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