,1108);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,184,570,185,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=e^2=b^2,b*a*b^-1=c*a*c=d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations