G = C14.342+ 1+4 order 448 = 26·7
34th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.342+ 1+4, C4⋊D4⋊8D7, C4⋊C4.90D14, (D4×Dic7)⋊17C2, Dic7⋊4D4⋊7C2, (C2×D4).153D14, Dic7.Q8⋊11C2, (C2×C28).36C23, C22⋊C4.48D14, Dic7⋊D4⋊29C2, D14.D4⋊16C2, (C2×C14).145C24, D14⋊C4.13C22, (C22×C4).220D14, C2.36(D4⋊6D14), Dic7.22(C4○D4), (D4×C14).119C22, C23.D14⋊15C2, C22.1(D4⋊2D7), C23.11D14⋊5C2, Dic7⋊C4.16C22, C4⋊Dic7.206C22, (C22×C14).16C23, (C4×Dic7).92C22, (C2×Dic7).66C23, (C22×D7).63C23, C23.179(C22×D7), C22.166(C23×D7), C23.D7.22C22, C23.18D14⋊20C2, (C22×C28).378C22, C7⋊6(C22.47C24), (C22×Dic7).106C22, (C7×C4⋊D4)⋊9C2, (C4×C7⋊D4)⋊53C2, C2.36(D7×C4○D4), C4⋊C4⋊D7⋊12C2, C14.81(C2×C4○D4), (C2×Dic7⋊C4)⋊40C2, C2.33(C2×D4⋊2D7), (C2×C4×D7).208C22, (C2×C14).21(C4○D4), (C7×C4⋊C4).141C22, (C2×C4).293(C22×D7), (C2×C7⋊D4).26C22, (C7×C22⋊C4).10C22, SmallGroup(448,1054)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.342+ 1+4
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=b2, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, dbd-1=ebe=a7b, cd=dc, ce=ec, ede=a7b2d >
Subgroups: 1004 in 238 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, Dic7, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C22.47C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23.11D14, C23.D14, Dic7⋊4D4, D14.D4, Dic7.Q8, C4⋊C4⋊D7, C2×Dic7⋊C4, C4×C7⋊D4, D4×Dic7, C23.18D14, Dic7⋊D4, C7×C4⋊D4, C14.342+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.47C24, D4⋊2D7, C23×D7, C2×D4⋊2D7, D4⋊6D14, D7×C4○D4, C14.342+ 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)
(1 152 45 92)(2 151 46 91)(3 150 47 90)(4 149 48 89)(5 148 49 88)(6 147 50 87)(7 146 51 86)(8 145 52 85)(9 144 53 98)(10 143 54 97)(11 142 55 96)(12 141 56 95)(13 154 43 94)(14 153 44 93)(15 77 62 131)(16 76 63 130)(17 75 64 129)(18 74 65 128)(19 73 66 127)(20 72 67 140)(21 71 68 139)(22 84 69 138)(23 83 70 137)(24 82 57 136)(25 81 58 135)(26 80 59 134)(27 79 60 133)(28 78 61 132)(29 193 157 176)(30 192 158 175)(31 191 159 174)(32 190 160 173)(33 189 161 172)(34 188 162 171)(35 187 163 170)(36 186 164 169)(37 185 165 182)(38 184 166 181)(39 183 167 180)(40 196 168 179)(41 195 155 178)(42 194 156 177)(99 212 114 209)(100 211 115 208)(101 224 116 207)(102 223 117 206)(103 222 118 205)(104 221 119 204)(105 220 120 203)(106 219 121 202)(107 218 122 201)(108 217 123 200)(109 216 124 199)(110 215 125 198)(111 214 126 197)(112 213 113 210)
(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)(15 110 22 103)(16 109 23 102)(17 108 24 101)(18 107 25 100)(19 106 26 99)(20 105 27 112)(21 104 28 111)(43 167 50 160)(44 166 51 159)(45 165 52 158)(46 164 53 157)(47 163 54 156)(48 162 55 155)(49 161 56 168)(57 116 64 123)(58 115 65 122)(59 114 66 121)(60 113 67 120)(61 126 68 119)(62 125 69 118)(63 124 70 117)(71 204 78 197)(72 203 79 210)(73 202 80 209)(74 201 81 208)(75 200 82 207)(76 199 83 206)(77 198 84 205)(85 192 92 185)(86 191 93 184)(87 190 94 183)(88 189 95 196)(89 188 96 195)(90 187 97 194)(91 186 98 193)(127 219 134 212)(128 218 135 211)(129 217 136 224)(130 216 137 223)(131 215 138 222)(132 214 139 221)(133 213 140 220)(141 179 148 172)(142 178 149 171)(143 177 150 170)(144 176 151 169)(145 175 152 182)(146 174 153 181)(147 173 154 180)
(1 139 45 71)(2 140 46 72)(3 127 47 73)(4 128 48 74)(5 129 49 75)(6 130 50 76)(7 131 51 77)(8 132 52 78)(9 133 53 79)(10 134 54 80)(11 135 55 81)(12 136 56 82)(13 137 43 83)(14 138 44 84)(15 93 62 153)(16 94 63 154)(17 95 64 141)(18 96 65 142)(19 97 66 143)(20 98 67 144)(21 85 68 145)(22 86 69 146)(23 87 70 147)(24 88 57 148)(25 89 58 149)(26 90 59 150)(27 91 60 151)(28 92 61 152)(29 213 157 210)(30 214 158 197)(31 215 159 198)(32 216 160 199)(33 217 161 200)(34 218 162 201)(35 219 163 202)(36 220 164 203)(37 221 165 204)(38 222 166 205)(39 223 167 206)(40 224 168 207)(41 211 155 208)(42 212 156 209)(99 187 114 170)(100 188 115 171)(101 189 116 172)(102 190 117 173)(103 191 118 174)(104 192 119 175)(105 193 120 176)(106 194 121 177)(107 195 122 178)(108 196 123 179)(109 183 124 180)(110 184 125 181)(111 185 126 182)(112 186 113 169)
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 99)(11 100)(12 101)(13 102)(14 103)(15 38)(16 39)(17 40)(18 41)(19 42)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)(28 37)(43 117)(44 118)(45 119)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 113)(54 114)(55 115)(56 116)(57 161)(58 162)(59 163)(60 164)(61 165)(62 166)(63 167)(64 168)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 185)(72 186)(73 187)(74 188)(75 189)(76 190)(77 191)(78 192)(79 193)(80 194)(81 195)(82 196)(83 183)(84 184)(85 204)(86 205)(87 206)(88 207)(89 208)(90 209)(91 210)(92 197)(93 198)(94 199)(95 200)(96 201)(97 202)(98 203)(127 170)(128 171)(129 172)(130 173)(131 174)(132 175)(133 176)(134 177)(135 178)(136 179)(137 180)(138 181)(139 182)(140 169)(141 217)(142 218)(143 219)(144 220)(145 221)(146 222)(147 223)(148 224)(149 211)(150 212)(151 213)(152 214)(153 215)(154 216)
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), (1,152,45,92)(2,151,46,91)(3,150,47,90)(4,149,48,89)(5,148,49,88)(6,147,50,87)(7,146,51,86)(8,145,52,85)(9,144,53,98)(10,143,54,97)(11,142,55,96)(12,141,56,95)(13,154,43,94)(14,153,44,93)(15,77,62,131)(16,76,63,130)(17,75,64,129)(18,74,65,128)(19,73,66,127)(20,72,67,140)(21,71,68,139)(22,84,69,138)(23,83,70,137)(24,82,57,136)(25,81,58,135)(26,80,59,134)(27,79,60,133)(28,78,61,132)(29,193,157,176)(30,192,158,175)(31,191,159,174)(32,190,160,173)(33,189,161,172)(34,188,162,171)(35,187,163,170)(36,186,164,169)(37,185,165,182)(38,184,166,181)(39,183,167,180)(40,196,168,179)(41,195,155,178)(42,194,156,177)(99,212,114,209)(100,211,115,208)(101,224,116,207)(102,223,117,206)(103,222,118,205)(104,221,119,204)(105,220,120,203)(106,219,121,202)(107,218,122,201)(108,217,123,200)(109,216,124,199)(110,215,125,198)(111,214,126,197)(112,213,113,210), (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)(15,110,22,103)(16,109,23,102)(17,108,24,101)(18,107,25,100)(19,106,26,99)(20,105,27,112)(21,104,28,111)(43,167,50,160)(44,166,51,159)(45,165,52,158)(46,164,53,157)(47,163,54,156)(48,162,55,155)(49,161,56,168)(57,116,64,123)(58,115,65,122)(59,114,66,121)(60,113,67,120)(61,126,68,119)(62,125,69,118)(63,124,70,117)(71,204,78,197)(72,203,79,210)(73,202,80,209)(74,201,81,208)(75,200,82,207)(76,199,83,206)(77,198,84,205)(85,192,92,185)(86,191,93,184)(87,190,94,183)(88,189,95,196)(89,188,96,195)(90,187,97,194)(91,186,98,193)(127,219,134,212)(128,218,135,211)(129,217,136,224)(130,216,137,223)(131,215,138,222)(132,214,139,221)(133,213,140,220)(141,179,148,172)(142,178,149,171)(143,177,150,170)(144,176,151,169)(145,175,152,182)(146,174,153,181)(147,173,154,180), (1,139,45,71)(2,140,46,72)(3,127,47,73)(4,128,48,74)(5,129,49,75)(6,130,50,76)(7,131,51,77)(8,132,52,78)(9,133,53,79)(10,134,54,80)(11,135,55,81)(12,136,56,82)(13,137,43,83)(14,138,44,84)(15,93,62,153)(16,94,63,154)(17,95,64,141)(18,96,65,142)(19,97,66,143)(20,98,67,144)(21,85,68,145)(22,86,69,146)(23,87,70,147)(24,88,57,148)(25,89,58,149)(26,90,59,150)(27,91,60,151)(28,92,61,152)(29,213,157,210)(30,214,158,197)(31,215,159,198)(32,216,160,199)(33,217,161,200)(34,218,162,201)(35,219,163,202)(36,220,164,203)(37,221,165,204)(38,222,166,205)(39,223,167,206)(40,224,168,207)(41,211,155,208)(42,212,156,209)(99,187,114,170)(100,188,115,171)(101,189,116,172)(102,190,117,173)(103,191,118,174)(104,192,119,175)(105,193,120,176)(106,194,121,177)(107,195,122,178)(108,196,123,179)(109,183,124,180)(110,184,125,181)(111,185,126,182)(112,186,113,169), (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,99)(11,100)(12,101)(13,102)(14,103)(15,38)(16,39)(17,40)(18,41)(19,42)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,113)(54,114)(55,115)(56,116)(57,161)(58,162)(59,163)(60,164)(61,165)(62,166)(63,167)(64,168)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,185)(72,186)(73,187)(74,188)(75,189)(76,190)(77,191)(78,192)(79,193)(80,194)(81,195)(82,196)(83,183)(84,184)(85,204)(86,205)(87,206)(88,207)(89,208)(90,209)(91,210)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(127,170)(128,171)(129,172)(130,173)(131,174)(132,175)(133,176)(134,177)(135,178)(136,179)(137,180)(138,181)(139,182)(140,169)(141,217)(142,218)(143,219)(144,220)(145,221)(146,222)(147,223)(148,224)(149,211)(150,212)(151,213)(152,214)(153,215)(154,216)>;
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), (1,152,45,92)(2,151,46,91)(3,150,47,90)(4,149,48,89)(5,148,49,88)(6,147,50,87)(7,146,51,86)(8,145,52,85)(9,144,53,98)(10,143,54,97)(11,142,55,96)(12,141,56,95)(13,154,43,94)(14,153,44,93)(15,77,62,131)(16,76,63,130)(17,75,64,129)(18,74,65,128)(19,73,66,127)(20,72,67,140)(21,71,68,139)(22,84,69,138)(23,83,70,137)(24,82,57,136)(25,81,58,135)(26,80,59,134)(27,79,60,133)(28,78,61,132)(29,193,157,176)(30,192,158,175)(31,191,159,174)(32,190,160,173)(33,189,161,172)(34,188,162,171)(35,187,163,170)(36,186,164,169)(37,185,165,182)(38,184,166,181)(39,183,167,180)(40,196,168,179)(41,195,155,178)(42,194,156,177)(99,212,114,209)(100,211,115,208)(101,224,116,207)(102,223,117,206)(103,222,118,205)(104,221,119,204)(105,220,120,203)(106,219,121,202)(107,218,122,201)(108,217,123,200)(109,216,124,199)(110,215,125,198)(111,214,126,197)(112,213,113,210), (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)(15,110,22,103)(16,109,23,102)(17,108,24,101)(18,107,25,100)(19,106,26,99)(20,105,27,112)(21,104,28,111)(43,167,50,160)(44,166,51,159)(45,165,52,158)(46,164,53,157)(47,163,54,156)(48,162,55,155)(49,161,56,168)(57,116,64,123)(58,115,65,122)(59,114,66,121)(60,113,67,120)(61,126,68,119)(62,125,69,118)(63,124,70,117)(71,204,78,197)(72,203,79,210)(73,202,80,209)(74,201,81,208)(75,200,82,207)(76,199,83,206)(77,198,84,205)(85,192,92,185)(86,191,93,184)(87,190,94,183)(88,189,95,196)(89,188,96,195)(90,187,97,194)(91,186,98,193)(127,219,134,212)(128,218,135,211)(129,217,136,224)(130,216,137,223)(131,215,138,222)(132,214,139,221)(133,213,140,220)(141,179,148,172)(142,178,149,171)(143,177,150,170)(144,176,151,169)(145,175,152,182)(146,174,153,181)(147,173,154,180), (1,139,45,71)(2,140,46,72)(3,127,47,73)(4,128,48,74)(5,129,49,75)(6,130,50,76)(7,131,51,77)(8,132,52,78)(9,133,53,79)(10,134,54,80)(11,135,55,81)(12,136,56,82)(13,137,43,83)(14,138,44,84)(15,93,62,153)(16,94,63,154)(17,95,64,141)(18,96,65,142)(19,97,66,143)(20,98,67,144)(21,85,68,145)(22,86,69,146)(23,87,70,147)(24,88,57,148)(25,89,58,149)(26,90,59,150)(27,91,60,151)(28,92,61,152)(29,213,157,210)(30,214,158,197)(31,215,159,198)(32,216,160,199)(33,217,161,200)(34,218,162,201)(35,219,163,202)(36,220,164,203)(37,221,165,204)(38,222,166,205)(39,223,167,206)(40,224,168,207)(41,211,155,208)(42,212,156,209)(99,187,114,170)(100,188,115,171)(101,189,116,172)(102,190,117,173)(103,191,118,174)(104,192,119,175)(105,193,120,176)(106,194,121,177)(107,195,122,178)(108,196,123,179)(109,183,124,180)(110,184,125,181)(111,185,126,182)(112,186,113,169), (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,99)(11,100)(12,101)(13,102)(14,103)(15,38)(16,39)(17,40)(18,41)(19,42)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,113)(54,114)(55,115)(56,116)(57,161)(58,162)(59,163)(60,164)(61,165)(62,166)(63,167)(64,168)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,185)(72,186)(73,187)(74,188)(75,189)(76,190)(77,191)(78,192)(79,193)(80,194)(81,195)(82,196)(83,183)(84,184)(85,204)(86,205)(87,206)(88,207)(89,208)(90,209)(91,210)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(127,170)(128,171)(129,172)(130,173)(131,174)(132,175)(133,176)(134,177)(135,178)(136,179)(137,180)(138,181)(139,182)(140,169)(141,217)(142,218)(143,219)(144,220)(145,221)(146,222)(147,223)(148,224)(149,211)(150,212)(151,213)(152,214)(153,215)(154,216) );
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)], [(1,152,45,92),(2,151,46,91),(3,150,47,90),(4,149,48,89),(5,148,49,88),(6,147,50,87),(7,146,51,86),(8,145,52,85),(9,144,53,98),(10,143,54,97),(11,142,55,96),(12,141,56,95),(13,154,43,94),(14,153,44,93),(15,77,62,131),(16,76,63,130),(17,75,64,129),(18,74,65,128),(19,73,66,127),(20,72,67,140),(21,71,68,139),(22,84,69,138),(23,83,70,137),(24,82,57,136),(25,81,58,135),(26,80,59,134),(27,79,60,133),(28,78,61,132),(29,193,157,176),(30,192,158,175),(31,191,159,174),(32,190,160,173),(33,189,161,172),(34,188,162,171),(35,187,163,170),(36,186,164,169),(37,185,165,182),(38,184,166,181),(39,183,167,180),(40,196,168,179),(41,195,155,178),(42,194,156,177),(99,212,114,209),(100,211,115,208),(101,224,116,207),(102,223,117,206),(103,222,118,205),(104,221,119,204),(105,220,120,203),(106,219,121,202),(107,218,122,201),(108,217,123,200),(109,216,124,199),(110,215,125,198),(111,214,126,197),(112,213,113,210)], [(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),(15,110,22,103),(16,109,23,102),(17,108,24,101),(18,107,25,100),(19,106,26,99),(20,105,27,112),(21,104,28,111),(43,167,50,160),(44,166,51,159),(45,165,52,158),(46,164,53,157),(47,163,54,156),(48,162,55,155),(49,161,56,168),(57,116,64,123),(58,115,65,122),(59,114,66,121),(60,113,67,120),(61,126,68,119),(62,125,69,118),(63,124,70,117),(71,204,78,197),(72,203,79,210),(73,202,80,209),(74,201,81,208),(75,200,82,207),(76,199,83,206),(77,198,84,205),(85,192,92,185),(86,191,93,184),(87,190,94,183),(88,189,95,196),(89,188,96,195),(90,187,97,194),(91,186,98,193),(127,219,134,212),(128,218,135,211),(129,217,136,224),(130,216,137,223),(131,215,138,222),(132,214,139,221),(133,213,140,220),(141,179,148,172),(142,178,149,171),(143,177,150,170),(144,176,151,169),(145,175,152,182),(146,174,153,181),(147,173,154,180)], [(1,139,45,71),(2,140,46,72),(3,127,47,73),(4,128,48,74),(5,129,49,75),(6,130,50,76),(7,131,51,77),(8,132,52,78),(9,133,53,79),(10,134,54,80),(11,135,55,81),(12,136,56,82),(13,137,43,83),(14,138,44,84),(15,93,62,153),(16,94,63,154),(17,95,64,141),(18,96,65,142),(19,97,66,143),(20,98,67,144),(21,85,68,145),(22,86,69,146),(23,87,70,147),(24,88,57,148),(25,89,58,149),(26,90,59,150),(27,91,60,151),(28,92,61,152),(29,213,157,210),(30,214,158,197),(31,215,159,198),(32,216,160,199),(33,217,161,200),(34,218,162,201),(35,219,163,202),(36,220,164,203),(37,221,165,204),(38,222,166,205),(39,223,167,206),(40,224,168,207),(41,211,155,208),(42,212,156,209),(99,187,114,170),(100,188,115,171),(101,189,116,172),(102,190,117,173),(103,191,118,174),(104,192,119,175),(105,193,120,176),(106,194,121,177),(107,195,122,178),(108,196,123,179),(109,183,124,180),(110,184,125,181),(111,185,126,182),(112,186,113,169)], [(1,104),(2,105),(3,106),(4,107),(5,108),(6,109),(7,110),(8,111),(9,112),(10,99),(11,100),(12,101),(13,102),(14,103),(15,38),(16,39),(17,40),(18,41),(19,42),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36),(28,37),(43,117),(44,118),(45,119),(46,120),(47,121),(48,122),(49,123),(50,124),(51,125),(52,126),(53,113),(54,114),(55,115),(56,116),(57,161),(58,162),(59,163),(60,164),(61,165),(62,166),(63,167),(64,168),(65,155),(66,156),(67,157),(68,158),(69,159),(70,160),(71,185),(72,186),(73,187),(74,188),(75,189),(76,190),(77,191),(78,192),(79,193),(80,194),(81,195),(82,196),(83,183),(84,184),(85,204),(86,205),(87,206),(88,207),(89,208),(90,209),(91,210),(92,197),(93,198),(94,199),(95,200),(96,201),(97,202),(98,203),(127,170),(128,171),(129,172),(130,173),(131,174),(132,175),(133,176),(134,177),(135,178),(136,179),(137,180),(138,181),(139,182),(140,169),(141,217),(142,218),(143,219),(144,220),(145,221),(146,222),(147,223),(148,224),(149,211),(150,212),(151,213),(152,214),(153,215),(154,216)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 28 | 2 | 2 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 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 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | D4⋊2D7 | D4⋊6D14 | D7×C4○D4 |
| kernel | C14.342+ 1+4 | C23.11D14 | C23.D14 | Dic7⋊4D4 | D14.D4 | Dic7.Q8 | C4⋊C4⋊D7 | C2×Dic7⋊C4 | C4×C7⋊D4 | D4×Dic7 | C23.18D14 | Dic7⋊D4 | C7×C4⋊D4 | C4⋊D4 | Dic7 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 1 | 3 | 4 | 4 | 6 | 3 | 3 | 9 | 1 | 6 | 6 | 6 |
Matrix representation of C14.342+ 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 21 |
| 0 | 0 | 0 | 0 | 17 | 19 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 26 |
| 0 | 0 | 0 | 0 | 19 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,17,0,0,0,0,21,19],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,10,0,0,0,0,3,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,28,0,0,0,0,0,0,0,0,19,0,0,0,0,26,0],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,17,0,0,0,0,12,0,0,0,0,0,0,0,0,17,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C14.342+ 1+4 in GAP, Magma, Sage, TeX
C_{14}._{34}2_+^{1+4} % in TeX
G:=Group("C14.34ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1054);
// by ID
G=gap.SmallGroup(448,1054);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,100,794,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=b^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,d*b*d^-1=e*b*e=a^7*b,c*d=d*c,c*e=e*c,e*d*e=a^7*b^2*d>;
// generators/relations