?
G = C14.452+ (1+4) order 448 = 26·7
45th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.452+ (1+4), C28⋊Q8⋊21C2, C4⋊D4⋊19D7, C28⋊2D4⋊25C2, C4⋊C4.183D14, (D4×Dic7)⋊23C2, (C2×D4).94D14, (C2×C28).43C23, C22⋊C4.51D14, Dic7⋊D4⋊15C2, C28.204(C4○D4), C4.97(D4⋊2D7), C28.17D4⋊19C2, (C2×C14).160C24, D14⋊C4.16C22, (C22×C4).227D14, C2.47(D4⋊6D14), C23.20(C22×D7), Dic7.23(C4○D4), Dic7.D4⋊21C2, (D4×C14).126C22, C23.11D14⋊7C2, Dic7⋊C4.19C22, C4⋊Dic7.373C22, (C4×Dic7).97C22, (C2×Dic7).79C23, (C22×D7).67C23, C22.181(C23×D7), C23.21D14⋊27C2, (C22×C14).189C23, (C22×C28).244C22, C7⋊3(C22.49C24), C23.D7.112C22, (C2×Dic14).156C22, (C22×Dic7).113C22, (C4×C7⋊D4)⋊20C2, C2.44(D7×C4○D4), C4⋊C4⋊7D7⋊21C2, (C7×C4⋊D4)⋊22C2, (C2×C4×D7).87C22, C14.157(C2×C4○D4), C2.39(C2×D4⋊2D7), (C7×C4⋊C4).148C22, (C2×C4).588(C22×D7), (C2×C7⋊D4).33C22, (C7×C22⋊C4).17C22, SmallGroup(448,1069)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1004 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×12], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×15], D4 [×8], Q8 [×2], C23, C23 [×2], C23, D7, C14 [×3], C14 [×3], C42 [×5], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×5], C22×C4, C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×3], C2×Q8 [×2], Dic7 [×2], Dic7 [×6], C28 [×2], C28 [×3], D14 [×3], C2×C14, C2×C14 [×9], C42⋊C2 [×4], C4×D4 [×2], C4⋊D4, C4⋊D4 [×3], C4.4D4 [×4], C4⋊Q8, Dic14 [×2], C4×D7 [×2], C2×Dic7 [×3], C2×Dic7 [×4], C2×Dic7 [×4], C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×2], C2×C28 [×2], C7×D4 [×4], C22×D7, C22×C14, C22×C14 [×2], C22.49C24, C4×Dic7, C4×Dic7 [×4], Dic7⋊C4, Dic7⋊C4 [×2], C4⋊Dic7 [×2], D14⋊C4, D14⋊C4 [×2], C23.D7, C23.D7 [×6], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14 [×2], C2×C4×D7, C22×Dic7 [×2], C2×C7⋊D4, C2×C7⋊D4 [×2], C22×C28, D4×C14, D4×C14 [×2], C23.11D14 [×2], Dic7.D4 [×2], C28⋊Q8, C4⋊C4⋊7D7, C23.21D14, C4×C7⋊D4, D4×Dic7, C28.17D4 [×2], C28⋊2D4, Dic7⋊D4 [×2], C7×C4⋊D4, C14.452+ (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], C22.49C24, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, D4⋊6D14, D7×C4○D4, C14.452+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=e2=1, d2=a7b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=a7c, 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)
(1 185 21 176)(2 186 22 177)(3 187 23 178)(4 188 24 179)(5 189 25 180)(6 190 26 181)(7 191 27 182)(8 192 28 169)(9 193 15 170)(10 194 16 171)(11 195 17 172)(12 196 18 173)(13 183 19 174)(14 184 20 175)(29 221 55 199)(30 222 56 200)(31 223 43 201)(32 224 44 202)(33 211 45 203)(34 212 46 204)(35 213 47 205)(36 214 48 206)(37 215 49 207)(38 216 50 208)(39 217 51 209)(40 218 52 210)(41 219 53 197)(42 220 54 198)(57 132 83 126)(58 133 84 113)(59 134 71 114)(60 135 72 115)(61 136 73 116)(62 137 74 117)(63 138 75 118)(64 139 76 119)(65 140 77 120)(66 127 78 121)(67 128 79 122)(68 129 80 123)(69 130 81 124)(70 131 82 125)(85 159 105 145)(86 160 106 146)(87 161 107 147)(88 162 108 148)(89 163 109 149)(90 164 110 150)(91 165 111 151)(92 166 112 152)(93 167 99 153)(94 168 100 154)(95 155 101 141)(96 156 102 142)(97 157 103 143)(98 158 104 144)
(1 125)(2 126)(3 113)(4 114)(5 115)(6 116)(7 117)(8 118)(9 119)(10 120)(11 121)(12 122)(13 123)(14 124)(15 139)(16 140)(17 127)(18 128)(19 129)(20 130)(21 131)(22 132)(23 133)(24 134)(25 135)(26 136)(27 137)(28 138)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 141)(36 142)(37 143)(38 144)(39 145)(40 146)(41 147)(42 148)(43 165)(44 166)(45 167)(46 168)(47 155)(48 156)(49 157)(50 158)(51 159)(52 160)(53 161)(54 162)(55 163)(56 164)(57 177)(58 178)(59 179)(60 180)(61 181)(62 182)(63 169)(64 170)(65 171)(66 172)(67 173)(68 174)(69 175)(70 176)(71 188)(72 189)(73 190)(74 191)(75 192)(76 193)(77 194)(78 195)(79 196)(80 183)(81 184)(82 185)(83 186)(84 187)(85 209)(86 210)(87 197)(88 198)(89 199)(90 200)(91 201)(92 202)(93 203)(94 204)(95 205)(96 206)(97 207)(98 208)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)
(1 54 28 35)(2 53 15 34)(3 52 16 33)(4 51 17 32)(5 50 18 31)(6 49 19 30)(7 48 20 29)(8 47 21 42)(9 46 22 41)(10 45 23 40)(11 44 24 39)(12 43 25 38)(13 56 26 37)(14 55 27 36)(57 107 76 94)(58 106 77 93)(59 105 78 92)(60 104 79 91)(61 103 80 90)(62 102 81 89)(63 101 82 88)(64 100 83 87)(65 99 84 86)(66 112 71 85)(67 111 72 98)(68 110 73 97)(69 109 74 96)(70 108 75 95)(113 160 140 153)(114 159 127 152)(115 158 128 151)(116 157 129 150)(117 156 130 149)(118 155 131 148)(119 168 132 147)(120 167 133 146)(121 166 134 145)(122 165 135 144)(123 164 136 143)(124 163 137 142)(125 162 138 141)(126 161 139 154)(169 213 185 198)(170 212 186 197)(171 211 187 210)(172 224 188 209)(173 223 189 208)(174 222 190 207)(175 221 191 206)(176 220 192 205)(177 219 193 204)(178 218 194 203)(179 217 195 202)(180 216 196 201)(181 215 183 200)(182 214 184 199)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 55)(16 56)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(57 89)(58 90)(59 91)(60 92)(61 93)(62 94)(63 95)(64 96)(65 97)(66 98)(67 85)(68 86)(69 87)(70 88)(71 111)(72 112)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 107)(82 108)(83 109)(84 110)(113 150)(114 151)(115 152)(116 153)(117 154)(118 141)(119 142)(120 143)(121 144)(122 145)(123 146)(124 147)(125 148)(126 149)(127 158)(128 159)(129 160)(130 161)(131 162)(132 163)(133 164)(134 165)(135 166)(136 167)(137 168)(138 155)(139 156)(140 157)(169 198)(170 199)(171 200)(172 201)(173 202)(174 203)(175 204)(176 205)(177 206)(178 207)(179 208)(180 209)(181 210)(182 197)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 219)(192 220)(193 221)(194 222)(195 223)(196 224)
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), (1,185,21,176)(2,186,22,177)(3,187,23,178)(4,188,24,179)(5,189,25,180)(6,190,26,181)(7,191,27,182)(8,192,28,169)(9,193,15,170)(10,194,16,171)(11,195,17,172)(12,196,18,173)(13,183,19,174)(14,184,20,175)(29,221,55,199)(30,222,56,200)(31,223,43,201)(32,224,44,202)(33,211,45,203)(34,212,46,204)(35,213,47,205)(36,214,48,206)(37,215,49,207)(38,216,50,208)(39,217,51,209)(40,218,52,210)(41,219,53,197)(42,220,54,198)(57,132,83,126)(58,133,84,113)(59,134,71,114)(60,135,72,115)(61,136,73,116)(62,137,74,117)(63,138,75,118)(64,139,76,119)(65,140,77,120)(66,127,78,121)(67,128,79,122)(68,129,80,123)(69,130,81,124)(70,131,82,125)(85,159,105,145)(86,160,106,146)(87,161,107,147)(88,162,108,148)(89,163,109,149)(90,164,110,150)(91,165,111,151)(92,166,112,152)(93,167,99,153)(94,168,100,154)(95,155,101,141)(96,156,102,142)(97,157,103,143)(98,158,104,144), (1,125)(2,126)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,121)(12,122)(13,123)(14,124)(15,139)(16,140)(17,127)(18,128)(19,129)(20,130)(21,131)(22,132)(23,133)(24,134)(25,135)(26,136)(27,137)(28,138)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,141)(36,142)(37,143)(38,144)(39,145)(40,146)(41,147)(42,148)(43,165)(44,166)(45,167)(46,168)(47,155)(48,156)(49,157)(50,158)(51,159)(52,160)(53,161)(54,162)(55,163)(56,164)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,169)(64,170)(65,171)(66,172)(67,173)(68,174)(69,175)(70,176)(71,188)(72,189)(73,190)(74,191)(75,192)(76,193)(77,194)(78,195)(79,196)(80,183)(81,184)(82,185)(83,186)(84,187)(85,209)(86,210)(87,197)(88,198)(89,199)(90,200)(91,201)(92,202)(93,203)(94,204)(95,205)(96,206)(97,207)(98,208)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (1,54,28,35)(2,53,15,34)(3,52,16,33)(4,51,17,32)(5,50,18,31)(6,49,19,30)(7,48,20,29)(8,47,21,42)(9,46,22,41)(10,45,23,40)(11,44,24,39)(12,43,25,38)(13,56,26,37)(14,55,27,36)(57,107,76,94)(58,106,77,93)(59,105,78,92)(60,104,79,91)(61,103,80,90)(62,102,81,89)(63,101,82,88)(64,100,83,87)(65,99,84,86)(66,112,71,85)(67,111,72,98)(68,110,73,97)(69,109,74,96)(70,108,75,95)(113,160,140,153)(114,159,127,152)(115,158,128,151)(116,157,129,150)(117,156,130,149)(118,155,131,148)(119,168,132,147)(120,167,133,146)(121,166,134,145)(122,165,135,144)(123,164,136,143)(124,163,137,142)(125,162,138,141)(126,161,139,154)(169,213,185,198)(170,212,186,197)(171,211,187,210)(172,224,188,209)(173,223,189,208)(174,222,190,207)(175,221,191,206)(176,220,192,205)(177,219,193,204)(178,218,194,203)(179,217,195,202)(180,216,196,201)(181,215,183,200)(182,214,184,199), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,55)(16,56)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,85)(68,86)(69,87)(70,88)(71,111)(72,112)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,109)(84,110)(113,150)(114,151)(115,152)(116,153)(117,154)(118,141)(119,142)(120,143)(121,144)(122,145)(123,146)(124,147)(125,148)(126,149)(127,158)(128,159)(129,160)(130,161)(131,162)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,155)(139,156)(140,157)(169,198)(170,199)(171,200)(172,201)(173,202)(174,203)(175,204)(176,205)(177,206)(178,207)(179,208)(180,209)(181,210)(182,197)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224)>;
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), (1,185,21,176)(2,186,22,177)(3,187,23,178)(4,188,24,179)(5,189,25,180)(6,190,26,181)(7,191,27,182)(8,192,28,169)(9,193,15,170)(10,194,16,171)(11,195,17,172)(12,196,18,173)(13,183,19,174)(14,184,20,175)(29,221,55,199)(30,222,56,200)(31,223,43,201)(32,224,44,202)(33,211,45,203)(34,212,46,204)(35,213,47,205)(36,214,48,206)(37,215,49,207)(38,216,50,208)(39,217,51,209)(40,218,52,210)(41,219,53,197)(42,220,54,198)(57,132,83,126)(58,133,84,113)(59,134,71,114)(60,135,72,115)(61,136,73,116)(62,137,74,117)(63,138,75,118)(64,139,76,119)(65,140,77,120)(66,127,78,121)(67,128,79,122)(68,129,80,123)(69,130,81,124)(70,131,82,125)(85,159,105,145)(86,160,106,146)(87,161,107,147)(88,162,108,148)(89,163,109,149)(90,164,110,150)(91,165,111,151)(92,166,112,152)(93,167,99,153)(94,168,100,154)(95,155,101,141)(96,156,102,142)(97,157,103,143)(98,158,104,144), (1,125)(2,126)(3,113)(4,114)(5,115)(6,116)(7,117)(8,118)(9,119)(10,120)(11,121)(12,122)(13,123)(14,124)(15,139)(16,140)(17,127)(18,128)(19,129)(20,130)(21,131)(22,132)(23,133)(24,134)(25,135)(26,136)(27,137)(28,138)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,141)(36,142)(37,143)(38,144)(39,145)(40,146)(41,147)(42,148)(43,165)(44,166)(45,167)(46,168)(47,155)(48,156)(49,157)(50,158)(51,159)(52,160)(53,161)(54,162)(55,163)(56,164)(57,177)(58,178)(59,179)(60,180)(61,181)(62,182)(63,169)(64,170)(65,171)(66,172)(67,173)(68,174)(69,175)(70,176)(71,188)(72,189)(73,190)(74,191)(75,192)(76,193)(77,194)(78,195)(79,196)(80,183)(81,184)(82,185)(83,186)(84,187)(85,209)(86,210)(87,197)(88,198)(89,199)(90,200)(91,201)(92,202)(93,203)(94,204)(95,205)(96,206)(97,207)(98,208)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (1,54,28,35)(2,53,15,34)(3,52,16,33)(4,51,17,32)(5,50,18,31)(6,49,19,30)(7,48,20,29)(8,47,21,42)(9,46,22,41)(10,45,23,40)(11,44,24,39)(12,43,25,38)(13,56,26,37)(14,55,27,36)(57,107,76,94)(58,106,77,93)(59,105,78,92)(60,104,79,91)(61,103,80,90)(62,102,81,89)(63,101,82,88)(64,100,83,87)(65,99,84,86)(66,112,71,85)(67,111,72,98)(68,110,73,97)(69,109,74,96)(70,108,75,95)(113,160,140,153)(114,159,127,152)(115,158,128,151)(116,157,129,150)(117,156,130,149)(118,155,131,148)(119,168,132,147)(120,167,133,146)(121,166,134,145)(122,165,135,144)(123,164,136,143)(124,163,137,142)(125,162,138,141)(126,161,139,154)(169,213,185,198)(170,212,186,197)(171,211,187,210)(172,224,188,209)(173,223,189,208)(174,222,190,207)(175,221,191,206)(176,220,192,205)(177,219,193,204)(178,218,194,203)(179,217,195,202)(180,216,196,201)(181,215,183,200)(182,214,184,199), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,55)(16,56)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(57,89)(58,90)(59,91)(60,92)(61,93)(62,94)(63,95)(64,96)(65,97)(66,98)(67,85)(68,86)(69,87)(70,88)(71,111)(72,112)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,109)(84,110)(113,150)(114,151)(115,152)(116,153)(117,154)(118,141)(119,142)(120,143)(121,144)(122,145)(123,146)(124,147)(125,148)(126,149)(127,158)(128,159)(129,160)(130,161)(131,162)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,155)(139,156)(140,157)(169,198)(170,199)(171,200)(172,201)(173,202)(174,203)(175,204)(176,205)(177,206)(178,207)(179,208)(180,209)(181,210)(182,197)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224) );
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)], [(1,185,21,176),(2,186,22,177),(3,187,23,178),(4,188,24,179),(5,189,25,180),(6,190,26,181),(7,191,27,182),(8,192,28,169),(9,193,15,170),(10,194,16,171),(11,195,17,172),(12,196,18,173),(13,183,19,174),(14,184,20,175),(29,221,55,199),(30,222,56,200),(31,223,43,201),(32,224,44,202),(33,211,45,203),(34,212,46,204),(35,213,47,205),(36,214,48,206),(37,215,49,207),(38,216,50,208),(39,217,51,209),(40,218,52,210),(41,219,53,197),(42,220,54,198),(57,132,83,126),(58,133,84,113),(59,134,71,114),(60,135,72,115),(61,136,73,116),(62,137,74,117),(63,138,75,118),(64,139,76,119),(65,140,77,120),(66,127,78,121),(67,128,79,122),(68,129,80,123),(69,130,81,124),(70,131,82,125),(85,159,105,145),(86,160,106,146),(87,161,107,147),(88,162,108,148),(89,163,109,149),(90,164,110,150),(91,165,111,151),(92,166,112,152),(93,167,99,153),(94,168,100,154),(95,155,101,141),(96,156,102,142),(97,157,103,143),(98,158,104,144)], [(1,125),(2,126),(3,113),(4,114),(5,115),(6,116),(7,117),(8,118),(9,119),(10,120),(11,121),(12,122),(13,123),(14,124),(15,139),(16,140),(17,127),(18,128),(19,129),(20,130),(21,131),(22,132),(23,133),(24,134),(25,135),(26,136),(27,137),(28,138),(29,149),(30,150),(31,151),(32,152),(33,153),(34,154),(35,141),(36,142),(37,143),(38,144),(39,145),(40,146),(41,147),(42,148),(43,165),(44,166),(45,167),(46,168),(47,155),(48,156),(49,157),(50,158),(51,159),(52,160),(53,161),(54,162),(55,163),(56,164),(57,177),(58,178),(59,179),(60,180),(61,181),(62,182),(63,169),(64,170),(65,171),(66,172),(67,173),(68,174),(69,175),(70,176),(71,188),(72,189),(73,190),(74,191),(75,192),(76,193),(77,194),(78,195),(79,196),(80,183),(81,184),(82,185),(83,186),(84,187),(85,209),(86,210),(87,197),(88,198),(89,199),(90,200),(91,201),(92,202),(93,203),(94,204),(95,205),(96,206),(97,207),(98,208),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,218),(107,219),(108,220),(109,221),(110,222),(111,223),(112,224)], [(1,54,28,35),(2,53,15,34),(3,52,16,33),(4,51,17,32),(5,50,18,31),(6,49,19,30),(7,48,20,29),(8,47,21,42),(9,46,22,41),(10,45,23,40),(11,44,24,39),(12,43,25,38),(13,56,26,37),(14,55,27,36),(57,107,76,94),(58,106,77,93),(59,105,78,92),(60,104,79,91),(61,103,80,90),(62,102,81,89),(63,101,82,88),(64,100,83,87),(65,99,84,86),(66,112,71,85),(67,111,72,98),(68,110,73,97),(69,109,74,96),(70,108,75,95),(113,160,140,153),(114,159,127,152),(115,158,128,151),(116,157,129,150),(117,156,130,149),(118,155,131,148),(119,168,132,147),(120,167,133,146),(121,166,134,145),(122,165,135,144),(123,164,136,143),(124,163,137,142),(125,162,138,141),(126,161,139,154),(169,213,185,198),(170,212,186,197),(171,211,187,210),(172,224,188,209),(173,223,189,208),(174,222,190,207),(175,221,191,206),(176,220,192,205),(177,219,193,204),(178,218,194,203),(179,217,195,202),(180,216,196,201),(181,215,183,200),(182,214,184,199)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,55),(16,56),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54),(57,89),(58,90),(59,91),(60,92),(61,93),(62,94),(63,95),(64,96),(65,97),(66,98),(67,85),(68,86),(69,87),(70,88),(71,111),(72,112),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104),(79,105),(80,106),(81,107),(82,108),(83,109),(84,110),(113,150),(114,151),(115,152),(116,153),(117,154),(118,141),(119,142),(120,143),(121,144),(122,145),(123,146),(124,147),(125,148),(126,149),(127,158),(128,159),(129,160),(130,161),(131,162),(132,163),(133,164),(134,165),(135,166),(136,167),(137,168),(138,155),(139,156),(140,157),(169,198),(170,199),(171,200),(172,201),(173,202),(174,203),(175,204),(176,205),(177,206),(178,207),(179,208),(180,209),(181,210),(182,197),(183,211),(184,212),(185,213),(186,214),(187,215),(188,216),(189,217),(190,218),(191,219),(192,220),(193,221),(194,222),(195,223),(196,224)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 21 | 0 | 0 |
| 0 | 0 | 16 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 18 | 12 |
| 17 | 11 | 0 | 0 | 0 | 0 |
| 16 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 13 | 0 | 0 | 0 | 0 |
| 11 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 26 | 0 | 0 |
| 0 | 0 | 5 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 16 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 13 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,20,16,0,0,0,0,21,27,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,18,0,0,0,0,0,12],[17,16,0,0,0,0,11,12,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,11,28],[28,11,0,0,0,0,13,1,0,0,0,0,0,0,4,5,0,0,0,0,26,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,16,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,13,0,0,0,0,0,28] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 4Q | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 4 | 4 | 4 | 28 | 2 | 2 | 2 | 2 | 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 | 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 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4⋊2D7 | D4⋊6D14 | D7×C4○D4 |
| kernel | C14.452+ (1+4) | C23.11D14 | Dic7.D4 | C28⋊Q8 | C4⋊C4⋊7D7 | C23.21D14 | C4×C7⋊D4 | D4×Dic7 | C28.17D4 | C28⋊2D4 | Dic7⋊D4 | C7×C4⋊D4 | C4⋊D4 | Dic7 | C28 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 4 | 4 | 6 | 3 | 3 | 9 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{45}2_+^{(1+4)} % in TeX
G:=Group("C14.45ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1069);
// by ID
G=gap.SmallGroup(448,1069);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,758,219,1571,570,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=e^2=1,d^2=a^7*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^7*c,e*d*e=a^7*b^2*d>;
// generators/relations
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