G = C14.782- 1+4 order 448 = 26·7
33rd non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.782- 1+4, C14.572+ 1+4, C22⋊Q8⋊22D7, C4⋊C4.100D14, (C2×Q8).78D14, D14⋊3Q8⋊24C2, C28⋊7D4.19C2, (C2×C28).65C23, C22⋊C4.65D14, C4.Dic14⋊26C2, Dic7⋊4D4⋊16C2, D14.5D4⋊23C2, (C2×C14).189C24, D14⋊C4.72C22, (C2×D28).31C22, (C22×C4).251D14, C2.59(D4⋊6D14), C22.D28⋊17C2, C4⋊Dic7.221C22, (Q8×C14).118C22, C22.5(Q8⋊2D7), (C2×Dic7).95C23, (C22×D7).80C23, C22.210(C23×D7), C23.197(C22×D7), Dic7⋊C4.119C22, (C22×C28).317C22, (C22×C14).217C23, C7⋊5(C22.33C24), (C4×Dic7).116C22, C2.38(D4.10D14), (C22×Dic7).125C22, C4⋊C4⋊D7⋊24C2, (C7×C22⋊Q8)⋊25C2, (C2×Dic7⋊C4)⋊30C2, C14.117(C2×C4○D4), C2.21(C2×Q8⋊2D7), (C2×C4×D7).105C22, (C2×C14).29(C4○D4), (C7×C4⋊C4).169C22, (C2×C4).186(C22×D7), (C2×C7⋊D4).41C22, (C7×C22⋊C4).44C22, SmallGroup(448,1098)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.782- 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=a7b2, e2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a7b-1, dbd-1=ebe-1=a7b, cd=dc, ce=ec, ede-1=a7b2d >
Subgroups: 988 in 218 conjugacy classes, 95 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×2], C22 [×8], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×2], D7 [×2], C14 [×3], C14 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×11], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8, Dic7 [×6], C28 [×6], D14 [×6], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4 [×4], C42.C2 [×2], C42⋊2C2 [×2], C4×D7 [×2], D28, C2×Dic7 [×6], C2×Dic7 [×3], C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×4], C2×C28, C7×Q8, C22×D7 [×2], C22×C14, C22.33C24, C4×Dic7 [×2], Dic7⋊C4 [×6], C4⋊Dic7, C4⋊Dic7 [×4], D14⋊C4 [×8], C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×C4×D7 [×2], C2×D28, C22×Dic7 [×2], C2×C7⋊D4 [×2], C22×C28, Q8×C14, Dic7⋊4D4 [×2], C22.D28 [×2], C4.Dic14 [×2], D14.5D4 [×2], C4⋊C4⋊D7 [×2], C2×Dic7⋊C4, C28⋊7D4, D14⋊3Q8 [×2], C7×C22⋊Q8, C14.782- 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D7 [×7], C22.33C24, Q8⋊2D7 [×2], C23×D7, D4⋊6D14, C2×Q8⋊2D7, D4.10D14, C14.782- 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 56 193 131)(2 43 194 132)(3 44 195 133)(4 45 196 134)(5 46 183 135)(6 47 184 136)(7 48 185 137)(8 49 186 138)(9 50 187 139)(10 51 188 140)(11 52 189 127)(12 53 190 128)(13 54 191 129)(14 55 192 130)(15 114 207 101)(16 115 208 102)(17 116 209 103)(18 117 210 104)(19 118 197 105)(20 119 198 106)(21 120 199 107)(22 121 200 108)(23 122 201 109)(24 123 202 110)(25 124 203 111)(26 125 204 112)(27 126 205 99)(28 113 206 100)(29 80 173 86)(30 81 174 87)(31 82 175 88)(32 83 176 89)(33 84 177 90)(34 71 178 91)(35 72 179 92)(36 73 180 93)(37 74 181 94)(38 75 182 95)(39 76 169 96)(40 77 170 97)(41 78 171 98)(42 79 172 85)(57 141 213 166)(58 142 214 167)(59 143 215 168)(60 144 216 155)(61 145 217 156)(62 146 218 157)(63 147 219 158)(64 148 220 159)(65 149 221 160)(66 150 222 161)(67 151 223 162)(68 152 224 163)(69 153 211 164)(70 154 212 165)
(15 200)(16 201)(17 202)(18 203)(19 204)(20 205)(21 206)(22 207)(23 208)(24 209)(25 210)(26 197)(27 198)(28 199)(29 180)(30 181)(31 182)(32 169)(33 170)(34 171)(35 172)(36 173)(37 174)(38 175)(39 176)(40 177)(41 178)(42 179)(43 139)(44 140)(45 127)(46 128)(47 129)(48 130)(49 131)(50 132)(51 133)(52 134)(53 135)(54 136)(55 137)(56 138)(141 159)(142 160)(143 161)(144 162)(145 163)(146 164)(147 165)(148 166)(149 167)(150 168)(151 155)(152 156)(153 157)(154 158)
(1 88 186 75)(2 87 187 74)(3 86 188 73)(4 85 189 72)(5 98 190 71)(6 97 191 84)(7 96 192 83)(8 95 193 82)(9 94 194 81)(10 93 195 80)(11 92 196 79)(12 91 183 78)(13 90 184 77)(14 89 185 76)(15 162 200 144)(16 161 201 143)(17 160 202 142)(18 159 203 141)(19 158 204 154)(20 157 205 153)(21 156 206 152)(22 155 207 151)(23 168 208 150)(24 167 209 149)(25 166 210 148)(26 165 197 147)(27 164 198 146)(28 163 199 145)(29 133 180 51)(30 132 181 50)(31 131 182 49)(32 130 169 48)(33 129 170 47)(34 128 171 46)(35 127 172 45)(36 140 173 44)(37 139 174 43)(38 138 175 56)(39 137 176 55)(40 136 177 54)(41 135 178 53)(42 134 179 52)(57 111 220 117)(58 110 221 116)(59 109 222 115)(60 108 223 114)(61 107 224 113)(62 106 211 126)(63 105 212 125)(64 104 213 124)(65 103 214 123)(66 102 215 122)(67 101 216 121)(68 100 217 120)(69 99 218 119)(70 112 219 118)
(1 109 193 122)(2 110 194 123)(3 111 195 124)(4 112 196 125)(5 99 183 126)(6 100 184 113)(7 101 185 114)(8 102 186 115)(9 103 187 116)(10 104 188 117)(11 105 189 118)(12 106 190 119)(13 107 191 120)(14 108 192 121)(15 130 207 55)(16 131 208 56)(17 132 209 43)(18 133 210 44)(19 134 197 45)(20 135 198 46)(21 136 199 47)(22 137 200 48)(23 138 201 49)(24 139 202 50)(25 140 203 51)(26 127 204 52)(27 128 205 53)(28 129 206 54)(29 148 173 159)(30 149 174 160)(31 150 175 161)(32 151 176 162)(33 152 177 163)(34 153 178 164)(35 154 179 165)(36 141 180 166)(37 142 181 167)(38 143 182 168)(39 144 169 155)(40 145 170 156)(41 146 171 157)(42 147 172 158)(57 80 213 86)(58 81 214 87)(59 82 215 88)(60 83 216 89)(61 84 217 90)(62 71 218 91)(63 72 219 92)(64 73 220 93)(65 74 221 94)(66 75 222 95)(67 76 223 96)(68 77 224 97)(69 78 211 98)(70 79 212 85)
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), (1,56,193,131)(2,43,194,132)(3,44,195,133)(4,45,196,134)(5,46,183,135)(6,47,184,136)(7,48,185,137)(8,49,186,138)(9,50,187,139)(10,51,188,140)(11,52,189,127)(12,53,190,128)(13,54,191,129)(14,55,192,130)(15,114,207,101)(16,115,208,102)(17,116,209,103)(18,117,210,104)(19,118,197,105)(20,119,198,106)(21,120,199,107)(22,121,200,108)(23,122,201,109)(24,123,202,110)(25,124,203,111)(26,125,204,112)(27,126,205,99)(28,113,206,100)(29,80,173,86)(30,81,174,87)(31,82,175,88)(32,83,176,89)(33,84,177,90)(34,71,178,91)(35,72,179,92)(36,73,180,93)(37,74,181,94)(38,75,182,95)(39,76,169,96)(40,77,170,97)(41,78,171,98)(42,79,172,85)(57,141,213,166)(58,142,214,167)(59,143,215,168)(60,144,216,155)(61,145,217,156)(62,146,218,157)(63,147,219,158)(64,148,220,159)(65,149,221,160)(66,150,222,161)(67,151,223,162)(68,152,224,163)(69,153,211,164)(70,154,212,165), (15,200)(16,201)(17,202)(18,203)(19,204)(20,205)(21,206)(22,207)(23,208)(24,209)(25,210)(26,197)(27,198)(28,199)(29,180)(30,181)(31,182)(32,169)(33,170)(34,171)(35,172)(36,173)(37,174)(38,175)(39,176)(40,177)(41,178)(42,179)(43,139)(44,140)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(141,159)(142,160)(143,161)(144,162)(145,163)(146,164)(147,165)(148,166)(149,167)(150,168)(151,155)(152,156)(153,157)(154,158), (1,88,186,75)(2,87,187,74)(3,86,188,73)(4,85,189,72)(5,98,190,71)(6,97,191,84)(7,96,192,83)(8,95,193,82)(9,94,194,81)(10,93,195,80)(11,92,196,79)(12,91,183,78)(13,90,184,77)(14,89,185,76)(15,162,200,144)(16,161,201,143)(17,160,202,142)(18,159,203,141)(19,158,204,154)(20,157,205,153)(21,156,206,152)(22,155,207,151)(23,168,208,150)(24,167,209,149)(25,166,210,148)(26,165,197,147)(27,164,198,146)(28,163,199,145)(29,133,180,51)(30,132,181,50)(31,131,182,49)(32,130,169,48)(33,129,170,47)(34,128,171,46)(35,127,172,45)(36,140,173,44)(37,139,174,43)(38,138,175,56)(39,137,176,55)(40,136,177,54)(41,135,178,53)(42,134,179,52)(57,111,220,117)(58,110,221,116)(59,109,222,115)(60,108,223,114)(61,107,224,113)(62,106,211,126)(63,105,212,125)(64,104,213,124)(65,103,214,123)(66,102,215,122)(67,101,216,121)(68,100,217,120)(69,99,218,119)(70,112,219,118), (1,109,193,122)(2,110,194,123)(3,111,195,124)(4,112,196,125)(5,99,183,126)(6,100,184,113)(7,101,185,114)(8,102,186,115)(9,103,187,116)(10,104,188,117)(11,105,189,118)(12,106,190,119)(13,107,191,120)(14,108,192,121)(15,130,207,55)(16,131,208,56)(17,132,209,43)(18,133,210,44)(19,134,197,45)(20,135,198,46)(21,136,199,47)(22,137,200,48)(23,138,201,49)(24,139,202,50)(25,140,203,51)(26,127,204,52)(27,128,205,53)(28,129,206,54)(29,148,173,159)(30,149,174,160)(31,150,175,161)(32,151,176,162)(33,152,177,163)(34,153,178,164)(35,154,179,165)(36,141,180,166)(37,142,181,167)(38,143,182,168)(39,144,169,155)(40,145,170,156)(41,146,171,157)(42,147,172,158)(57,80,213,86)(58,81,214,87)(59,82,215,88)(60,83,216,89)(61,84,217,90)(62,71,218,91)(63,72,219,92)(64,73,220,93)(65,74,221,94)(66,75,222,95)(67,76,223,96)(68,77,224,97)(69,78,211,98)(70,79,212,85)>;
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), (1,56,193,131)(2,43,194,132)(3,44,195,133)(4,45,196,134)(5,46,183,135)(6,47,184,136)(7,48,185,137)(8,49,186,138)(9,50,187,139)(10,51,188,140)(11,52,189,127)(12,53,190,128)(13,54,191,129)(14,55,192,130)(15,114,207,101)(16,115,208,102)(17,116,209,103)(18,117,210,104)(19,118,197,105)(20,119,198,106)(21,120,199,107)(22,121,200,108)(23,122,201,109)(24,123,202,110)(25,124,203,111)(26,125,204,112)(27,126,205,99)(28,113,206,100)(29,80,173,86)(30,81,174,87)(31,82,175,88)(32,83,176,89)(33,84,177,90)(34,71,178,91)(35,72,179,92)(36,73,180,93)(37,74,181,94)(38,75,182,95)(39,76,169,96)(40,77,170,97)(41,78,171,98)(42,79,172,85)(57,141,213,166)(58,142,214,167)(59,143,215,168)(60,144,216,155)(61,145,217,156)(62,146,218,157)(63,147,219,158)(64,148,220,159)(65,149,221,160)(66,150,222,161)(67,151,223,162)(68,152,224,163)(69,153,211,164)(70,154,212,165), (15,200)(16,201)(17,202)(18,203)(19,204)(20,205)(21,206)(22,207)(23,208)(24,209)(25,210)(26,197)(27,198)(28,199)(29,180)(30,181)(31,182)(32,169)(33,170)(34,171)(35,172)(36,173)(37,174)(38,175)(39,176)(40,177)(41,178)(42,179)(43,139)(44,140)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(141,159)(142,160)(143,161)(144,162)(145,163)(146,164)(147,165)(148,166)(149,167)(150,168)(151,155)(152,156)(153,157)(154,158), (1,88,186,75)(2,87,187,74)(3,86,188,73)(4,85,189,72)(5,98,190,71)(6,97,191,84)(7,96,192,83)(8,95,193,82)(9,94,194,81)(10,93,195,80)(11,92,196,79)(12,91,183,78)(13,90,184,77)(14,89,185,76)(15,162,200,144)(16,161,201,143)(17,160,202,142)(18,159,203,141)(19,158,204,154)(20,157,205,153)(21,156,206,152)(22,155,207,151)(23,168,208,150)(24,167,209,149)(25,166,210,148)(26,165,197,147)(27,164,198,146)(28,163,199,145)(29,133,180,51)(30,132,181,50)(31,131,182,49)(32,130,169,48)(33,129,170,47)(34,128,171,46)(35,127,172,45)(36,140,173,44)(37,139,174,43)(38,138,175,56)(39,137,176,55)(40,136,177,54)(41,135,178,53)(42,134,179,52)(57,111,220,117)(58,110,221,116)(59,109,222,115)(60,108,223,114)(61,107,224,113)(62,106,211,126)(63,105,212,125)(64,104,213,124)(65,103,214,123)(66,102,215,122)(67,101,216,121)(68,100,217,120)(69,99,218,119)(70,112,219,118), (1,109,193,122)(2,110,194,123)(3,111,195,124)(4,112,196,125)(5,99,183,126)(6,100,184,113)(7,101,185,114)(8,102,186,115)(9,103,187,116)(10,104,188,117)(11,105,189,118)(12,106,190,119)(13,107,191,120)(14,108,192,121)(15,130,207,55)(16,131,208,56)(17,132,209,43)(18,133,210,44)(19,134,197,45)(20,135,198,46)(21,136,199,47)(22,137,200,48)(23,138,201,49)(24,139,202,50)(25,140,203,51)(26,127,204,52)(27,128,205,53)(28,129,206,54)(29,148,173,159)(30,149,174,160)(31,150,175,161)(32,151,176,162)(33,152,177,163)(34,153,178,164)(35,154,179,165)(36,141,180,166)(37,142,181,167)(38,143,182,168)(39,144,169,155)(40,145,170,156)(41,146,171,157)(42,147,172,158)(57,80,213,86)(58,81,214,87)(59,82,215,88)(60,83,216,89)(61,84,217,90)(62,71,218,91)(63,72,219,92)(64,73,220,93)(65,74,221,94)(66,75,222,95)(67,76,223,96)(68,77,224,97)(69,78,211,98)(70,79,212,85) );
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)], [(1,56,193,131),(2,43,194,132),(3,44,195,133),(4,45,196,134),(5,46,183,135),(6,47,184,136),(7,48,185,137),(8,49,186,138),(9,50,187,139),(10,51,188,140),(11,52,189,127),(12,53,190,128),(13,54,191,129),(14,55,192,130),(15,114,207,101),(16,115,208,102),(17,116,209,103),(18,117,210,104),(19,118,197,105),(20,119,198,106),(21,120,199,107),(22,121,200,108),(23,122,201,109),(24,123,202,110),(25,124,203,111),(26,125,204,112),(27,126,205,99),(28,113,206,100),(29,80,173,86),(30,81,174,87),(31,82,175,88),(32,83,176,89),(33,84,177,90),(34,71,178,91),(35,72,179,92),(36,73,180,93),(37,74,181,94),(38,75,182,95),(39,76,169,96),(40,77,170,97),(41,78,171,98),(42,79,172,85),(57,141,213,166),(58,142,214,167),(59,143,215,168),(60,144,216,155),(61,145,217,156),(62,146,218,157),(63,147,219,158),(64,148,220,159),(65,149,221,160),(66,150,222,161),(67,151,223,162),(68,152,224,163),(69,153,211,164),(70,154,212,165)], [(15,200),(16,201),(17,202),(18,203),(19,204),(20,205),(21,206),(22,207),(23,208),(24,209),(25,210),(26,197),(27,198),(28,199),(29,180),(30,181),(31,182),(32,169),(33,170),(34,171),(35,172),(36,173),(37,174),(38,175),(39,176),(40,177),(41,178),(42,179),(43,139),(44,140),(45,127),(46,128),(47,129),(48,130),(49,131),(50,132),(51,133),(52,134),(53,135),(54,136),(55,137),(56,138),(141,159),(142,160),(143,161),(144,162),(145,163),(146,164),(147,165),(148,166),(149,167),(150,168),(151,155),(152,156),(153,157),(154,158)], [(1,88,186,75),(2,87,187,74),(3,86,188,73),(4,85,189,72),(5,98,190,71),(6,97,191,84),(7,96,192,83),(8,95,193,82),(9,94,194,81),(10,93,195,80),(11,92,196,79),(12,91,183,78),(13,90,184,77),(14,89,185,76),(15,162,200,144),(16,161,201,143),(17,160,202,142),(18,159,203,141),(19,158,204,154),(20,157,205,153),(21,156,206,152),(22,155,207,151),(23,168,208,150),(24,167,209,149),(25,166,210,148),(26,165,197,147),(27,164,198,146),(28,163,199,145),(29,133,180,51),(30,132,181,50),(31,131,182,49),(32,130,169,48),(33,129,170,47),(34,128,171,46),(35,127,172,45),(36,140,173,44),(37,139,174,43),(38,138,175,56),(39,137,176,55),(40,136,177,54),(41,135,178,53),(42,134,179,52),(57,111,220,117),(58,110,221,116),(59,109,222,115),(60,108,223,114),(61,107,224,113),(62,106,211,126),(63,105,212,125),(64,104,213,124),(65,103,214,123),(66,102,215,122),(67,101,216,121),(68,100,217,120),(69,99,218,119),(70,112,219,118)], [(1,109,193,122),(2,110,194,123),(3,111,195,124),(4,112,196,125),(5,99,183,126),(6,100,184,113),(7,101,185,114),(8,102,186,115),(9,103,187,116),(10,104,188,117),(11,105,189,118),(12,106,190,119),(13,107,191,120),(14,108,192,121),(15,130,207,55),(16,131,208,56),(17,132,209,43),(18,133,210,44),(19,134,197,45),(20,135,198,46),(21,136,199,47),(22,137,200,48),(23,138,201,49),(24,139,202,50),(25,140,203,51),(26,127,204,52),(27,128,205,53),(28,129,206,54),(29,148,173,159),(30,149,174,160),(31,150,175,161),(32,151,176,162),(33,152,177,163),(34,153,178,164),(35,154,179,165),(36,141,180,166),(37,142,181,167),(38,143,182,168),(39,144,169,155),(40,145,170,156),(41,146,171,157),(42,147,172,158),(57,80,213,86),(58,81,214,87),(59,82,215,88),(60,83,216,89),(61,84,217,90),(62,71,218,91),(63,72,219,92),(64,73,220,93),(65,74,221,94),(66,75,222,95),(67,76,223,96),(68,77,224,97),(69,78,211,98),(70,79,212,85)])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 28 | 28 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | 2- 1+4 | Q8⋊2D7 | D4⋊6D14 | D4.10D14 |
| kernel | C14.782- 1+4 | Dic7⋊4D4 | C22.D28 | C4.Dic14 | D14.5D4 | C4⋊C4⋊D7 | C2×Dic7⋊C4 | C28⋊7D4 | D14⋊3Q8 | C7×C22⋊Q8 | C22⋊Q8 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 3 | 4 | 6 | 9 | 3 | 3 | 1 | 1 | 6 | 6 | 6 |
Matrix representation of C14.782- 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 7 | 22 | 0 | 0 |
| 0 | 0 | 25 | 0 | 4 | 4 |
| 0 | 0 | 11 | 4 | 25 | 18 |
| 22 | 22 | 0 | 0 | 0 | 0 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 27 | 13 | 13 |
| 0 | 0 | 12 | 9 | 0 | 28 |
| 0 | 0 | 27 | 0 | 11 | 2 |
| 0 | 0 | 18 | 27 | 11 | 2 |
| 28 | 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 | 24 | 18 | 28 | 0 |
| 0 | 0 | 24 | 18 | 0 | 28 |
| 12 | 23 | 0 | 0 | 0 | 0 |
| 19 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 26 | 0 | 0 |
| 0 | 0 | 24 | 19 | 0 | 0 |
| 0 | 0 | 11 | 0 | 26 | 18 |
| 0 | 0 | 7 | 26 | 22 | 3 |
| 1 | 14 | 0 | 0 | 0 | 0 |
| 4 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 27 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 9 | 27 | 11 | 2 |
| 0 | 0 | 2 | 0 | 27 | 18 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,7,25,11,0,0,4,22,0,4,0,0,0,0,4,25,0,0,0,0,4,18],[22,3,0,0,0,0,22,7,0,0,0,0,0,0,7,12,27,18,0,0,27,9,0,27,0,0,13,0,11,11,0,0,13,28,2,2],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,24,24,0,0,0,1,18,18,0,0,0,0,28,0,0,0,0,0,0,28],[12,19,0,0,0,0,23,17,0,0,0,0,0,0,10,24,11,7,0,0,26,19,0,26,0,0,0,0,26,22,0,0,0,0,18,3],[1,4,0,0,0,0,14,28,0,0,0,0,0,0,20,11,9,2,0,0,27,9,27,0,0,0,0,0,11,27,0,0,0,0,2,18] >;
C14.782- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{78}2_-^{1+4} % in TeX
G:=Group("C14.78ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1098);
// by ID
G=gap.SmallGroup(448,1098);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,387,100,675,409,80,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=a^7*b^2,e^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=a^7*b^-1,d*b*d^-1=e*b*e^-1=a^7*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations