?
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, C23.11D14⋊5C2, C22.1(D4⋊2D7), C23.D14⋊15C2, 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
Subgroups: 1004 in 238 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×5], C4 [×12], C22, C22 [×2], C22 [×11], C7, C2×C4 [×4], C2×C4 [×15], D4 [×10], C23 [×3], C23, D7, C14 [×3], C14 [×4], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×5], C2×D4 [×3], C2×D4 [×3], Dic7 [×2], Dic7 [×6], C28 [×4], D14 [×3], C2×C14, C2×C14 [×2], C2×C14 [×8], C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4, C4⋊D4 [×3], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C4×D7, C2×Dic7 [×7], C2×Dic7 [×6], C7⋊D4 [×5], C2×C28 [×4], C2×C28, C7×D4 [×5], C22×D7, C22×C14 [×3], C22.47C24, C4×Dic7 [×3], Dic7⋊C4 [×7], C4⋊Dic7 [×2], D14⋊C4 [×3], C23.D7 [×5], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×C4×D7, C22×Dic7 [×4], C2×C7⋊D4 [×3], C22×C28, D4×C14 [×3], C23.11D14, C23.D14, Dic7⋊4D4, D14.D4, Dic7.Q8, C4⋊C4⋊D7, C2×Dic7⋊C4, C4×C7⋊D4, D4×Dic7 [×2], C23.18D14, Dic7⋊D4 [×3], C7×C4⋊D4, C14.342+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×4], C24, D14 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D7 [×7], C22.47C24, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, D4⋊6D14, D7×C4○D4, C14.342+ (1+4)
Generators and relations
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 >
►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)
(1 164 73 187)(2 163 74 186)(3 162 75 185)(4 161 76 184)(5 160 77 183)(6 159 78 196)(7 158 79 195)(8 157 80 194)(9 156 81 193)(10 155 82 192)(11 168 83 191)(12 167 84 190)(13 166 71 189)(14 165 72 188)(15 200 223 49)(16 199 224 48)(17 198 211 47)(18 197 212 46)(19 210 213 45)(20 209 214 44)(21 208 215 43)(22 207 216 56)(23 206 217 55)(24 205 218 54)(25 204 219 53)(26 203 220 52)(27 202 221 51)(28 201 222 50)(29 133 113 70)(30 132 114 69)(31 131 115 68)(32 130 116 67)(33 129 117 66)(34 128 118 65)(35 127 119 64)(36 140 120 63)(37 139 121 62)(38 138 122 61)(39 137 123 60)(40 136 124 59)(41 135 125 58)(42 134 126 57)(85 141 170 109)(86 154 171 108)(87 153 172 107)(88 152 173 106)(89 151 174 105)(90 150 175 104)(91 149 176 103)(92 148 177 102)(93 147 178 101)(94 146 179 100)(95 145 180 99)(96 144 181 112)(97 143 182 111)(98 142 169 110)
(1 134 8 127)(2 133 9 140)(3 132 10 139)(4 131 11 138)(5 130 12 137)(6 129 13 136)(7 128 14 135)(15 178 22 171)(16 177 23 170)(17 176 24 169)(18 175 25 182)(19 174 26 181)(20 173 27 180)(21 172 28 179)(29 156 36 163)(30 155 37 162)(31 168 38 161)(32 167 39 160)(33 166 40 159)(34 165 41 158)(35 164 42 157)(43 107 50 100)(44 106 51 99)(45 105 52 112)(46 104 53 111)(47 103 54 110)(48 102 55 109)(49 101 56 108)(57 80 64 73)(58 79 65 72)(59 78 66 71)(60 77 67 84)(61 76 68 83)(62 75 69 82)(63 74 70 81)(85 224 92 217)(86 223 93 216)(87 222 94 215)(88 221 95 214)(89 220 96 213)(90 219 97 212)(91 218 98 211)(113 193 120 186)(114 192 121 185)(115 191 122 184)(116 190 123 183)(117 189 124 196)(118 188 125 195)(119 187 126 194)(141 199 148 206)(142 198 149 205)(143 197 150 204)(144 210 151 203)(145 209 152 202)(146 208 153 201)(147 207 154 200)
(1 111 73 143)(2 112 74 144)(3 99 75 145)(4 100 76 146)(5 101 77 147)(6 102 78 148)(7 103 79 149)(8 104 80 150)(9 105 81 151)(10 106 82 152)(11 107 83 153)(12 108 84 154)(13 109 71 141)(14 110 72 142)(15 32 223 116)(16 33 224 117)(17 34 211 118)(18 35 212 119)(19 36 213 120)(20 37 214 121)(21 38 215 122)(22 39 216 123)(23 40 217 124)(24 41 218 125)(25 42 219 126)(26 29 220 113)(27 30 221 114)(28 31 222 115)(43 68 208 131)(44 69 209 132)(45 70 210 133)(46 57 197 134)(47 58 198 135)(48 59 199 136)(49 60 200 137)(50 61 201 138)(51 62 202 139)(52 63 203 140)(53 64 204 127)(54 65 205 128)(55 66 206 129)(56 67 207 130)(85 196 170 159)(86 183 171 160)(87 184 172 161)(88 185 173 162)(89 186 174 163)(90 187 175 164)(91 188 176 165)(92 189 177 166)(93 190 178 167)(94 191 179 168)(95 192 180 155)(96 193 181 156)(97 194 182 157)(98 195 169 158)
(1 212)(2 213)(3 214)(4 215)(5 216)(6 217)(7 218)(8 219)(9 220)(10 221)(11 222)(12 223)(13 224)(14 211)(15 84)(16 71)(17 72)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 112)(30 99)(31 100)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(41 110)(42 111)(43 168)(44 155)(45 156)(46 157)(47 158)(48 159)(49 160)(50 161)(51 162)(52 163)(53 164)(54 165)(55 166)(56 167)(57 175)(58 176)(59 177)(60 178)(61 179)(62 180)(63 181)(64 182)(65 169)(66 170)(67 171)(68 172)(69 173)(70 174)(85 129)(86 130)(87 131)(88 132)(89 133)(90 134)(91 135)(92 136)(93 137)(94 138)(95 139)(96 140)(97 127)(98 128)(113 144)(114 145)(115 146)(116 147)(117 148)(118 149)(119 150)(120 151)(121 152)(122 153)(123 154)(124 141)(125 142)(126 143)(183 200)(184 201)(185 202)(186 203)(187 204)(188 205)(189 206)(190 207)(191 208)(192 209)(193 210)(194 197)(195 198)(196 199)
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), (1,164,73,187)(2,163,74,186)(3,162,75,185)(4,161,76,184)(5,160,77,183)(6,159,78,196)(7,158,79,195)(8,157,80,194)(9,156,81,193)(10,155,82,192)(11,168,83,191)(12,167,84,190)(13,166,71,189)(14,165,72,188)(15,200,223,49)(16,199,224,48)(17,198,211,47)(18,197,212,46)(19,210,213,45)(20,209,214,44)(21,208,215,43)(22,207,216,56)(23,206,217,55)(24,205,218,54)(25,204,219,53)(26,203,220,52)(27,202,221,51)(28,201,222,50)(29,133,113,70)(30,132,114,69)(31,131,115,68)(32,130,116,67)(33,129,117,66)(34,128,118,65)(35,127,119,64)(36,140,120,63)(37,139,121,62)(38,138,122,61)(39,137,123,60)(40,136,124,59)(41,135,125,58)(42,134,126,57)(85,141,170,109)(86,154,171,108)(87,153,172,107)(88,152,173,106)(89,151,174,105)(90,150,175,104)(91,149,176,103)(92,148,177,102)(93,147,178,101)(94,146,179,100)(95,145,180,99)(96,144,181,112)(97,143,182,111)(98,142,169,110), (1,134,8,127)(2,133,9,140)(3,132,10,139)(4,131,11,138)(5,130,12,137)(6,129,13,136)(7,128,14,135)(15,178,22,171)(16,177,23,170)(17,176,24,169)(18,175,25,182)(19,174,26,181)(20,173,27,180)(21,172,28,179)(29,156,36,163)(30,155,37,162)(31,168,38,161)(32,167,39,160)(33,166,40,159)(34,165,41,158)(35,164,42,157)(43,107,50,100)(44,106,51,99)(45,105,52,112)(46,104,53,111)(47,103,54,110)(48,102,55,109)(49,101,56,108)(57,80,64,73)(58,79,65,72)(59,78,66,71)(60,77,67,84)(61,76,68,83)(62,75,69,82)(63,74,70,81)(85,224,92,217)(86,223,93,216)(87,222,94,215)(88,221,95,214)(89,220,96,213)(90,219,97,212)(91,218,98,211)(113,193,120,186)(114,192,121,185)(115,191,122,184)(116,190,123,183)(117,189,124,196)(118,188,125,195)(119,187,126,194)(141,199,148,206)(142,198,149,205)(143,197,150,204)(144,210,151,203)(145,209,152,202)(146,208,153,201)(147,207,154,200), (1,111,73,143)(2,112,74,144)(3,99,75,145)(4,100,76,146)(5,101,77,147)(6,102,78,148)(7,103,79,149)(8,104,80,150)(9,105,81,151)(10,106,82,152)(11,107,83,153)(12,108,84,154)(13,109,71,141)(14,110,72,142)(15,32,223,116)(16,33,224,117)(17,34,211,118)(18,35,212,119)(19,36,213,120)(20,37,214,121)(21,38,215,122)(22,39,216,123)(23,40,217,124)(24,41,218,125)(25,42,219,126)(26,29,220,113)(27,30,221,114)(28,31,222,115)(43,68,208,131)(44,69,209,132)(45,70,210,133)(46,57,197,134)(47,58,198,135)(48,59,199,136)(49,60,200,137)(50,61,201,138)(51,62,202,139)(52,63,203,140)(53,64,204,127)(54,65,205,128)(55,66,206,129)(56,67,207,130)(85,196,170,159)(86,183,171,160)(87,184,172,161)(88,185,173,162)(89,186,174,163)(90,187,175,164)(91,188,176,165)(92,189,177,166)(93,190,178,167)(94,191,179,168)(95,192,180,155)(96,193,181,156)(97,194,182,157)(98,195,169,158), (1,212)(2,213)(3,214)(4,215)(5,216)(6,217)(7,218)(8,219)(9,220)(10,221)(11,222)(12,223)(13,224)(14,211)(15,84)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,112)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,168)(44,155)(45,156)(46,157)(47,158)(48,159)(49,160)(50,161)(51,162)(52,163)(53,164)(54,165)(55,166)(56,167)(57,175)(58,176)(59,177)(60,178)(61,179)(62,180)(63,181)(64,182)(65,169)(66,170)(67,171)(68,172)(69,173)(70,174)(85,129)(86,130)(87,131)(88,132)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(97,127)(98,128)(113,144)(114,145)(115,146)(116,147)(117,148)(118,149)(119,150)(120,151)(121,152)(122,153)(123,154)(124,141)(125,142)(126,143)(183,200)(184,201)(185,202)(186,203)(187,204)(188,205)(189,206)(190,207)(191,208)(192,209)(193,210)(194,197)(195,198)(196,199)>;
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), (1,164,73,187)(2,163,74,186)(3,162,75,185)(4,161,76,184)(5,160,77,183)(6,159,78,196)(7,158,79,195)(8,157,80,194)(9,156,81,193)(10,155,82,192)(11,168,83,191)(12,167,84,190)(13,166,71,189)(14,165,72,188)(15,200,223,49)(16,199,224,48)(17,198,211,47)(18,197,212,46)(19,210,213,45)(20,209,214,44)(21,208,215,43)(22,207,216,56)(23,206,217,55)(24,205,218,54)(25,204,219,53)(26,203,220,52)(27,202,221,51)(28,201,222,50)(29,133,113,70)(30,132,114,69)(31,131,115,68)(32,130,116,67)(33,129,117,66)(34,128,118,65)(35,127,119,64)(36,140,120,63)(37,139,121,62)(38,138,122,61)(39,137,123,60)(40,136,124,59)(41,135,125,58)(42,134,126,57)(85,141,170,109)(86,154,171,108)(87,153,172,107)(88,152,173,106)(89,151,174,105)(90,150,175,104)(91,149,176,103)(92,148,177,102)(93,147,178,101)(94,146,179,100)(95,145,180,99)(96,144,181,112)(97,143,182,111)(98,142,169,110), (1,134,8,127)(2,133,9,140)(3,132,10,139)(4,131,11,138)(5,130,12,137)(6,129,13,136)(7,128,14,135)(15,178,22,171)(16,177,23,170)(17,176,24,169)(18,175,25,182)(19,174,26,181)(20,173,27,180)(21,172,28,179)(29,156,36,163)(30,155,37,162)(31,168,38,161)(32,167,39,160)(33,166,40,159)(34,165,41,158)(35,164,42,157)(43,107,50,100)(44,106,51,99)(45,105,52,112)(46,104,53,111)(47,103,54,110)(48,102,55,109)(49,101,56,108)(57,80,64,73)(58,79,65,72)(59,78,66,71)(60,77,67,84)(61,76,68,83)(62,75,69,82)(63,74,70,81)(85,224,92,217)(86,223,93,216)(87,222,94,215)(88,221,95,214)(89,220,96,213)(90,219,97,212)(91,218,98,211)(113,193,120,186)(114,192,121,185)(115,191,122,184)(116,190,123,183)(117,189,124,196)(118,188,125,195)(119,187,126,194)(141,199,148,206)(142,198,149,205)(143,197,150,204)(144,210,151,203)(145,209,152,202)(146,208,153,201)(147,207,154,200), (1,111,73,143)(2,112,74,144)(3,99,75,145)(4,100,76,146)(5,101,77,147)(6,102,78,148)(7,103,79,149)(8,104,80,150)(9,105,81,151)(10,106,82,152)(11,107,83,153)(12,108,84,154)(13,109,71,141)(14,110,72,142)(15,32,223,116)(16,33,224,117)(17,34,211,118)(18,35,212,119)(19,36,213,120)(20,37,214,121)(21,38,215,122)(22,39,216,123)(23,40,217,124)(24,41,218,125)(25,42,219,126)(26,29,220,113)(27,30,221,114)(28,31,222,115)(43,68,208,131)(44,69,209,132)(45,70,210,133)(46,57,197,134)(47,58,198,135)(48,59,199,136)(49,60,200,137)(50,61,201,138)(51,62,202,139)(52,63,203,140)(53,64,204,127)(54,65,205,128)(55,66,206,129)(56,67,207,130)(85,196,170,159)(86,183,171,160)(87,184,172,161)(88,185,173,162)(89,186,174,163)(90,187,175,164)(91,188,176,165)(92,189,177,166)(93,190,178,167)(94,191,179,168)(95,192,180,155)(96,193,181,156)(97,194,182,157)(98,195,169,158), (1,212)(2,213)(3,214)(4,215)(5,216)(6,217)(7,218)(8,219)(9,220)(10,221)(11,222)(12,223)(13,224)(14,211)(15,84)(16,71)(17,72)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,112)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,168)(44,155)(45,156)(46,157)(47,158)(48,159)(49,160)(50,161)(51,162)(52,163)(53,164)(54,165)(55,166)(56,167)(57,175)(58,176)(59,177)(60,178)(61,179)(62,180)(63,181)(64,182)(65,169)(66,170)(67,171)(68,172)(69,173)(70,174)(85,129)(86,130)(87,131)(88,132)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(97,127)(98,128)(113,144)(114,145)(115,146)(116,147)(117,148)(118,149)(119,150)(120,151)(121,152)(122,153)(123,154)(124,141)(125,142)(126,143)(183,200)(184,201)(185,202)(186,203)(187,204)(188,205)(189,206)(190,207)(191,208)(192,209)(193,210)(194,197)(195,198)(196,199) );
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)], [(1,164,73,187),(2,163,74,186),(3,162,75,185),(4,161,76,184),(5,160,77,183),(6,159,78,196),(7,158,79,195),(8,157,80,194),(9,156,81,193),(10,155,82,192),(11,168,83,191),(12,167,84,190),(13,166,71,189),(14,165,72,188),(15,200,223,49),(16,199,224,48),(17,198,211,47),(18,197,212,46),(19,210,213,45),(20,209,214,44),(21,208,215,43),(22,207,216,56),(23,206,217,55),(24,205,218,54),(25,204,219,53),(26,203,220,52),(27,202,221,51),(28,201,222,50),(29,133,113,70),(30,132,114,69),(31,131,115,68),(32,130,116,67),(33,129,117,66),(34,128,118,65),(35,127,119,64),(36,140,120,63),(37,139,121,62),(38,138,122,61),(39,137,123,60),(40,136,124,59),(41,135,125,58),(42,134,126,57),(85,141,170,109),(86,154,171,108),(87,153,172,107),(88,152,173,106),(89,151,174,105),(90,150,175,104),(91,149,176,103),(92,148,177,102),(93,147,178,101),(94,146,179,100),(95,145,180,99),(96,144,181,112),(97,143,182,111),(98,142,169,110)], [(1,134,8,127),(2,133,9,140),(3,132,10,139),(4,131,11,138),(5,130,12,137),(6,129,13,136),(7,128,14,135),(15,178,22,171),(16,177,23,170),(17,176,24,169),(18,175,25,182),(19,174,26,181),(20,173,27,180),(21,172,28,179),(29,156,36,163),(30,155,37,162),(31,168,38,161),(32,167,39,160),(33,166,40,159),(34,165,41,158),(35,164,42,157),(43,107,50,100),(44,106,51,99),(45,105,52,112),(46,104,53,111),(47,103,54,110),(48,102,55,109),(49,101,56,108),(57,80,64,73),(58,79,65,72),(59,78,66,71),(60,77,67,84),(61,76,68,83),(62,75,69,82),(63,74,70,81),(85,224,92,217),(86,223,93,216),(87,222,94,215),(88,221,95,214),(89,220,96,213),(90,219,97,212),(91,218,98,211),(113,193,120,186),(114,192,121,185),(115,191,122,184),(116,190,123,183),(117,189,124,196),(118,188,125,195),(119,187,126,194),(141,199,148,206),(142,198,149,205),(143,197,150,204),(144,210,151,203),(145,209,152,202),(146,208,153,201),(147,207,154,200)], [(1,111,73,143),(2,112,74,144),(3,99,75,145),(4,100,76,146),(5,101,77,147),(6,102,78,148),(7,103,79,149),(8,104,80,150),(9,105,81,151),(10,106,82,152),(11,107,83,153),(12,108,84,154),(13,109,71,141),(14,110,72,142),(15,32,223,116),(16,33,224,117),(17,34,211,118),(18,35,212,119),(19,36,213,120),(20,37,214,121),(21,38,215,122),(22,39,216,123),(23,40,217,124),(24,41,218,125),(25,42,219,126),(26,29,220,113),(27,30,221,114),(28,31,222,115),(43,68,208,131),(44,69,209,132),(45,70,210,133),(46,57,197,134),(47,58,198,135),(48,59,199,136),(49,60,200,137),(50,61,201,138),(51,62,202,139),(52,63,203,140),(53,64,204,127),(54,65,205,128),(55,66,206,129),(56,67,207,130),(85,196,170,159),(86,183,171,160),(87,184,172,161),(88,185,173,162),(89,186,174,163),(90,187,175,164),(91,188,176,165),(92,189,177,166),(93,190,178,167),(94,191,179,168),(95,192,180,155),(96,193,181,156),(97,194,182,157),(98,195,169,158)], [(1,212),(2,213),(3,214),(4,215),(5,216),(6,217),(7,218),(8,219),(9,220),(10,221),(11,222),(12,223),(13,224),(14,211),(15,84),(16,71),(17,72),(18,73),(19,74),(20,75),(21,76),(22,77),(23,78),(24,79),(25,80),(26,81),(27,82),(28,83),(29,112),(30,99),(31,100),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(41,110),(42,111),(43,168),(44,155),(45,156),(46,157),(47,158),(48,159),(49,160),(50,161),(51,162),(52,163),(53,164),(54,165),(55,166),(56,167),(57,175),(58,176),(59,177),(60,178),(61,179),(62,180),(63,181),(64,182),(65,169),(66,170),(67,171),(68,172),(69,173),(70,174),(85,129),(86,130),(87,131),(88,132),(89,133),(90,134),(91,135),(92,136),(93,137),(94,138),(95,139),(96,140),(97,127),(98,128),(113,144),(114,145),(115,146),(116,147),(117,148),(118,149),(119,150),(120,151),(121,152),(122,153),(123,154),(124,141),(125,142),(126,143),(183,200),(184,201),(185,202),(186,203),(187,204),(188,205),(189,206),(190,207),(191,208),(192,209),(193,210),(194,197),(195,198),(196,199)])
Matrix representation ►G ⊆ 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] >;
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 |
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
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