?
G = C14.812- (1+4) order 448 = 26·7
36th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.812- (1+4), C14.602+ (1+4), C28⋊Q8⋊31C2, C4⋊C4.105D14, (C2×D4).97D14, Dic7.Q8⋊26C2, C22⋊C4.26D14, C28.48D4⋊14C2, (C2×C28).178C23, (C2×C14).195C24, C28.17D4.8C2, (C22×C4).256D14, C2.62(D4⋊6D14), C22.D4.3D7, C22⋊Dic14⋊30C2, (D4×C14).133C22, C23.D14⋊29C2, Dic7⋊C4.40C22, C4⋊Dic7.226C22, (C22×C28).86C22, C22.216(C23×D7), C23.128(C22×D7), C23.D7.41C22, (C22×C14).220C23, C7⋊2(C22.57C24), (C2×Dic14).36C22, (C2×Dic7).100C23, (C4×Dic7).122C22, C23.18D14.3C2, C2.42(D4.10D14), (C22×Dic7).128C22, (C2×C4).59(C22×D7), (C7×C4⋊C4).175C22, (C7×C22⋊C4).50C22, (C7×C22.D4).3C2, SmallGroup(448,1104)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 780 in 196 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C7, C2×C4, C2×C4 [×4], C2×C4 [×10], D4, Q8 [×3], C23 [×2], C14, C14 [×2], C14 [×2], C42 [×3], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×14], C22×C4, C22×C4, C2×D4, C2×Q8 [×3], Dic7 [×8], C28 [×5], C2×C14, C2×C14 [×6], C22⋊Q8 [×4], C22.D4, C22.D4, C4.4D4, C42.C2 [×2], C42⋊2C2 [×4], C4⋊Q8 [×2], Dic14 [×3], C2×Dic7 [×8], C2×Dic7, C2×C28, C2×C28 [×4], C2×C28, C7×D4, C22×C14 [×2], C22.57C24, C4×Dic7, C4×Dic7 [×2], Dic7⋊C4 [×10], C4⋊Dic7 [×4], C23.D7, C23.D7 [×6], C7×C22⋊C4, C7×C22⋊C4 [×2], C7×C4⋊C4 [×2], C2×Dic14, C2×Dic14 [×2], C22×Dic7, C22×C28, D4×C14, C22⋊Dic14 [×2], C23.D14 [×4], C28⋊Q8 [×2], Dic7.Q8 [×2], C28.48D4 [×2], C23.18D14, C28.17D4, C7×C22.D4, C14.812- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ (1+4), 2- (1+4) [×2], C22×D7 [×7], C22.57C24, C23×D7, D4⋊6D14, D4.10D14 [×2], C14.812- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=e2=a7b2, bab-1=cac-1=eae-1=a-1, ad=da, cbc-1=b-1, bd=db, ebe-1=a7b, dcd-1=a7c, ce=ec, ede-1=a7b2d >
►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 80 130 216)(2 79 131 215)(3 78 132 214)(4 77 133 213)(5 76 134 212)(6 75 135 211)(7 74 136 224)(8 73 137 223)(9 72 138 222)(10 71 139 221)(11 84 140 220)(12 83 127 219)(13 82 128 218)(14 81 129 217)(15 91 206 37)(16 90 207 36)(17 89 208 35)(18 88 209 34)(19 87 210 33)(20 86 197 32)(21 85 198 31)(22 98 199 30)(23 97 200 29)(24 96 201 42)(25 95 202 41)(26 94 203 40)(27 93 204 39)(28 92 205 38)(43 157 102 144)(44 156 103 143)(45 155 104 142)(46 168 105 141)(47 167 106 154)(48 166 107 153)(49 165 108 152)(50 164 109 151)(51 163 110 150)(52 162 111 149)(53 161 112 148)(54 160 99 147)(55 159 100 146)(56 158 101 145)(57 124 196 182)(58 123 183 181)(59 122 184 180)(60 121 185 179)(61 120 186 178)(62 119 187 177)(63 118 188 176)(64 117 189 175)(65 116 190 174)(66 115 191 173)(67 114 192 172)(68 113 193 171)(69 126 194 170)(70 125 195 169)
(1 143 8 150)(2 142 9 149)(3 141 10 148)(4 154 11 147)(5 153 12 146)(6 152 13 145)(7 151 14 144)(15 177 22 170)(16 176 23 169)(17 175 24 182)(18 174 25 181)(19 173 26 180)(20 172 27 179)(21 171 28 178)(29 70 36 63)(30 69 37 62)(31 68 38 61)(32 67 39 60)(33 66 40 59)(34 65 41 58)(35 64 42 57)(43 224 50 217)(44 223 51 216)(45 222 52 215)(46 221 53 214)(47 220 54 213)(48 219 55 212)(49 218 56 211)(71 112 78 105)(72 111 79 104)(73 110 80 103)(74 109 81 102)(75 108 82 101)(76 107 83 100)(77 106 84 99)(85 193 92 186)(86 192 93 185)(87 191 94 184)(88 190 95 183)(89 189 96 196)(90 188 97 195)(91 187 98 194)(113 205 120 198)(114 204 121 197)(115 203 122 210)(116 202 123 209)(117 201 124 208)(118 200 125 207)(119 199 126 206)(127 159 134 166)(128 158 135 165)(129 157 136 164)(130 156 137 163)(131 155 138 162)(132 168 139 161)(133 167 140 160)
(1 34 137 95)(2 35 138 96)(3 36 139 97)(4 37 140 98)(5 38 127 85)(6 39 128 86)(7 40 129 87)(8 41 130 88)(9 42 131 89)(10 29 132 90)(11 30 133 91)(12 31 134 92)(13 32 135 93)(14 33 136 94)(15 220 199 77)(16 221 200 78)(17 222 201 79)(18 223 202 80)(19 224 203 81)(20 211 204 82)(21 212 205 83)(22 213 206 84)(23 214 207 71)(24 215 208 72)(25 216 209 73)(26 217 210 74)(27 218 197 75)(28 219 198 76)(43 122 109 173)(44 123 110 174)(45 124 111 175)(46 125 112 176)(47 126 99 177)(48 113 100 178)(49 114 101 179)(50 115 102 180)(51 116 103 181)(52 117 104 182)(53 118 105 169)(54 119 106 170)(55 120 107 171)(56 121 108 172)(57 162 189 142)(58 163 190 143)(59 164 191 144)(60 165 192 145)(61 166 193 146)(62 167 194 147)(63 168 195 148)(64 155 196 149)(65 156 183 150)(66 157 184 151)(67 158 185 152)(68 159 186 153)(69 160 187 154)(70 161 188 141)
(1 209 137 25)(2 208 138 24)(3 207 139 23)(4 206 140 22)(5 205 127 21)(6 204 128 20)(7 203 129 19)(8 202 130 18)(9 201 131 17)(10 200 132 16)(11 199 133 15)(12 198 134 28)(13 197 135 27)(14 210 136 26)(29 221 90 78)(30 220 91 77)(31 219 92 76)(32 218 93 75)(33 217 94 74)(34 216 95 73)(35 215 96 72)(36 214 97 71)(37 213 98 84)(38 212 85 83)(39 211 86 82)(40 224 87 81)(41 223 88 80)(42 222 89 79)(43 184 109 66)(44 183 110 65)(45 196 111 64)(46 195 112 63)(47 194 99 62)(48 193 100 61)(49 192 101 60)(50 191 102 59)(51 190 103 58)(52 189 104 57)(53 188 105 70)(54 187 106 69)(55 186 107 68)(56 185 108 67)(113 166 178 146)(114 165 179 145)(115 164 180 144)(116 163 181 143)(117 162 182 142)(118 161 169 141)(119 160 170 154)(120 159 171 153)(121 158 172 152)(122 157 173 151)(123 156 174 150)(124 155 175 149)(125 168 176 148)(126 167 177 147)
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), (1,80,130,216)(2,79,131,215)(3,78,132,214)(4,77,133,213)(5,76,134,212)(6,75,135,211)(7,74,136,224)(8,73,137,223)(9,72,138,222)(10,71,139,221)(11,84,140,220)(12,83,127,219)(13,82,128,218)(14,81,129,217)(15,91,206,37)(16,90,207,36)(17,89,208,35)(18,88,209,34)(19,87,210,33)(20,86,197,32)(21,85,198,31)(22,98,199,30)(23,97,200,29)(24,96,201,42)(25,95,202,41)(26,94,203,40)(27,93,204,39)(28,92,205,38)(43,157,102,144)(44,156,103,143)(45,155,104,142)(46,168,105,141)(47,167,106,154)(48,166,107,153)(49,165,108,152)(50,164,109,151)(51,163,110,150)(52,162,111,149)(53,161,112,148)(54,160,99,147)(55,159,100,146)(56,158,101,145)(57,124,196,182)(58,123,183,181)(59,122,184,180)(60,121,185,179)(61,120,186,178)(62,119,187,177)(63,118,188,176)(64,117,189,175)(65,116,190,174)(66,115,191,173)(67,114,192,172)(68,113,193,171)(69,126,194,170)(70,125,195,169), (1,143,8,150)(2,142,9,149)(3,141,10,148)(4,154,11,147)(5,153,12,146)(6,152,13,145)(7,151,14,144)(15,177,22,170)(16,176,23,169)(17,175,24,182)(18,174,25,181)(19,173,26,180)(20,172,27,179)(21,171,28,178)(29,70,36,63)(30,69,37,62)(31,68,38,61)(32,67,39,60)(33,66,40,59)(34,65,41,58)(35,64,42,57)(43,224,50,217)(44,223,51,216)(45,222,52,215)(46,221,53,214)(47,220,54,213)(48,219,55,212)(49,218,56,211)(71,112,78,105)(72,111,79,104)(73,110,80,103)(74,109,81,102)(75,108,82,101)(76,107,83,100)(77,106,84,99)(85,193,92,186)(86,192,93,185)(87,191,94,184)(88,190,95,183)(89,189,96,196)(90,188,97,195)(91,187,98,194)(113,205,120,198)(114,204,121,197)(115,203,122,210)(116,202,123,209)(117,201,124,208)(118,200,125,207)(119,199,126,206)(127,159,134,166)(128,158,135,165)(129,157,136,164)(130,156,137,163)(131,155,138,162)(132,168,139,161)(133,167,140,160), (1,34,137,95)(2,35,138,96)(3,36,139,97)(4,37,140,98)(5,38,127,85)(6,39,128,86)(7,40,129,87)(8,41,130,88)(9,42,131,89)(10,29,132,90)(11,30,133,91)(12,31,134,92)(13,32,135,93)(14,33,136,94)(15,220,199,77)(16,221,200,78)(17,222,201,79)(18,223,202,80)(19,224,203,81)(20,211,204,82)(21,212,205,83)(22,213,206,84)(23,214,207,71)(24,215,208,72)(25,216,209,73)(26,217,210,74)(27,218,197,75)(28,219,198,76)(43,122,109,173)(44,123,110,174)(45,124,111,175)(46,125,112,176)(47,126,99,177)(48,113,100,178)(49,114,101,179)(50,115,102,180)(51,116,103,181)(52,117,104,182)(53,118,105,169)(54,119,106,170)(55,120,107,171)(56,121,108,172)(57,162,189,142)(58,163,190,143)(59,164,191,144)(60,165,192,145)(61,166,193,146)(62,167,194,147)(63,168,195,148)(64,155,196,149)(65,156,183,150)(66,157,184,151)(67,158,185,152)(68,159,186,153)(69,160,187,154)(70,161,188,141), (1,209,137,25)(2,208,138,24)(3,207,139,23)(4,206,140,22)(5,205,127,21)(6,204,128,20)(7,203,129,19)(8,202,130,18)(9,201,131,17)(10,200,132,16)(11,199,133,15)(12,198,134,28)(13,197,135,27)(14,210,136,26)(29,221,90,78)(30,220,91,77)(31,219,92,76)(32,218,93,75)(33,217,94,74)(34,216,95,73)(35,215,96,72)(36,214,97,71)(37,213,98,84)(38,212,85,83)(39,211,86,82)(40,224,87,81)(41,223,88,80)(42,222,89,79)(43,184,109,66)(44,183,110,65)(45,196,111,64)(46,195,112,63)(47,194,99,62)(48,193,100,61)(49,192,101,60)(50,191,102,59)(51,190,103,58)(52,189,104,57)(53,188,105,70)(54,187,106,69)(55,186,107,68)(56,185,108,67)(113,166,178,146)(114,165,179,145)(115,164,180,144)(116,163,181,143)(117,162,182,142)(118,161,169,141)(119,160,170,154)(120,159,171,153)(121,158,172,152)(122,157,173,151)(123,156,174,150)(124,155,175,149)(125,168,176,148)(126,167,177,147)>;
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), (1,80,130,216)(2,79,131,215)(3,78,132,214)(4,77,133,213)(5,76,134,212)(6,75,135,211)(7,74,136,224)(8,73,137,223)(9,72,138,222)(10,71,139,221)(11,84,140,220)(12,83,127,219)(13,82,128,218)(14,81,129,217)(15,91,206,37)(16,90,207,36)(17,89,208,35)(18,88,209,34)(19,87,210,33)(20,86,197,32)(21,85,198,31)(22,98,199,30)(23,97,200,29)(24,96,201,42)(25,95,202,41)(26,94,203,40)(27,93,204,39)(28,92,205,38)(43,157,102,144)(44,156,103,143)(45,155,104,142)(46,168,105,141)(47,167,106,154)(48,166,107,153)(49,165,108,152)(50,164,109,151)(51,163,110,150)(52,162,111,149)(53,161,112,148)(54,160,99,147)(55,159,100,146)(56,158,101,145)(57,124,196,182)(58,123,183,181)(59,122,184,180)(60,121,185,179)(61,120,186,178)(62,119,187,177)(63,118,188,176)(64,117,189,175)(65,116,190,174)(66,115,191,173)(67,114,192,172)(68,113,193,171)(69,126,194,170)(70,125,195,169), (1,143,8,150)(2,142,9,149)(3,141,10,148)(4,154,11,147)(5,153,12,146)(6,152,13,145)(7,151,14,144)(15,177,22,170)(16,176,23,169)(17,175,24,182)(18,174,25,181)(19,173,26,180)(20,172,27,179)(21,171,28,178)(29,70,36,63)(30,69,37,62)(31,68,38,61)(32,67,39,60)(33,66,40,59)(34,65,41,58)(35,64,42,57)(43,224,50,217)(44,223,51,216)(45,222,52,215)(46,221,53,214)(47,220,54,213)(48,219,55,212)(49,218,56,211)(71,112,78,105)(72,111,79,104)(73,110,80,103)(74,109,81,102)(75,108,82,101)(76,107,83,100)(77,106,84,99)(85,193,92,186)(86,192,93,185)(87,191,94,184)(88,190,95,183)(89,189,96,196)(90,188,97,195)(91,187,98,194)(113,205,120,198)(114,204,121,197)(115,203,122,210)(116,202,123,209)(117,201,124,208)(118,200,125,207)(119,199,126,206)(127,159,134,166)(128,158,135,165)(129,157,136,164)(130,156,137,163)(131,155,138,162)(132,168,139,161)(133,167,140,160), (1,34,137,95)(2,35,138,96)(3,36,139,97)(4,37,140,98)(5,38,127,85)(6,39,128,86)(7,40,129,87)(8,41,130,88)(9,42,131,89)(10,29,132,90)(11,30,133,91)(12,31,134,92)(13,32,135,93)(14,33,136,94)(15,220,199,77)(16,221,200,78)(17,222,201,79)(18,223,202,80)(19,224,203,81)(20,211,204,82)(21,212,205,83)(22,213,206,84)(23,214,207,71)(24,215,208,72)(25,216,209,73)(26,217,210,74)(27,218,197,75)(28,219,198,76)(43,122,109,173)(44,123,110,174)(45,124,111,175)(46,125,112,176)(47,126,99,177)(48,113,100,178)(49,114,101,179)(50,115,102,180)(51,116,103,181)(52,117,104,182)(53,118,105,169)(54,119,106,170)(55,120,107,171)(56,121,108,172)(57,162,189,142)(58,163,190,143)(59,164,191,144)(60,165,192,145)(61,166,193,146)(62,167,194,147)(63,168,195,148)(64,155,196,149)(65,156,183,150)(66,157,184,151)(67,158,185,152)(68,159,186,153)(69,160,187,154)(70,161,188,141), (1,209,137,25)(2,208,138,24)(3,207,139,23)(4,206,140,22)(5,205,127,21)(6,204,128,20)(7,203,129,19)(8,202,130,18)(9,201,131,17)(10,200,132,16)(11,199,133,15)(12,198,134,28)(13,197,135,27)(14,210,136,26)(29,221,90,78)(30,220,91,77)(31,219,92,76)(32,218,93,75)(33,217,94,74)(34,216,95,73)(35,215,96,72)(36,214,97,71)(37,213,98,84)(38,212,85,83)(39,211,86,82)(40,224,87,81)(41,223,88,80)(42,222,89,79)(43,184,109,66)(44,183,110,65)(45,196,111,64)(46,195,112,63)(47,194,99,62)(48,193,100,61)(49,192,101,60)(50,191,102,59)(51,190,103,58)(52,189,104,57)(53,188,105,70)(54,187,106,69)(55,186,107,68)(56,185,108,67)(113,166,178,146)(114,165,179,145)(115,164,180,144)(116,163,181,143)(117,162,182,142)(118,161,169,141)(119,160,170,154)(120,159,171,153)(121,158,172,152)(122,157,173,151)(123,156,174,150)(124,155,175,149)(125,168,176,148)(126,167,177,147) );
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)], [(1,80,130,216),(2,79,131,215),(3,78,132,214),(4,77,133,213),(5,76,134,212),(6,75,135,211),(7,74,136,224),(8,73,137,223),(9,72,138,222),(10,71,139,221),(11,84,140,220),(12,83,127,219),(13,82,128,218),(14,81,129,217),(15,91,206,37),(16,90,207,36),(17,89,208,35),(18,88,209,34),(19,87,210,33),(20,86,197,32),(21,85,198,31),(22,98,199,30),(23,97,200,29),(24,96,201,42),(25,95,202,41),(26,94,203,40),(27,93,204,39),(28,92,205,38),(43,157,102,144),(44,156,103,143),(45,155,104,142),(46,168,105,141),(47,167,106,154),(48,166,107,153),(49,165,108,152),(50,164,109,151),(51,163,110,150),(52,162,111,149),(53,161,112,148),(54,160,99,147),(55,159,100,146),(56,158,101,145),(57,124,196,182),(58,123,183,181),(59,122,184,180),(60,121,185,179),(61,120,186,178),(62,119,187,177),(63,118,188,176),(64,117,189,175),(65,116,190,174),(66,115,191,173),(67,114,192,172),(68,113,193,171),(69,126,194,170),(70,125,195,169)], [(1,143,8,150),(2,142,9,149),(3,141,10,148),(4,154,11,147),(5,153,12,146),(6,152,13,145),(7,151,14,144),(15,177,22,170),(16,176,23,169),(17,175,24,182),(18,174,25,181),(19,173,26,180),(20,172,27,179),(21,171,28,178),(29,70,36,63),(30,69,37,62),(31,68,38,61),(32,67,39,60),(33,66,40,59),(34,65,41,58),(35,64,42,57),(43,224,50,217),(44,223,51,216),(45,222,52,215),(46,221,53,214),(47,220,54,213),(48,219,55,212),(49,218,56,211),(71,112,78,105),(72,111,79,104),(73,110,80,103),(74,109,81,102),(75,108,82,101),(76,107,83,100),(77,106,84,99),(85,193,92,186),(86,192,93,185),(87,191,94,184),(88,190,95,183),(89,189,96,196),(90,188,97,195),(91,187,98,194),(113,205,120,198),(114,204,121,197),(115,203,122,210),(116,202,123,209),(117,201,124,208),(118,200,125,207),(119,199,126,206),(127,159,134,166),(128,158,135,165),(129,157,136,164),(130,156,137,163),(131,155,138,162),(132,168,139,161),(133,167,140,160)], [(1,34,137,95),(2,35,138,96),(3,36,139,97),(4,37,140,98),(5,38,127,85),(6,39,128,86),(7,40,129,87),(8,41,130,88),(9,42,131,89),(10,29,132,90),(11,30,133,91),(12,31,134,92),(13,32,135,93),(14,33,136,94),(15,220,199,77),(16,221,200,78),(17,222,201,79),(18,223,202,80),(19,224,203,81),(20,211,204,82),(21,212,205,83),(22,213,206,84),(23,214,207,71),(24,215,208,72),(25,216,209,73),(26,217,210,74),(27,218,197,75),(28,219,198,76),(43,122,109,173),(44,123,110,174),(45,124,111,175),(46,125,112,176),(47,126,99,177),(48,113,100,178),(49,114,101,179),(50,115,102,180),(51,116,103,181),(52,117,104,182),(53,118,105,169),(54,119,106,170),(55,120,107,171),(56,121,108,172),(57,162,189,142),(58,163,190,143),(59,164,191,144),(60,165,192,145),(61,166,193,146),(62,167,194,147),(63,168,195,148),(64,155,196,149),(65,156,183,150),(66,157,184,151),(67,158,185,152),(68,159,186,153),(69,160,187,154),(70,161,188,141)], [(1,209,137,25),(2,208,138,24),(3,207,139,23),(4,206,140,22),(5,205,127,21),(6,204,128,20),(7,203,129,19),(8,202,130,18),(9,201,131,17),(10,200,132,16),(11,199,133,15),(12,198,134,28),(13,197,135,27),(14,210,136,26),(29,221,90,78),(30,220,91,77),(31,219,92,76),(32,218,93,75),(33,217,94,74),(34,216,95,73),(35,215,96,72),(36,214,97,71),(37,213,98,84),(38,212,85,83),(39,211,86,82),(40,224,87,81),(41,223,88,80),(42,222,89,79),(43,184,109,66),(44,183,110,65),(45,196,111,64),(46,195,112,63),(47,194,99,62),(48,193,100,61),(49,192,101,60),(50,191,102,59),(51,190,103,58),(52,189,104,57),(53,188,105,70),(54,187,106,69),(55,186,107,68),(56,185,108,67),(113,166,178,146),(114,165,179,145),(115,164,180,144),(116,163,181,143),(117,162,182,142),(118,161,169,141),(119,160,170,154),(120,159,171,153),(121,158,172,152),(122,157,173,151),(123,156,174,150),(124,155,175,149),(125,168,176,148),(126,167,177,147)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 6 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 6 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 13 |
| 9 | 20 | 13 | 22 | 0 | 0 | 0 | 0 |
| 15 | 7 | 14 | 27 | 0 | 0 | 0 | 0 |
| 5 | 24 | 20 | 22 | 0 | 0 | 0 | 0 |
| 10 | 24 | 11 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 23 | 12 | 24 |
| 0 | 0 | 0 | 0 | 0 | 8 | 26 | 26 |
| 0 | 0 | 0 | 0 | 12 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 12 | 21 | 21 |
| 20 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 10 | 0 | 0 | 0 | 0 |
| 24 | 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 24 | 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 20 | 0 | 7 |
| 0 | 0 | 0 | 0 | 28 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 9 |
| 28 | 1 | 21 | 17 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 14 | 1 | 2 |
| 0 | 0 | 0 | 0 | 9 | 0 | 28 | 28 |
| 9 | 20 | 1 | 5 | 0 | 0 | 0 | 0 |
| 15 | 7 | 19 | 10 | 0 | 0 | 0 | 0 |
| 5 | 24 | 20 | 22 | 0 | 0 | 0 | 0 |
| 10 | 24 | 11 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 21 | 3 | 3 |
| 0 | 0 | 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 17 | 8 | 8 |
G:=sub<GL(8,GF(29))| [6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,4,0,5,0,0,0,0,0,0,13,0,5,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,28,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13],[9,15,5,10,0,0,0,0,20,7,24,24,0,0,0,0,13,14,20,11,0,0,0,0,22,27,22,22,0,0,0,0,0,0,0,0,0,0,12,17,0,0,0,0,23,8,9,12,0,0,0,0,12,26,0,21,0,0,0,0,24,26,0,21],[20,0,24,0,0,0,0,0,0,7,0,24,0,0,0,0,28,0,9,0,0,0,0,0,0,10,0,22,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,14,20,21,9,0,0,0,0,1,0,0,0,0,0,0,0,2,7,0,9],[28,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,21,0,1,0,0,0,0,0,17,26,28,28,0,0,0,0,0,0,0,0,1,7,0,9,0,0,0,0,8,28,14,0,0,0,0,0,0,0,1,28,0,0,0,0,0,0,2,28],[9,15,5,10,0,0,0,0,20,7,24,24,0,0,0,0,1,19,20,11,0,0,0,0,5,10,22,22,0,0,0,0,0,0,0,0,0,0,17,12,0,0,0,0,0,21,0,17,0,0,0,0,17,3,0,8,0,0,0,0,0,3,0,8] >;
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4E | 4F | ··· | 4M | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | D4⋊6D14 | D4.10D14 |
| kernel | C14.812- (1+4) | C22⋊Dic14 | C23.D14 | C28⋊Q8 | Dic7.Q8 | C28.48D4 | C23.18D14 | C28.17D4 | C7×C22.D4 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 4 | 2 | 2 | 2 | 1 | 1 | 1 | 3 | 9 | 6 | 3 | 3 | 1 | 2 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_{14}._{81}2_-^{(1+4)} % in TeX
G:=Group("C14.81ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1104);
// by ID
G=gap.SmallGroup(448,1104);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,219,184,1571,570,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=a^7,d^2=e^2=a^7*b^2,b*a*b^-1=c*a*c^-1=e*a*e^-1=a^-1,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e^-1=a^7*b,d*c*d^-1=a^7*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations
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