G = C14.502+ 1+4 order 448 = 26·7
50th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.502+ 1+4, C14.752- 1+4, C28⋊Q8⋊23C2, C4⋊C4.94D14, (C2×Dic7)⋊4Q8, C22⋊Q8.8D7, C22.6(Q8×D7), (C2×Q8).74D14, Dic7.4(C2×Q8), Dic7.Q8⋊16C2, (C2×C28).49C23, C22⋊C4.53D14, Dic7⋊Q8⋊12C2, C14.33(C22×Q8), (C2×C14).167C24, (C22×C4).232D14, C4⋊Dic7.47C22, C2.52(D4⋊6D14), C28.48D4.19C2, C22⋊Dic14.3C2, (Q8×C14).102C22, C23.185(C22×D7), C22.188(C23×D7), C23.D7.31C22, Dic7⋊C4.161C22, (C22×C28).314C22, (C22×C14).195C23, C7⋊3(C23.41C23), (C2×Dic7).231C23, (C4×Dic7).101C22, C23.11D14.2C2, C2.33(D4.10D14), (C2×Dic14).158C22, (C22×Dic7).117C22, C2.16(C2×Q8×D7), (C2×C14).6(C2×Q8), (C7×C22⋊Q8).8C2, (C7×C4⋊C4).153C22, (C2×Dic7⋊C4).23C2, (C2×C4).181(C22×D7), (C7×C22⋊C4).22C22, SmallGroup(448,1076)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.502+ 1+4
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=a7b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=a7c, ce=ec, ede=b2d >
Subgroups: 796 in 206 conjugacy classes, 103 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C4 [×16], C22, C22 [×2], C22 [×2], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], Q8 [×4], C23, C14 [×3], C14 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×17], C22×C4, C22×C4 [×2], C2×Q8, C2×Q8 [×3], Dic7 [×4], Dic7 [×6], C28 [×6], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8, C22⋊Q8 [×3], C42.C2 [×4], C4⋊Q8 [×4], Dic14 [×3], C2×Dic7 [×12], C2×Dic7, C2×C28 [×2], C2×C28 [×4], C2×C28, C7×Q8, C22×C14, C23.41C23, C4×Dic7 [×4], Dic7⋊C4 [×14], C4⋊Dic7, C4⋊Dic7 [×2], C23.D7 [×2], C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×Dic14, C2×Dic14 [×2], C22×Dic7 [×2], C22×C28, Q8×C14, C23.11D14 [×2], C22⋊Dic14 [×2], C28⋊Q8 [×2], Dic7.Q8 [×4], C2×Dic7⋊C4, C28.48D4, Dic7⋊Q8 [×2], C7×C22⋊Q8, C14.502+ 1+4
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D7, C2×Q8 [×6], C24, D14 [×7], C22×Q8, 2+ 1+4, 2- 1+4, C22×D7 [×7], C23.41C23, Q8×D7 [×2], C23×D7, D4⋊6D14, C2×Q8×D7, D4.10D14, C14.502+ 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 38 27 222)(2 37 28 221)(3 36 15 220)(4 35 16 219)(5 34 17 218)(6 33 18 217)(7 32 19 216)(8 31 20 215)(9 30 21 214)(10 29 22 213)(11 42 23 212)(12 41 24 211)(13 40 25 224)(14 39 26 223)(43 63 71 88)(44 62 72 87)(45 61 73 86)(46 60 74 85)(47 59 75 98)(48 58 76 97)(49 57 77 96)(50 70 78 95)(51 69 79 94)(52 68 80 93)(53 67 81 92)(54 66 82 91)(55 65 83 90)(56 64 84 89)(99 125 156 132)(100 124 157 131)(101 123 158 130)(102 122 159 129)(103 121 160 128)(104 120 161 127)(105 119 162 140)(106 118 163 139)(107 117 164 138)(108 116 165 137)(109 115 166 136)(110 114 167 135)(111 113 168 134)(112 126 155 133)(141 207 179 195)(142 206 180 194)(143 205 181 193)(144 204 182 192)(145 203 169 191)(146 202 170 190)(147 201 171 189)(148 200 172 188)(149 199 173 187)(150 198 174 186)(151 197 175 185)(152 210 176 184)(153 209 177 183)(154 208 178 196)
(1 173 8 180)(2 174 9 181)(3 175 10 182)(4 176 11 169)(5 177 12 170)(6 178 13 171)(7 179 14 172)(15 151 22 144)(16 152 23 145)(17 153 24 146)(18 154 25 147)(19 141 26 148)(20 142 27 149)(21 143 28 150)(29 204 36 197)(30 205 37 198)(31 206 38 199)(32 207 39 200)(33 208 40 201)(34 209 41 202)(35 210 42 203)(43 139 50 132)(44 140 51 133)(45 127 52 134)(46 128 53 135)(47 129 54 136)(48 130 55 137)(49 131 56 138)(57 157 64 164)(58 158 65 165)(59 159 66 166)(60 160 67 167)(61 161 68 168)(62 162 69 155)(63 163 70 156)(71 118 78 125)(72 119 79 126)(73 120 80 113)(74 121 81 114)(75 122 82 115)(76 123 83 116)(77 124 84 117)(85 103 92 110)(86 104 93 111)(87 105 94 112)(88 106 95 99)(89 107 96 100)(90 108 97 101)(91 109 98 102)(183 211 190 218)(184 212 191 219)(185 213 192 220)(186 214 193 221)(187 215 194 222)(188 216 195 223)(189 217 196 224)
(1 92 20 60)(2 93 21 61)(3 94 22 62)(4 95 23 63)(5 96 24 64)(6 97 25 65)(7 98 26 66)(8 85 27 67)(9 86 28 68)(10 87 15 69)(11 88 16 70)(12 89 17 57)(13 90 18 58)(14 91 19 59)(29 44 220 79)(30 45 221 80)(31 46 222 81)(32 47 223 82)(33 48 224 83)(34 49 211 84)(35 50 212 71)(36 51 213 72)(37 52 214 73)(38 53 215 74)(39 54 216 75)(40 55 217 76)(41 56 218 77)(42 43 219 78)(99 152 163 169)(100 153 164 170)(101 154 165 171)(102 141 166 172)(103 142 167 173)(104 143 168 174)(105 144 155 175)(106 145 156 176)(107 146 157 177)(108 147 158 178)(109 148 159 179)(110 149 160 180)(111 150 161 181)(112 151 162 182)(113 198 127 193)(114 199 128 194)(115 200 129 195)(116 201 130 196)(117 202 131 183)(118 203 132 184)(119 204 133 185)(120 205 134 186)(121 206 135 187)(122 207 136 188)(123 208 137 189)(124 209 138 190)(125 210 139 191)(126 197 140 192)
(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(57 96)(58 97)(59 98)(60 85)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(99 156)(100 157)(101 158)(102 159)(103 160)(104 161)(105 162)(106 163)(107 164)(108 165)(109 166)(110 167)(111 168)(112 155)(113 134)(114 135)(115 136)(116 137)(117 138)(118 139)(119 140)(120 127)(121 128)(122 129)(123 130)(124 131)(125 132)(126 133)
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), (1,38,27,222)(2,37,28,221)(3,36,15,220)(4,35,16,219)(5,34,17,218)(6,33,18,217)(7,32,19,216)(8,31,20,215)(9,30,21,214)(10,29,22,213)(11,42,23,212)(12,41,24,211)(13,40,25,224)(14,39,26,223)(43,63,71,88)(44,62,72,87)(45,61,73,86)(46,60,74,85)(47,59,75,98)(48,58,76,97)(49,57,77,96)(50,70,78,95)(51,69,79,94)(52,68,80,93)(53,67,81,92)(54,66,82,91)(55,65,83,90)(56,64,84,89)(99,125,156,132)(100,124,157,131)(101,123,158,130)(102,122,159,129)(103,121,160,128)(104,120,161,127)(105,119,162,140)(106,118,163,139)(107,117,164,138)(108,116,165,137)(109,115,166,136)(110,114,167,135)(111,113,168,134)(112,126,155,133)(141,207,179,195)(142,206,180,194)(143,205,181,193)(144,204,182,192)(145,203,169,191)(146,202,170,190)(147,201,171,189)(148,200,172,188)(149,199,173,187)(150,198,174,186)(151,197,175,185)(152,210,176,184)(153,209,177,183)(154,208,178,196), (1,173,8,180)(2,174,9,181)(3,175,10,182)(4,176,11,169)(5,177,12,170)(6,178,13,171)(7,179,14,172)(15,151,22,144)(16,152,23,145)(17,153,24,146)(18,154,25,147)(19,141,26,148)(20,142,27,149)(21,143,28,150)(29,204,36,197)(30,205,37,198)(31,206,38,199)(32,207,39,200)(33,208,40,201)(34,209,41,202)(35,210,42,203)(43,139,50,132)(44,140,51,133)(45,127,52,134)(46,128,53,135)(47,129,54,136)(48,130,55,137)(49,131,56,138)(57,157,64,164)(58,158,65,165)(59,159,66,166)(60,160,67,167)(61,161,68,168)(62,162,69,155)(63,163,70,156)(71,118,78,125)(72,119,79,126)(73,120,80,113)(74,121,81,114)(75,122,82,115)(76,123,83,116)(77,124,84,117)(85,103,92,110)(86,104,93,111)(87,105,94,112)(88,106,95,99)(89,107,96,100)(90,108,97,101)(91,109,98,102)(183,211,190,218)(184,212,191,219)(185,213,192,220)(186,214,193,221)(187,215,194,222)(188,216,195,223)(189,217,196,224), (1,92,20,60)(2,93,21,61)(3,94,22,62)(4,95,23,63)(5,96,24,64)(6,97,25,65)(7,98,26,66)(8,85,27,67)(9,86,28,68)(10,87,15,69)(11,88,16,70)(12,89,17,57)(13,90,18,58)(14,91,19,59)(29,44,220,79)(30,45,221,80)(31,46,222,81)(32,47,223,82)(33,48,224,83)(34,49,211,84)(35,50,212,71)(36,51,213,72)(37,52,214,73)(38,53,215,74)(39,54,216,75)(40,55,217,76)(41,56,218,77)(42,43,219,78)(99,152,163,169)(100,153,164,170)(101,154,165,171)(102,141,166,172)(103,142,167,173)(104,143,168,174)(105,144,155,175)(106,145,156,176)(107,146,157,177)(108,147,158,178)(109,148,159,179)(110,149,160,180)(111,150,161,181)(112,151,162,182)(113,198,127,193)(114,199,128,194)(115,200,129,195)(116,201,130,196)(117,202,131,183)(118,203,132,184)(119,204,133,185)(120,205,134,186)(121,206,135,187)(122,207,136,188)(123,208,137,189)(124,209,138,190)(125,210,139,191)(126,197,140,192), (43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(57,96)(58,97)(59,98)(60,85)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(99,156)(100,157)(101,158)(102,159)(103,160)(104,161)(105,162)(106,163)(107,164)(108,165)(109,166)(110,167)(111,168)(112,155)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)>;
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), (1,38,27,222)(2,37,28,221)(3,36,15,220)(4,35,16,219)(5,34,17,218)(6,33,18,217)(7,32,19,216)(8,31,20,215)(9,30,21,214)(10,29,22,213)(11,42,23,212)(12,41,24,211)(13,40,25,224)(14,39,26,223)(43,63,71,88)(44,62,72,87)(45,61,73,86)(46,60,74,85)(47,59,75,98)(48,58,76,97)(49,57,77,96)(50,70,78,95)(51,69,79,94)(52,68,80,93)(53,67,81,92)(54,66,82,91)(55,65,83,90)(56,64,84,89)(99,125,156,132)(100,124,157,131)(101,123,158,130)(102,122,159,129)(103,121,160,128)(104,120,161,127)(105,119,162,140)(106,118,163,139)(107,117,164,138)(108,116,165,137)(109,115,166,136)(110,114,167,135)(111,113,168,134)(112,126,155,133)(141,207,179,195)(142,206,180,194)(143,205,181,193)(144,204,182,192)(145,203,169,191)(146,202,170,190)(147,201,171,189)(148,200,172,188)(149,199,173,187)(150,198,174,186)(151,197,175,185)(152,210,176,184)(153,209,177,183)(154,208,178,196), (1,173,8,180)(2,174,9,181)(3,175,10,182)(4,176,11,169)(5,177,12,170)(6,178,13,171)(7,179,14,172)(15,151,22,144)(16,152,23,145)(17,153,24,146)(18,154,25,147)(19,141,26,148)(20,142,27,149)(21,143,28,150)(29,204,36,197)(30,205,37,198)(31,206,38,199)(32,207,39,200)(33,208,40,201)(34,209,41,202)(35,210,42,203)(43,139,50,132)(44,140,51,133)(45,127,52,134)(46,128,53,135)(47,129,54,136)(48,130,55,137)(49,131,56,138)(57,157,64,164)(58,158,65,165)(59,159,66,166)(60,160,67,167)(61,161,68,168)(62,162,69,155)(63,163,70,156)(71,118,78,125)(72,119,79,126)(73,120,80,113)(74,121,81,114)(75,122,82,115)(76,123,83,116)(77,124,84,117)(85,103,92,110)(86,104,93,111)(87,105,94,112)(88,106,95,99)(89,107,96,100)(90,108,97,101)(91,109,98,102)(183,211,190,218)(184,212,191,219)(185,213,192,220)(186,214,193,221)(187,215,194,222)(188,216,195,223)(189,217,196,224), (1,92,20,60)(2,93,21,61)(3,94,22,62)(4,95,23,63)(5,96,24,64)(6,97,25,65)(7,98,26,66)(8,85,27,67)(9,86,28,68)(10,87,15,69)(11,88,16,70)(12,89,17,57)(13,90,18,58)(14,91,19,59)(29,44,220,79)(30,45,221,80)(31,46,222,81)(32,47,223,82)(33,48,224,83)(34,49,211,84)(35,50,212,71)(36,51,213,72)(37,52,214,73)(38,53,215,74)(39,54,216,75)(40,55,217,76)(41,56,218,77)(42,43,219,78)(99,152,163,169)(100,153,164,170)(101,154,165,171)(102,141,166,172)(103,142,167,173)(104,143,168,174)(105,144,155,175)(106,145,156,176)(107,146,157,177)(108,147,158,178)(109,148,159,179)(110,149,160,180)(111,150,161,181)(112,151,162,182)(113,198,127,193)(114,199,128,194)(115,200,129,195)(116,201,130,196)(117,202,131,183)(118,203,132,184)(119,204,133,185)(120,205,134,186)(121,206,135,187)(122,207,136,188)(123,208,137,189)(124,209,138,190)(125,210,139,191)(126,197,140,192), (43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(57,96)(58,97)(59,98)(60,85)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(99,156)(100,157)(101,158)(102,159)(103,160)(104,161)(105,162)(106,163)(107,164)(108,165)(109,166)(110,167)(111,168)(112,155)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133) );
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)], [(1,38,27,222),(2,37,28,221),(3,36,15,220),(4,35,16,219),(5,34,17,218),(6,33,18,217),(7,32,19,216),(8,31,20,215),(9,30,21,214),(10,29,22,213),(11,42,23,212),(12,41,24,211),(13,40,25,224),(14,39,26,223),(43,63,71,88),(44,62,72,87),(45,61,73,86),(46,60,74,85),(47,59,75,98),(48,58,76,97),(49,57,77,96),(50,70,78,95),(51,69,79,94),(52,68,80,93),(53,67,81,92),(54,66,82,91),(55,65,83,90),(56,64,84,89),(99,125,156,132),(100,124,157,131),(101,123,158,130),(102,122,159,129),(103,121,160,128),(104,120,161,127),(105,119,162,140),(106,118,163,139),(107,117,164,138),(108,116,165,137),(109,115,166,136),(110,114,167,135),(111,113,168,134),(112,126,155,133),(141,207,179,195),(142,206,180,194),(143,205,181,193),(144,204,182,192),(145,203,169,191),(146,202,170,190),(147,201,171,189),(148,200,172,188),(149,199,173,187),(150,198,174,186),(151,197,175,185),(152,210,176,184),(153,209,177,183),(154,208,178,196)], [(1,173,8,180),(2,174,9,181),(3,175,10,182),(4,176,11,169),(5,177,12,170),(6,178,13,171),(7,179,14,172),(15,151,22,144),(16,152,23,145),(17,153,24,146),(18,154,25,147),(19,141,26,148),(20,142,27,149),(21,143,28,150),(29,204,36,197),(30,205,37,198),(31,206,38,199),(32,207,39,200),(33,208,40,201),(34,209,41,202),(35,210,42,203),(43,139,50,132),(44,140,51,133),(45,127,52,134),(46,128,53,135),(47,129,54,136),(48,130,55,137),(49,131,56,138),(57,157,64,164),(58,158,65,165),(59,159,66,166),(60,160,67,167),(61,161,68,168),(62,162,69,155),(63,163,70,156),(71,118,78,125),(72,119,79,126),(73,120,80,113),(74,121,81,114),(75,122,82,115),(76,123,83,116),(77,124,84,117),(85,103,92,110),(86,104,93,111),(87,105,94,112),(88,106,95,99),(89,107,96,100),(90,108,97,101),(91,109,98,102),(183,211,190,218),(184,212,191,219),(185,213,192,220),(186,214,193,221),(187,215,194,222),(188,216,195,223),(189,217,196,224)], [(1,92,20,60),(2,93,21,61),(3,94,22,62),(4,95,23,63),(5,96,24,64),(6,97,25,65),(7,98,26,66),(8,85,27,67),(9,86,28,68),(10,87,15,69),(11,88,16,70),(12,89,17,57),(13,90,18,58),(14,91,19,59),(29,44,220,79),(30,45,221,80),(31,46,222,81),(32,47,223,82),(33,48,224,83),(34,49,211,84),(35,50,212,71),(36,51,213,72),(37,52,214,73),(38,53,215,74),(39,54,216,75),(40,55,217,76),(41,56,218,77),(42,43,219,78),(99,152,163,169),(100,153,164,170),(101,154,165,171),(102,141,166,172),(103,142,167,173),(104,143,168,174),(105,144,155,175),(106,145,156,176),(107,146,157,177),(108,147,158,178),(109,148,159,179),(110,149,160,180),(111,150,161,181),(112,151,162,182),(113,198,127,193),(114,199,128,194),(115,200,129,195),(116,201,130,196),(117,202,131,183),(118,203,132,184),(119,204,133,185),(120,205,134,186),(121,206,135,187),(122,207,136,188),(123,208,137,189),(124,209,138,190),(125,210,139,191),(126,197,140,192)], [(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(50,78),(51,79),(52,80),(53,81),(54,82),(55,83),(56,84),(57,96),(58,97),(59,98),(60,85),(61,86),(62,87),(63,88),(64,89),(65,90),(66,91),(67,92),(68,93),(69,94),(70,95),(99,156),(100,157),(101,158),(102,159),(103,160),(104,161),(105,162),(106,163),(107,164),(108,165),(109,166),(110,167),(111,168),(112,155),(113,134),(114,135),(115,136),(116,137),(117,138),(118,139),(119,140),(120,127),(121,128),(122,129),(123,130),(124,131),(125,132),(126,133)])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 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 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D7 | D14 | D14 | D14 | D14 | 2+ 1+4 | 2- 1+4 | Q8×D7 | D4⋊6D14 | D4.10D14 |
| kernel | C14.502+ 1+4 | C23.11D14 | C22⋊Dic14 | C28⋊Q8 | Dic7.Q8 | C2×Dic7⋊C4 | C28.48D4 | Dic7⋊Q8 | C7×C22⋊Q8 | C2×Dic7 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 1 | 4 | 3 | 6 | 9 | 3 | 3 | 1 | 1 | 6 | 6 | 6 |
Matrix representation of C14.502+ 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 21 | 0 | 0 |
| 0 | 0 | 8 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 21 |
| 0 | 0 | 0 | 0 | 8 | 21 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 5 | 0 | 0 |
| 0 | 0 | 28 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 5 |
| 0 | 0 | 0 | 0 | 28 | 27 |
| 8 | 5 | 0 | 0 | 0 | 0 |
| 16 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 16 | 0 | 0 |
| 0 | 0 | 13 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 16 |
| 0 | 0 | 0 | 0 | 13 | 5 |
| 20 | 27 | 0 | 0 | 0 | 0 |
| 12 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 28 | 0 | 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 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,26,8,0,0,0,0,21,21,0,0,0,0,0,0,26,8,0,0,0,0,21,21],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,28,0,0,0,0,5,27,0,0,0,0,0,0,2,28,0,0,0,0,5,27],[8,16,0,0,0,0,5,21,0,0,0,0,0,0,24,13,0,0,0,0,16,5,0,0,0,0,0,0,24,13,0,0,0,0,16,5],[20,12,0,0,0,0,27,9,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,1,0,0,0,0,0,0,1,0,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28] >;
C14.502+ 1+4 in GAP, Magma, Sage, TeX
C_{14}._{50}2_+^{1+4} % in TeX
G:=Group("C14.50ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1076);
// by ID
G=gap.SmallGroup(448,1076);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,387,100,1123,185,80,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=a^7*b^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=a^7*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations