G = C14.112+ 1+4 order 448 = 26·7
11st non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.112+ 1+4, C4⋊D28⋊10C2, C28⋊7D4⋊28C2, C4⋊C4.307D14, D28⋊C4⋊10C2, D14.5D4⋊2C2, C4.94(C4○D28), (C2×C14).58C24, C28.3Q8⋊10C2, C28.196(C4○D4), (C2×C28).619C23, (C22×C4).183D14, C2.14(D4⋊6D14), C22.92(C23×D7), D14⋊C4.142C22, (C2×D28).136C22, C4⋊Dic7.194C22, (C4×Dic7).66C22, (C2×Dic7).19C23, (C22×D7).16C23, C23.229(C22×D7), Dic7⋊C4.151C22, (C22×C14).407C23, (C22×C28).220C22, C7⋊1(C22.47C24), C22.11(Q8⋊2D7), C23.D7.143C22, (C2×C4⋊C4)⋊23D7, (C14×C4⋊C4)⋊20C2, C4⋊C4⋊D7⋊2C2, (C4×C7⋊D4)⋊11C2, C4⋊C4⋊7D7⋊10C2, C14.25(C2×C4○D4), C2.27(C2×C4○D28), C2.9(C2×Q8⋊2D7), (C2×C4×D7).57C22, (C7×C4⋊C4).299C22, (C2×C4).146(C22×D7), (C2×C7⋊D4).95C22, (C2×C14).198(C4○D4), SmallGroup(448,967)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.112+ 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=a7b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >
Subgroups: 1124 in 238 conjugacy classes, 99 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C42⋊2C2, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22.47C24, C4×Dic7, C4×Dic7, Dic7⋊C4, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, D14⋊C4, D14⋊C4, C23.D7, C7×C4⋊C4, C7×C4⋊C4, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C22×C28, C28.3Q8, C4⋊C4⋊7D7, D28⋊C4, D14.5D4, C4⋊D28, C4⋊C4⋊D7, C4×C7⋊D4, C4×C7⋊D4, C28⋊7D4, C28⋊7D4, C14×C4⋊C4, C14.112+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.47C24, C4○D28, Q8⋊2D7, C23×D7, C2×C4○D28, D4⋊6D14, C2×Q8⋊2D7, C14.112+ 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)
(1 161 130 111)(2 160 131 110)(3 159 132 109)(4 158 133 108)(5 157 134 107)(6 156 135 106)(7 155 136 105)(8 168 137 104)(9 167 138 103)(10 166 139 102)(11 165 140 101)(12 164 127 100)(13 163 128 99)(14 162 129 112)(15 98 31 194)(16 97 32 193)(17 96 33 192)(18 95 34 191)(19 94 35 190)(20 93 36 189)(21 92 37 188)(22 91 38 187)(23 90 39 186)(24 89 40 185)(25 88 41 184)(26 87 42 183)(27 86 29 196)(28 85 30 195)(43 200 142 213)(44 199 143 212)(45 198 144 211)(46 197 145 224)(47 210 146 223)(48 209 147 222)(49 208 148 221)(50 207 149 220)(51 206 150 219)(52 205 151 218)(53 204 152 217)(54 203 153 216)(55 202 154 215)(56 201 141 214)(57 74 123 181)(58 73 124 180)(59 72 125 179)(60 71 126 178)(61 84 113 177)(62 83 114 176)(63 82 115 175)(64 81 116 174)(65 80 117 173)(66 79 118 172)(67 78 119 171)(68 77 120 170)(69 76 121 169)(70 75 122 182)
(1 130)(2 131)(3 132)(4 133)(5 134)(6 135)(7 136)(8 137)(9 138)(10 139)(11 140)(12 127)(13 128)(14 129)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 29)(28 30)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 123)(58 124)(59 125)(60 126)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 121)(70 122)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(155 162)(156 163)(157 164)(158 165)(159 166)(160 167)(161 168)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)(197 224)(198 211)(199 212)(200 213)(201 214)(202 215)(203 216)(204 217)(205 218)(206 219)(207 220)(208 221)(209 222)(210 223)
(1 58 130 124)(2 59 131 125)(3 60 132 126)(4 61 133 113)(5 62 134 114)(6 63 135 115)(7 64 136 116)(8 65 137 117)(9 66 138 118)(10 67 139 119)(11 68 140 120)(12 69 127 121)(13 70 128 122)(14 57 129 123)(15 208 31 221)(16 209 32 222)(17 210 33 223)(18 197 34 224)(19 198 35 211)(20 199 36 212)(21 200 37 213)(22 201 38 214)(23 202 39 215)(24 203 40 216)(25 204 41 217)(26 205 42 218)(27 206 29 219)(28 207 30 220)(43 92 142 188)(44 93 143 189)(45 94 144 190)(46 95 145 191)(47 96 146 192)(48 97 147 193)(49 98 148 194)(50 85 149 195)(51 86 150 196)(52 87 151 183)(53 88 152 184)(54 89 153 185)(55 90 154 186)(56 91 141 187)(71 109 178 159)(72 110 179 160)(73 111 180 161)(74 112 181 162)(75 99 182 163)(76 100 169 164)(77 101 170 165)(78 102 171 166)(79 103 172 167)(80 104 173 168)(81 105 174 155)(82 106 175 156)(83 107 176 157)(84 108 177 158)
(1 32 8 39)(2 33 9 40)(3 34 10 41)(4 35 11 42)(5 36 12 29)(6 37 13 30)(7 38 14 31)(15 136 22 129)(16 137 23 130)(17 138 24 131)(18 139 25 132)(19 140 26 133)(20 127 27 134)(21 128 28 135)(43 182 50 175)(44 169 51 176)(45 170 52 177)(46 171 53 178)(47 172 54 179)(48 173 55 180)(49 174 56 181)(57 208 64 201)(58 209 65 202)(59 210 66 203)(60 197 67 204)(61 198 68 205)(62 199 69 206)(63 200 70 207)(71 145 78 152)(72 146 79 153)(73 147 80 154)(74 148 81 141)(75 149 82 142)(76 150 83 143)(77 151 84 144)(85 106 92 99)(86 107 93 100)(87 108 94 101)(88 109 95 102)(89 110 96 103)(90 111 97 104)(91 112 98 105)(113 211 120 218)(114 212 121 219)(115 213 122 220)(116 214 123 221)(117 215 124 222)(118 216 125 223)(119 217 126 224)(155 187 162 194)(156 188 163 195)(157 189 164 196)(158 190 165 183)(159 191 166 184)(160 192 167 185)(161 193 168 186)
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), (1,161,130,111)(2,160,131,110)(3,159,132,109)(4,158,133,108)(5,157,134,107)(6,156,135,106)(7,155,136,105)(8,168,137,104)(9,167,138,103)(10,166,139,102)(11,165,140,101)(12,164,127,100)(13,163,128,99)(14,162,129,112)(15,98,31,194)(16,97,32,193)(17,96,33,192)(18,95,34,191)(19,94,35,190)(20,93,36,189)(21,92,37,188)(22,91,38,187)(23,90,39,186)(24,89,40,185)(25,88,41,184)(26,87,42,183)(27,86,29,196)(28,85,30,195)(43,200,142,213)(44,199,143,212)(45,198,144,211)(46,197,145,224)(47,210,146,223)(48,209,147,222)(49,208,148,221)(50,207,149,220)(51,206,150,219)(52,205,151,218)(53,204,152,217)(54,203,153,216)(55,202,154,215)(56,201,141,214)(57,74,123,181)(58,73,124,180)(59,72,125,179)(60,71,126,178)(61,84,113,177)(62,83,114,176)(63,82,115,175)(64,81,116,174)(65,80,117,173)(66,79,118,172)(67,78,119,171)(68,77,120,170)(69,76,121,169)(70,75,122,182), (1,130)(2,131)(3,132)(4,133)(5,134)(6,135)(7,136)(8,137)(9,138)(10,139)(11,140)(12,127)(13,128)(14,129)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,29)(28,30)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,123)(58,124)(59,125)(60,126)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,121)(70,122)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(197,224)(198,211)(199,212)(200,213)(201,214)(202,215)(203,216)(204,217)(205,218)(206,219)(207,220)(208,221)(209,222)(210,223), (1,58,130,124)(2,59,131,125)(3,60,132,126)(4,61,133,113)(5,62,134,114)(6,63,135,115)(7,64,136,116)(8,65,137,117)(9,66,138,118)(10,67,139,119)(11,68,140,120)(12,69,127,121)(13,70,128,122)(14,57,129,123)(15,208,31,221)(16,209,32,222)(17,210,33,223)(18,197,34,224)(19,198,35,211)(20,199,36,212)(21,200,37,213)(22,201,38,214)(23,202,39,215)(24,203,40,216)(25,204,41,217)(26,205,42,218)(27,206,29,219)(28,207,30,220)(43,92,142,188)(44,93,143,189)(45,94,144,190)(46,95,145,191)(47,96,146,192)(48,97,147,193)(49,98,148,194)(50,85,149,195)(51,86,150,196)(52,87,151,183)(53,88,152,184)(54,89,153,185)(55,90,154,186)(56,91,141,187)(71,109,178,159)(72,110,179,160)(73,111,180,161)(74,112,181,162)(75,99,182,163)(76,100,169,164)(77,101,170,165)(78,102,171,166)(79,103,172,167)(80,104,173,168)(81,105,174,155)(82,106,175,156)(83,107,176,157)(84,108,177,158), (1,32,8,39)(2,33,9,40)(3,34,10,41)(4,35,11,42)(5,36,12,29)(6,37,13,30)(7,38,14,31)(15,136,22,129)(16,137,23,130)(17,138,24,131)(18,139,25,132)(19,140,26,133)(20,127,27,134)(21,128,28,135)(43,182,50,175)(44,169,51,176)(45,170,52,177)(46,171,53,178)(47,172,54,179)(48,173,55,180)(49,174,56,181)(57,208,64,201)(58,209,65,202)(59,210,66,203)(60,197,67,204)(61,198,68,205)(62,199,69,206)(63,200,70,207)(71,145,78,152)(72,146,79,153)(73,147,80,154)(74,148,81,141)(75,149,82,142)(76,150,83,143)(77,151,84,144)(85,106,92,99)(86,107,93,100)(87,108,94,101)(88,109,95,102)(89,110,96,103)(90,111,97,104)(91,112,98,105)(113,211,120,218)(114,212,121,219)(115,213,122,220)(116,214,123,221)(117,215,124,222)(118,216,125,223)(119,217,126,224)(155,187,162,194)(156,188,163,195)(157,189,164,196)(158,190,165,183)(159,191,166,184)(160,192,167,185)(161,193,168,186)>;
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), (1,161,130,111)(2,160,131,110)(3,159,132,109)(4,158,133,108)(5,157,134,107)(6,156,135,106)(7,155,136,105)(8,168,137,104)(9,167,138,103)(10,166,139,102)(11,165,140,101)(12,164,127,100)(13,163,128,99)(14,162,129,112)(15,98,31,194)(16,97,32,193)(17,96,33,192)(18,95,34,191)(19,94,35,190)(20,93,36,189)(21,92,37,188)(22,91,38,187)(23,90,39,186)(24,89,40,185)(25,88,41,184)(26,87,42,183)(27,86,29,196)(28,85,30,195)(43,200,142,213)(44,199,143,212)(45,198,144,211)(46,197,145,224)(47,210,146,223)(48,209,147,222)(49,208,148,221)(50,207,149,220)(51,206,150,219)(52,205,151,218)(53,204,152,217)(54,203,153,216)(55,202,154,215)(56,201,141,214)(57,74,123,181)(58,73,124,180)(59,72,125,179)(60,71,126,178)(61,84,113,177)(62,83,114,176)(63,82,115,175)(64,81,116,174)(65,80,117,173)(66,79,118,172)(67,78,119,171)(68,77,120,170)(69,76,121,169)(70,75,122,182), (1,130)(2,131)(3,132)(4,133)(5,134)(6,135)(7,136)(8,137)(9,138)(10,139)(11,140)(12,127)(13,128)(14,129)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,29)(28,30)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,123)(58,124)(59,125)(60,126)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,121)(70,122)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(197,224)(198,211)(199,212)(200,213)(201,214)(202,215)(203,216)(204,217)(205,218)(206,219)(207,220)(208,221)(209,222)(210,223), (1,58,130,124)(2,59,131,125)(3,60,132,126)(4,61,133,113)(5,62,134,114)(6,63,135,115)(7,64,136,116)(8,65,137,117)(9,66,138,118)(10,67,139,119)(11,68,140,120)(12,69,127,121)(13,70,128,122)(14,57,129,123)(15,208,31,221)(16,209,32,222)(17,210,33,223)(18,197,34,224)(19,198,35,211)(20,199,36,212)(21,200,37,213)(22,201,38,214)(23,202,39,215)(24,203,40,216)(25,204,41,217)(26,205,42,218)(27,206,29,219)(28,207,30,220)(43,92,142,188)(44,93,143,189)(45,94,144,190)(46,95,145,191)(47,96,146,192)(48,97,147,193)(49,98,148,194)(50,85,149,195)(51,86,150,196)(52,87,151,183)(53,88,152,184)(54,89,153,185)(55,90,154,186)(56,91,141,187)(71,109,178,159)(72,110,179,160)(73,111,180,161)(74,112,181,162)(75,99,182,163)(76,100,169,164)(77,101,170,165)(78,102,171,166)(79,103,172,167)(80,104,173,168)(81,105,174,155)(82,106,175,156)(83,107,176,157)(84,108,177,158), (1,32,8,39)(2,33,9,40)(3,34,10,41)(4,35,11,42)(5,36,12,29)(6,37,13,30)(7,38,14,31)(15,136,22,129)(16,137,23,130)(17,138,24,131)(18,139,25,132)(19,140,26,133)(20,127,27,134)(21,128,28,135)(43,182,50,175)(44,169,51,176)(45,170,52,177)(46,171,53,178)(47,172,54,179)(48,173,55,180)(49,174,56,181)(57,208,64,201)(58,209,65,202)(59,210,66,203)(60,197,67,204)(61,198,68,205)(62,199,69,206)(63,200,70,207)(71,145,78,152)(72,146,79,153)(73,147,80,154)(74,148,81,141)(75,149,82,142)(76,150,83,143)(77,151,84,144)(85,106,92,99)(86,107,93,100)(87,108,94,101)(88,109,95,102)(89,110,96,103)(90,111,97,104)(91,112,98,105)(113,211,120,218)(114,212,121,219)(115,213,122,220)(116,214,123,221)(117,215,124,222)(118,216,125,223)(119,217,126,224)(155,187,162,194)(156,188,163,195)(157,189,164,196)(158,190,165,183)(159,191,166,184)(160,192,167,185)(161,193,168,186) );
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)], [(1,161,130,111),(2,160,131,110),(3,159,132,109),(4,158,133,108),(5,157,134,107),(6,156,135,106),(7,155,136,105),(8,168,137,104),(9,167,138,103),(10,166,139,102),(11,165,140,101),(12,164,127,100),(13,163,128,99),(14,162,129,112),(15,98,31,194),(16,97,32,193),(17,96,33,192),(18,95,34,191),(19,94,35,190),(20,93,36,189),(21,92,37,188),(22,91,38,187),(23,90,39,186),(24,89,40,185),(25,88,41,184),(26,87,42,183),(27,86,29,196),(28,85,30,195),(43,200,142,213),(44,199,143,212),(45,198,144,211),(46,197,145,224),(47,210,146,223),(48,209,147,222),(49,208,148,221),(50,207,149,220),(51,206,150,219),(52,205,151,218),(53,204,152,217),(54,203,153,216),(55,202,154,215),(56,201,141,214),(57,74,123,181),(58,73,124,180),(59,72,125,179),(60,71,126,178),(61,84,113,177),(62,83,114,176),(63,82,115,175),(64,81,116,174),(65,80,117,173),(66,79,118,172),(67,78,119,171),(68,77,120,170),(69,76,121,169),(70,75,122,182)], [(1,130),(2,131),(3,132),(4,133),(5,134),(6,135),(7,136),(8,137),(9,138),(10,139),(11,140),(12,127),(13,128),(14,129),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,29),(28,30),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,123),(58,124),(59,125),(60,126),(61,113),(62,114),(63,115),(64,116),(65,117),(66,118),(67,119),(68,120),(69,121),(70,122),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(141,148),(142,149),(143,150),(144,151),(145,152),(146,153),(147,154),(155,162),(156,163),(157,164),(158,165),(159,166),(160,167),(161,168),(169,176),(170,177),(171,178),(172,179),(173,180),(174,181),(175,182),(183,190),(184,191),(185,192),(186,193),(187,194),(188,195),(189,196),(197,224),(198,211),(199,212),(200,213),(201,214),(202,215),(203,216),(204,217),(205,218),(206,219),(207,220),(208,221),(209,222),(210,223)], [(1,58,130,124),(2,59,131,125),(3,60,132,126),(4,61,133,113),(5,62,134,114),(6,63,135,115),(7,64,136,116),(8,65,137,117),(9,66,138,118),(10,67,139,119),(11,68,140,120),(12,69,127,121),(13,70,128,122),(14,57,129,123),(15,208,31,221),(16,209,32,222),(17,210,33,223),(18,197,34,224),(19,198,35,211),(20,199,36,212),(21,200,37,213),(22,201,38,214),(23,202,39,215),(24,203,40,216),(25,204,41,217),(26,205,42,218),(27,206,29,219),(28,207,30,220),(43,92,142,188),(44,93,143,189),(45,94,144,190),(46,95,145,191),(47,96,146,192),(48,97,147,193),(49,98,148,194),(50,85,149,195),(51,86,150,196),(52,87,151,183),(53,88,152,184),(54,89,153,185),(55,90,154,186),(56,91,141,187),(71,109,178,159),(72,110,179,160),(73,111,180,161),(74,112,181,162),(75,99,182,163),(76,100,169,164),(77,101,170,165),(78,102,171,166),(79,103,172,167),(80,104,173,168),(81,105,174,155),(82,106,175,156),(83,107,176,157),(84,108,177,158)], [(1,32,8,39),(2,33,9,40),(3,34,10,41),(4,35,11,42),(5,36,12,29),(6,37,13,30),(7,38,14,31),(15,136,22,129),(16,137,23,130),(17,138,24,131),(18,139,25,132),(19,140,26,133),(20,127,27,134),(21,128,28,135),(43,182,50,175),(44,169,51,176),(45,170,52,177),(46,171,53,178),(47,172,54,179),(48,173,55,180),(49,174,56,181),(57,208,64,201),(58,209,65,202),(59,210,66,203),(60,197,67,204),(61,198,68,205),(62,199,69,206),(63,200,70,207),(71,145,78,152),(72,146,79,153),(73,147,80,154),(74,148,81,141),(75,149,82,142),(76,150,83,143),(77,151,84,144),(85,106,92,99),(86,107,93,100),(87,108,94,101),(88,109,95,102),(89,110,96,103),(90,111,97,104),(91,112,98,105),(113,211,120,218),(114,212,121,219),(115,213,122,220),(116,214,123,221),(117,215,124,222),(118,216,125,223),(119,217,126,224),(155,187,162,194),(156,188,163,195),(157,189,164,196),(158,190,165,183),(159,191,166,184),(160,192,167,185),(161,193,168,186)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | D14 | D14 | C4○D28 | 2+ 1+4 | Q8⋊2D7 | D4⋊6D14 |
| kernel | C14.112+ 1+4 | C28.3Q8 | C4⋊C4⋊7D7 | D28⋊C4 | D14.5D4 | C4⋊D28 | C4⋊C4⋊D7 | C4×C7⋊D4 | C28⋊7D4 | C14×C4⋊C4 | C2×C4⋊C4 | C28 | C2×C14 | C4⋊C4 | C22×C4 | C4 | C14 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 3 | 3 | 1 | 3 | 4 | 4 | 12 | 9 | 24 | 1 | 6 | 6 |
Matrix representation of C14.112+ 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 9 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 20 | 21 | 0 | 0 | 0 | 0 |
| 10 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 27 | 0 | 0 |
| 0 | 0 | 6 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 5 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 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 | 20 | 8 |
| 0 | 0 | 0 | 0 | 26 | 9 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 25 |
| 0 | 0 | 0 | 0 | 23 | 2 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,16,9,0,0,0,0,0,20,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[20,10,0,0,0,0,21,9,0,0,0,0,0,0,10,6,0,0,0,0,27,19,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[1,5,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,20,26,0,0,0,0,8,9],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,27,23,0,0,0,0,25,2] >;
C14.112+ 1+4 in GAP, Magma, Sage, TeX
C_{14}._{11}2_+^{1+4} % in TeX
G:=Group("C14.11ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,967);
// by ID
G=gap.SmallGroup(448,967);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,387,100,1571,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=b^2,e^2=a^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^7*b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations