?
G = C14.542+ (1+4) order 448 = 26·7
54th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.542+ (1+4), C14.202- (1+4), C22⋊Q8⋊15D7, C4⋊C4.193D14, (C2×Q8).76D14, D28⋊C4⋊29C2, D14⋊3Q8⋊20C2, D14.5(C4○D4), D14⋊D4.1C2, Dic7.Q8⋊20C2, C22⋊C4.18D14, D14.D4⋊26C2, D14.5D4⋊20C2, (C2×C28).627C23, (C2×C14).182C24, (C22×C4).244D14, C2.56(D4⋊6D14), D14⋊C4.148C22, (C2×D28).151C22, C23.D14⋊22C2, Dic7⋊C4.80C22, C4⋊Dic7.218C22, (Q8×C14).112C22, C23.121(C22×D7), C22.203(C23×D7), C23.23D14⋊24C2, (C22×C14).210C23, (C22×C28).382C22, C7⋊4(C22.33C24), (C4×Dic7).110C22, (C2×Dic7).237C23, (C22×D7).203C23, C23.D7.122C22, C2.21(Q8.10D14), (D7×C4⋊C4)⋊29C2, (C4×C7⋊D4)⋊58C2, C2.53(D7×C4○D4), C4⋊C4⋊D7⋊18C2, (C7×C22⋊Q8)⋊18C2, C14.165(C2×C4○D4), (C2×C4×D7).210C22, (C2×C4).52(C22×D7), (C7×C4⋊C4).163C22, (C2×C7⋊D4).129C22, (C7×C22⋊C4).37C22, SmallGroup(448,1091)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 988 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×10], C7, C2×C4 [×6], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×2], D7 [×3], C14 [×3], C14, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×3], C4⋊C4 [×11], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8, Dic7 [×6], C28 [×6], D14 [×2], D14 [×5], C2×C14, C2×C14 [×3], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4 [×4], C42.C2 [×2], C42⋊2C2 [×2], C4×D7 [×5], D28 [×2], C2×Dic7 [×6], C7⋊D4 [×3], C2×C28 [×6], C2×C28, C7×Q8, C22×D7 [×2], C22×C14, C22.33C24, C4×Dic7 [×2], Dic7⋊C4 [×8], C4⋊Dic7 [×3], D14⋊C4 [×6], C23.D7 [×2], C7×C22⋊C4 [×2], C7×C4⋊C4 [×3], C2×C4×D7 [×4], C2×D28, C2×C7⋊D4 [×2], C22×C28, Q8×C14, C23.D14, D14.D4 [×2], D14⋊D4, Dic7.Q8 [×2], D7×C4⋊C4, D28⋊C4, D14.5D4, C4⋊C4⋊D7, C4×C7⋊D4, C23.23D14, D14⋊3Q8 [×2], C7×C22⋊Q8, C14.542+ (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, C23×D7, D4⋊6D14, Q8.10D14, D7×C4○D4, C14.542+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=1, c2=e2=a7, d2=a7b2, bab-1=dad-1=eae-1=a-1, ac=ca, cbc-1=a7b-1, dbd-1=a7b, be=eb, 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 56 151 133)(2 55 152 132)(3 54 153 131)(4 53 154 130)(5 52 141 129)(6 51 142 128)(7 50 143 127)(8 49 144 140)(9 48 145 139)(10 47 146 138)(11 46 147 137)(12 45 148 136)(13 44 149 135)(14 43 150 134)(15 217 119 162)(16 216 120 161)(17 215 121 160)(18 214 122 159)(19 213 123 158)(20 212 124 157)(21 211 125 156)(22 224 126 155)(23 223 113 168)(24 222 114 167)(25 221 115 166)(26 220 116 165)(27 219 117 164)(28 218 118 163)(29 78 95 183)(30 77 96 196)(31 76 97 195)(32 75 98 194)(33 74 85 193)(34 73 86 192)(35 72 87 191)(36 71 88 190)(37 84 89 189)(38 83 90 188)(39 82 91 187)(40 81 92 186)(41 80 93 185)(42 79 94 184)(57 107 201 177)(58 106 202 176)(59 105 203 175)(60 104 204 174)(61 103 205 173)(62 102 206 172)(63 101 207 171)(64 100 208 170)(65 99 209 169)(66 112 210 182)(67 111 197 181)(68 110 198 180)(69 109 199 179)(70 108 200 178)
(1 140 8 133)(2 127 9 134)(3 128 10 135)(4 129 11 136)(5 130 12 137)(6 131 13 138)(7 132 14 139)(15 157 22 164)(16 158 23 165)(17 159 24 166)(18 160 25 167)(19 161 26 168)(20 162 27 155)(21 163 28 156)(29 188 36 195)(30 189 37 196)(31 190 38 183)(32 191 39 184)(33 192 40 185)(34 193 41 186)(35 194 42 187)(43 152 50 145)(44 153 51 146)(45 154 52 147)(46 141 53 148)(47 142 54 149)(48 143 55 150)(49 144 56 151)(57 105 64 112)(58 106 65 99)(59 107 66 100)(60 108 67 101)(61 109 68 102)(62 110 69 103)(63 111 70 104)(71 90 78 97)(72 91 79 98)(73 92 80 85)(74 93 81 86)(75 94 82 87)(76 95 83 88)(77 96 84 89)(113 220 120 213)(114 221 121 214)(115 222 122 215)(116 223 123 216)(117 224 124 217)(118 211 125 218)(119 212 126 219)(169 202 176 209)(170 203 177 210)(171 204 178 197)(172 205 179 198)(173 206 180 199)(174 207 181 200)(175 208 182 201)
(1 169 144 106)(2 182 145 105)(3 181 146 104)(4 180 147 103)(5 179 148 102)(6 178 149 101)(7 177 150 100)(8 176 151 99)(9 175 152 112)(10 174 153 111)(11 173 154 110)(12 172 141 109)(13 171 142 108)(14 170 143 107)(15 36 126 95)(16 35 113 94)(17 34 114 93)(18 33 115 92)(19 32 116 91)(20 31 117 90)(21 30 118 89)(22 29 119 88)(23 42 120 87)(24 41 121 86)(25 40 122 85)(26 39 123 98)(27 38 124 97)(28 37 125 96)(43 57 127 208)(44 70 128 207)(45 69 129 206)(46 68 130 205)(47 67 131 204)(48 66 132 203)(49 65 133 202)(50 64 134 201)(51 63 135 200)(52 62 136 199)(53 61 137 198)(54 60 138 197)(55 59 139 210)(56 58 140 209)(71 162 183 224)(72 161 184 223)(73 160 185 222)(74 159 186 221)(75 158 187 220)(76 157 188 219)(77 156 189 218)(78 155 190 217)(79 168 191 216)(80 167 192 215)(81 166 193 214)(82 165 194 213)(83 164 195 212)(84 163 196 211)
(1 37 8 30)(2 36 9 29)(3 35 10 42)(4 34 11 41)(5 33 12 40)(6 32 13 39)(7 31 14 38)(15 175 22 182)(16 174 23 181)(17 173 24 180)(18 172 25 179)(19 171 26 178)(20 170 27 177)(21 169 28 176)(43 83 50 76)(44 82 51 75)(45 81 52 74)(46 80 53 73)(47 79 54 72)(48 78 55 71)(49 77 56 84)(57 212 64 219)(58 211 65 218)(59 224 66 217)(60 223 67 216)(61 222 68 215)(62 221 69 214)(63 220 70 213)(85 148 92 141)(86 147 93 154)(87 146 94 153)(88 145 95 152)(89 144 96 151)(90 143 97 150)(91 142 98 149)(99 118 106 125)(100 117 107 124)(101 116 108 123)(102 115 109 122)(103 114 110 121)(104 113 111 120)(105 126 112 119)(127 195 134 188)(128 194 135 187)(129 193 136 186)(130 192 137 185)(131 191 138 184)(132 190 139 183)(133 189 140 196)(155 210 162 203)(156 209 163 202)(157 208 164 201)(158 207 165 200)(159 206 166 199)(160 205 167 198)(161 204 168 197)
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), (1,56,151,133)(2,55,152,132)(3,54,153,131)(4,53,154,130)(5,52,141,129)(6,51,142,128)(7,50,143,127)(8,49,144,140)(9,48,145,139)(10,47,146,138)(11,46,147,137)(12,45,148,136)(13,44,149,135)(14,43,150,134)(15,217,119,162)(16,216,120,161)(17,215,121,160)(18,214,122,159)(19,213,123,158)(20,212,124,157)(21,211,125,156)(22,224,126,155)(23,223,113,168)(24,222,114,167)(25,221,115,166)(26,220,116,165)(27,219,117,164)(28,218,118,163)(29,78,95,183)(30,77,96,196)(31,76,97,195)(32,75,98,194)(33,74,85,193)(34,73,86,192)(35,72,87,191)(36,71,88,190)(37,84,89,189)(38,83,90,188)(39,82,91,187)(40,81,92,186)(41,80,93,185)(42,79,94,184)(57,107,201,177)(58,106,202,176)(59,105,203,175)(60,104,204,174)(61,103,205,173)(62,102,206,172)(63,101,207,171)(64,100,208,170)(65,99,209,169)(66,112,210,182)(67,111,197,181)(68,110,198,180)(69,109,199,179)(70,108,200,178), (1,140,8,133)(2,127,9,134)(3,128,10,135)(4,129,11,136)(5,130,12,137)(6,131,13,138)(7,132,14,139)(15,157,22,164)(16,158,23,165)(17,159,24,166)(18,160,25,167)(19,161,26,168)(20,162,27,155)(21,163,28,156)(29,188,36,195)(30,189,37,196)(31,190,38,183)(32,191,39,184)(33,192,40,185)(34,193,41,186)(35,194,42,187)(43,152,50,145)(44,153,51,146)(45,154,52,147)(46,141,53,148)(47,142,54,149)(48,143,55,150)(49,144,56,151)(57,105,64,112)(58,106,65,99)(59,107,66,100)(60,108,67,101)(61,109,68,102)(62,110,69,103)(63,111,70,104)(71,90,78,97)(72,91,79,98)(73,92,80,85)(74,93,81,86)(75,94,82,87)(76,95,83,88)(77,96,84,89)(113,220,120,213)(114,221,121,214)(115,222,122,215)(116,223,123,216)(117,224,124,217)(118,211,125,218)(119,212,126,219)(169,202,176,209)(170,203,177,210)(171,204,178,197)(172,205,179,198)(173,206,180,199)(174,207,181,200)(175,208,182,201), (1,169,144,106)(2,182,145,105)(3,181,146,104)(4,180,147,103)(5,179,148,102)(6,178,149,101)(7,177,150,100)(8,176,151,99)(9,175,152,112)(10,174,153,111)(11,173,154,110)(12,172,141,109)(13,171,142,108)(14,170,143,107)(15,36,126,95)(16,35,113,94)(17,34,114,93)(18,33,115,92)(19,32,116,91)(20,31,117,90)(21,30,118,89)(22,29,119,88)(23,42,120,87)(24,41,121,86)(25,40,122,85)(26,39,123,98)(27,38,124,97)(28,37,125,96)(43,57,127,208)(44,70,128,207)(45,69,129,206)(46,68,130,205)(47,67,131,204)(48,66,132,203)(49,65,133,202)(50,64,134,201)(51,63,135,200)(52,62,136,199)(53,61,137,198)(54,60,138,197)(55,59,139,210)(56,58,140,209)(71,162,183,224)(72,161,184,223)(73,160,185,222)(74,159,186,221)(75,158,187,220)(76,157,188,219)(77,156,189,218)(78,155,190,217)(79,168,191,216)(80,167,192,215)(81,166,193,214)(82,165,194,213)(83,164,195,212)(84,163,196,211), (1,37,8,30)(2,36,9,29)(3,35,10,42)(4,34,11,41)(5,33,12,40)(6,32,13,39)(7,31,14,38)(15,175,22,182)(16,174,23,181)(17,173,24,180)(18,172,25,179)(19,171,26,178)(20,170,27,177)(21,169,28,176)(43,83,50,76)(44,82,51,75)(45,81,52,74)(46,80,53,73)(47,79,54,72)(48,78,55,71)(49,77,56,84)(57,212,64,219)(58,211,65,218)(59,224,66,217)(60,223,67,216)(61,222,68,215)(62,221,69,214)(63,220,70,213)(85,148,92,141)(86,147,93,154)(87,146,94,153)(88,145,95,152)(89,144,96,151)(90,143,97,150)(91,142,98,149)(99,118,106,125)(100,117,107,124)(101,116,108,123)(102,115,109,122)(103,114,110,121)(104,113,111,120)(105,126,112,119)(127,195,134,188)(128,194,135,187)(129,193,136,186)(130,192,137,185)(131,191,138,184)(132,190,139,183)(133,189,140,196)(155,210,162,203)(156,209,163,202)(157,208,164,201)(158,207,165,200)(159,206,166,199)(160,205,167,198)(161,204,168,197)>;
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), (1,56,151,133)(2,55,152,132)(3,54,153,131)(4,53,154,130)(5,52,141,129)(6,51,142,128)(7,50,143,127)(8,49,144,140)(9,48,145,139)(10,47,146,138)(11,46,147,137)(12,45,148,136)(13,44,149,135)(14,43,150,134)(15,217,119,162)(16,216,120,161)(17,215,121,160)(18,214,122,159)(19,213,123,158)(20,212,124,157)(21,211,125,156)(22,224,126,155)(23,223,113,168)(24,222,114,167)(25,221,115,166)(26,220,116,165)(27,219,117,164)(28,218,118,163)(29,78,95,183)(30,77,96,196)(31,76,97,195)(32,75,98,194)(33,74,85,193)(34,73,86,192)(35,72,87,191)(36,71,88,190)(37,84,89,189)(38,83,90,188)(39,82,91,187)(40,81,92,186)(41,80,93,185)(42,79,94,184)(57,107,201,177)(58,106,202,176)(59,105,203,175)(60,104,204,174)(61,103,205,173)(62,102,206,172)(63,101,207,171)(64,100,208,170)(65,99,209,169)(66,112,210,182)(67,111,197,181)(68,110,198,180)(69,109,199,179)(70,108,200,178), (1,140,8,133)(2,127,9,134)(3,128,10,135)(4,129,11,136)(5,130,12,137)(6,131,13,138)(7,132,14,139)(15,157,22,164)(16,158,23,165)(17,159,24,166)(18,160,25,167)(19,161,26,168)(20,162,27,155)(21,163,28,156)(29,188,36,195)(30,189,37,196)(31,190,38,183)(32,191,39,184)(33,192,40,185)(34,193,41,186)(35,194,42,187)(43,152,50,145)(44,153,51,146)(45,154,52,147)(46,141,53,148)(47,142,54,149)(48,143,55,150)(49,144,56,151)(57,105,64,112)(58,106,65,99)(59,107,66,100)(60,108,67,101)(61,109,68,102)(62,110,69,103)(63,111,70,104)(71,90,78,97)(72,91,79,98)(73,92,80,85)(74,93,81,86)(75,94,82,87)(76,95,83,88)(77,96,84,89)(113,220,120,213)(114,221,121,214)(115,222,122,215)(116,223,123,216)(117,224,124,217)(118,211,125,218)(119,212,126,219)(169,202,176,209)(170,203,177,210)(171,204,178,197)(172,205,179,198)(173,206,180,199)(174,207,181,200)(175,208,182,201), (1,169,144,106)(2,182,145,105)(3,181,146,104)(4,180,147,103)(5,179,148,102)(6,178,149,101)(7,177,150,100)(8,176,151,99)(9,175,152,112)(10,174,153,111)(11,173,154,110)(12,172,141,109)(13,171,142,108)(14,170,143,107)(15,36,126,95)(16,35,113,94)(17,34,114,93)(18,33,115,92)(19,32,116,91)(20,31,117,90)(21,30,118,89)(22,29,119,88)(23,42,120,87)(24,41,121,86)(25,40,122,85)(26,39,123,98)(27,38,124,97)(28,37,125,96)(43,57,127,208)(44,70,128,207)(45,69,129,206)(46,68,130,205)(47,67,131,204)(48,66,132,203)(49,65,133,202)(50,64,134,201)(51,63,135,200)(52,62,136,199)(53,61,137,198)(54,60,138,197)(55,59,139,210)(56,58,140,209)(71,162,183,224)(72,161,184,223)(73,160,185,222)(74,159,186,221)(75,158,187,220)(76,157,188,219)(77,156,189,218)(78,155,190,217)(79,168,191,216)(80,167,192,215)(81,166,193,214)(82,165,194,213)(83,164,195,212)(84,163,196,211), (1,37,8,30)(2,36,9,29)(3,35,10,42)(4,34,11,41)(5,33,12,40)(6,32,13,39)(7,31,14,38)(15,175,22,182)(16,174,23,181)(17,173,24,180)(18,172,25,179)(19,171,26,178)(20,170,27,177)(21,169,28,176)(43,83,50,76)(44,82,51,75)(45,81,52,74)(46,80,53,73)(47,79,54,72)(48,78,55,71)(49,77,56,84)(57,212,64,219)(58,211,65,218)(59,224,66,217)(60,223,67,216)(61,222,68,215)(62,221,69,214)(63,220,70,213)(85,148,92,141)(86,147,93,154)(87,146,94,153)(88,145,95,152)(89,144,96,151)(90,143,97,150)(91,142,98,149)(99,118,106,125)(100,117,107,124)(101,116,108,123)(102,115,109,122)(103,114,110,121)(104,113,111,120)(105,126,112,119)(127,195,134,188)(128,194,135,187)(129,193,136,186)(130,192,137,185)(131,191,138,184)(132,190,139,183)(133,189,140,196)(155,210,162,203)(156,209,163,202)(157,208,164,201)(158,207,165,200)(159,206,166,199)(160,205,167,198)(161,204,168,197) );
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)], [(1,56,151,133),(2,55,152,132),(3,54,153,131),(4,53,154,130),(5,52,141,129),(6,51,142,128),(7,50,143,127),(8,49,144,140),(9,48,145,139),(10,47,146,138),(11,46,147,137),(12,45,148,136),(13,44,149,135),(14,43,150,134),(15,217,119,162),(16,216,120,161),(17,215,121,160),(18,214,122,159),(19,213,123,158),(20,212,124,157),(21,211,125,156),(22,224,126,155),(23,223,113,168),(24,222,114,167),(25,221,115,166),(26,220,116,165),(27,219,117,164),(28,218,118,163),(29,78,95,183),(30,77,96,196),(31,76,97,195),(32,75,98,194),(33,74,85,193),(34,73,86,192),(35,72,87,191),(36,71,88,190),(37,84,89,189),(38,83,90,188),(39,82,91,187),(40,81,92,186),(41,80,93,185),(42,79,94,184),(57,107,201,177),(58,106,202,176),(59,105,203,175),(60,104,204,174),(61,103,205,173),(62,102,206,172),(63,101,207,171),(64,100,208,170),(65,99,209,169),(66,112,210,182),(67,111,197,181),(68,110,198,180),(69,109,199,179),(70,108,200,178)], [(1,140,8,133),(2,127,9,134),(3,128,10,135),(4,129,11,136),(5,130,12,137),(6,131,13,138),(7,132,14,139),(15,157,22,164),(16,158,23,165),(17,159,24,166),(18,160,25,167),(19,161,26,168),(20,162,27,155),(21,163,28,156),(29,188,36,195),(30,189,37,196),(31,190,38,183),(32,191,39,184),(33,192,40,185),(34,193,41,186),(35,194,42,187),(43,152,50,145),(44,153,51,146),(45,154,52,147),(46,141,53,148),(47,142,54,149),(48,143,55,150),(49,144,56,151),(57,105,64,112),(58,106,65,99),(59,107,66,100),(60,108,67,101),(61,109,68,102),(62,110,69,103),(63,111,70,104),(71,90,78,97),(72,91,79,98),(73,92,80,85),(74,93,81,86),(75,94,82,87),(76,95,83,88),(77,96,84,89),(113,220,120,213),(114,221,121,214),(115,222,122,215),(116,223,123,216),(117,224,124,217),(118,211,125,218),(119,212,126,219),(169,202,176,209),(170,203,177,210),(171,204,178,197),(172,205,179,198),(173,206,180,199),(174,207,181,200),(175,208,182,201)], [(1,169,144,106),(2,182,145,105),(3,181,146,104),(4,180,147,103),(5,179,148,102),(6,178,149,101),(7,177,150,100),(8,176,151,99),(9,175,152,112),(10,174,153,111),(11,173,154,110),(12,172,141,109),(13,171,142,108),(14,170,143,107),(15,36,126,95),(16,35,113,94),(17,34,114,93),(18,33,115,92),(19,32,116,91),(20,31,117,90),(21,30,118,89),(22,29,119,88),(23,42,120,87),(24,41,121,86),(25,40,122,85),(26,39,123,98),(27,38,124,97),(28,37,125,96),(43,57,127,208),(44,70,128,207),(45,69,129,206),(46,68,130,205),(47,67,131,204),(48,66,132,203),(49,65,133,202),(50,64,134,201),(51,63,135,200),(52,62,136,199),(53,61,137,198),(54,60,138,197),(55,59,139,210),(56,58,140,209),(71,162,183,224),(72,161,184,223),(73,160,185,222),(74,159,186,221),(75,158,187,220),(76,157,188,219),(77,156,189,218),(78,155,190,217),(79,168,191,216),(80,167,192,215),(81,166,193,214),(82,165,194,213),(83,164,195,212),(84,163,196,211)], [(1,37,8,30),(2,36,9,29),(3,35,10,42),(4,34,11,41),(5,33,12,40),(6,32,13,39),(7,31,14,38),(15,175,22,182),(16,174,23,181),(17,173,24,180),(18,172,25,179),(19,171,26,178),(20,170,27,177),(21,169,28,176),(43,83,50,76),(44,82,51,75),(45,81,52,74),(46,80,53,73),(47,79,54,72),(48,78,55,71),(49,77,56,84),(57,212,64,219),(58,211,65,218),(59,224,66,217),(60,223,67,216),(61,222,68,215),(62,221,69,214),(63,220,70,213),(85,148,92,141),(86,147,93,154),(87,146,94,153),(88,145,95,152),(89,144,96,151),(90,143,97,150),(91,142,98,149),(99,118,106,125),(100,117,107,124),(101,116,108,123),(102,115,109,122),(103,114,110,121),(104,113,111,120),(105,126,112,119),(127,195,134,188),(128,194,135,187),(129,193,136,186),(130,192,137,185),(131,191,138,184),(132,190,139,183),(133,189,140,196),(155,210,162,203),(156,209,163,202),(157,208,164,201),(158,207,165,200),(159,206,166,199),(160,205,167,198),(161,204,168,197)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 10 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 21 | 21 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 26 |
| 9 | 0 | 9 | 16 | 0 | 0 | 0 | 0 |
| 14 | 20 | 21 | 19 | 0 | 0 | 0 | 0 |
| 17 | 4 | 7 | 19 | 0 | 0 | 0 | 0 |
| 27 | 5 | 20 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 22 | 16 | 0 |
| 0 | 0 | 0 | 0 | 10 | 4 | 19 | 13 |
| 0 | 0 | 0 | 0 | 16 | 0 | 25 | 22 |
| 0 | 0 | 0 | 0 | 19 | 13 | 10 | 4 |
| 9 | 0 | 9 | 16 | 0 | 0 | 0 | 0 |
| 0 | 9 | 22 | 22 | 0 | 0 | 0 | 0 |
| 20 | 25 | 20 | 0 | 0 | 0 | 0 | 0 |
| 9 | 24 | 0 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 22 | 16 | 0 |
| 0 | 0 | 0 | 0 | 7 | 4 | 0 | 16 |
| 0 | 0 | 0 | 0 | 13 | 0 | 4 | 7 |
| 0 | 0 | 0 | 0 | 0 | 13 | 22 | 25 |
| 20 | 0 | 20 | 13 | 0 | 0 | 0 | 0 |
| 15 | 9 | 8 | 10 | 0 | 0 | 0 | 0 |
| 11 | 6 | 22 | 10 | 0 | 0 | 0 | 0 |
| 26 | 22 | 9 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 25 | 22 |
| 0 | 0 | 0 | 0 | 19 | 13 | 10 | 4 |
| 0 | 0 | 0 | 0 | 25 | 22 | 16 | 0 |
| 0 | 0 | 0 | 0 | 10 | 4 | 19 | 13 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 24 |
| 0 | 0 | 0 | 0 | 24 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 5 | 0 | 0 |
G:=sub<GL(8,GF(29))| [10,21,0,0,0,0,0,0,17,1,0,0,0,0,0,0,0,0,11,21,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,21,8,0,0,0,0,0,0,21,26,0,0,0,0,0,0,0,0,21,8,0,0,0,0,0,0,21,26],[9,14,17,27,0,0,0,0,0,20,4,5,0,0,0,0,9,21,7,20,0,0,0,0,16,19,19,22,0,0,0,0,0,0,0,0,25,10,16,19,0,0,0,0,22,4,0,13,0,0,0,0,16,19,25,10,0,0,0,0,0,13,22,4],[9,0,20,9,0,0,0,0,0,9,25,24,0,0,0,0,9,22,20,0,0,0,0,0,16,22,0,20,0,0,0,0,0,0,0,0,25,7,13,0,0,0,0,0,22,4,0,13,0,0,0,0,16,0,4,22,0,0,0,0,0,16,7,25],[20,15,11,26,0,0,0,0,0,9,6,22,0,0,0,0,20,8,22,9,0,0,0,0,13,10,10,7,0,0,0,0,0,0,0,0,16,19,25,10,0,0,0,0,0,13,22,4,0,0,0,0,25,10,16,19,0,0,0,0,22,4,0,13],[12,9,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,10,12,0,0,0,0,0,0,23,19,0,0,0,0,0,0,0,0,0,0,24,27,0,0,0,0,0,0,13,5,0,0,0,0,5,2,0,0,0,0,0,0,16,24,0,0] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | ··· | 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 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 14 | 14 | 28 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 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 | 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 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | D4⋊6D14 | Q8.10D14 | D7×C4○D4 |
| kernel | C14.542+ (1+4) | C23.D14 | D14.D4 | D14⋊D4 | Dic7.Q8 | D7×C4⋊C4 | D28⋊C4 | D14.5D4 | C4⋊C4⋊D7 | C4×C7⋊D4 | C23.23D14 | D14⋊3Q8 | C7×C22⋊Q8 | C22⋊Q8 | D14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C14 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 4 | 6 | 9 | 3 | 3 | 1 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{54}2_+^{(1+4)} % in TeX
G:=Group("C14.54ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1091);
// by ID
G=gap.SmallGroup(448,1091);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,268,675,570,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=e^2=a^7,d^2=a^7*b^2,b*a*b^-1=d*a*d^-1=e*a*e^-1=a^-1,a*c=c*a,c*b*c^-1=a^7*b^-1,d*b*d^-1=a^7*b,b*e=e*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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