?
G = C14.442+ (1+4) order 448 = 26·7
44th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.442+ (1+4), C4⋊D4⋊18D7, C4⋊C4.92D14, C28⋊2D4⋊24C2, C28⋊D4⋊18C2, (C2×D4).93D14, D14⋊D4⋊22C2, Dic7.Q8⋊13C2, C22⋊C4.50D14, Dic7⋊D4⋊31C2, Dic7⋊4D4⋊12C2, D14.5D4⋊13C2, D14.D4⋊21C2, (C2×C14).159C24, (C2×C28).597C23, D14⋊C4.71C22, Dic7.5(C4○D4), (C22×C4).226D14, C2.46(D4⋊6D14), C23.19(C22×D7), (D4×C14).125C22, (C2×D28).144C22, C23.11D14⋊6C2, C4⋊Dic7.208C22, (C22×C14).26C23, (C2×Dic7).78C23, (C22×D7).66C23, C22.180(C23×D7), C23.D7.27C22, C23.18D14⋊12C2, C23.23D14⋊23C2, Dic7⋊C4.160C22, (C22×C28).379C22, C7⋊3(C22.34C24), (C4×Dic7).211C22, (C22×Dic7).112C22, (C4×C7⋊D4)⋊56C2, C2.43(D7×C4○D4), (C7×C4⋊D4)⋊21C2, C14.156(C2×C4○D4), (C2×C4×D7).209C22, (C7×C4⋊C4).147C22, (C2×C4).179(C22×D7), (C2×C7⋊D4).32C22, (C7×C22⋊C4).16C22, SmallGroup(448,1068)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1164 in 240 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×5], C4 [×11], C22, C22 [×15], C7, C2×C4 [×4], C2×C4 [×12], D4 [×12], C23 [×3], C23 [×2], D7 [×2], C14 [×3], C14 [×3], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×7], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×D4 [×7], Dic7 [×2], Dic7 [×5], C28 [×4], D14 [×6], C2×C14, C2×C14 [×9], C42⋊C2, C4×D4 [×2], C4⋊D4, C4⋊D4 [×5], C22.D4 [×4], C42.C2, C4⋊1D4, C4×D7 [×2], D28, C2×Dic7 [×6], C2×Dic7 [×3], C7⋊D4 [×8], C2×C28 [×4], C2×C28, C7×D4 [×3], C22×D7 [×2], C22×C14 [×3], C22.34C24, C4×Dic7 [×2], Dic7⋊C4 [×6], C4⋊Dic7, D14⋊C4 [×4], C23.D7 [×4], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×C4×D7 [×2], C2×D28, C22×Dic7 [×2], C2×C7⋊D4 [×6], C22×C28, D4×C14 [×3], C23.11D14, Dic7⋊4D4, D14.D4, D14⋊D4, Dic7.Q8, D14.5D4, C4×C7⋊D4, C23.23D14, C23.18D14, C28⋊2D4, Dic7⋊D4 [×3], C28⋊D4, C7×C4⋊D4, C14.442+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D7 [×7], C22.34C24, C23×D7, D4⋊6D14 [×2], D7×C4○D4, C14.442+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=1, c2=e2=a7, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=a7b-1, bd=db, ebe-1=a7b, cd=dc, ce=ec, ede-1=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 17 169)(2 186 18 170)(3 187 19 171)(4 188 20 172)(5 189 21 173)(6 190 22 174)(7 191 23 175)(8 192 24 176)(9 193 25 177)(10 194 26 178)(11 195 27 179)(12 196 28 180)(13 183 15 181)(14 184 16 182)(29 215 52 208)(30 216 53 209)(31 217 54 210)(32 218 55 197)(33 219 56 198)(34 220 43 199)(35 221 44 200)(36 222 45 201)(37 223 46 202)(38 224 47 203)(39 211 48 204)(40 212 49 205)(41 213 50 206)(42 214 51 207)(57 137 75 125)(58 138 76 126)(59 139 77 113)(60 140 78 114)(61 127 79 115)(62 128 80 116)(63 129 81 117)(64 130 82 118)(65 131 83 119)(66 132 84 120)(67 133 71 121)(68 134 72 122)(69 135 73 123)(70 136 74 124)(85 165 101 149)(86 166 102 150)(87 167 103 151)(88 168 104 152)(89 155 105 153)(90 156 106 154)(91 157 107 141)(92 158 108 142)(93 159 109 143)(94 160 110 144)(95 161 111 145)(96 162 112 146)(97 163 99 147)(98 164 100 148)
(1 121 8 114)(2 122 9 115)(3 123 10 116)(4 124 11 117)(5 125 12 118)(6 126 13 119)(7 113 14 120)(15 131 22 138)(16 132 23 139)(17 133 24 140)(18 134 25 127)(19 135 26 128)(20 136 27 129)(21 137 28 130)(29 150 36 143)(30 151 37 144)(31 152 38 145)(32 153 39 146)(33 154 40 147)(34 141 41 148)(35 142 42 149)(43 157 50 164)(44 158 51 165)(45 159 52 166)(46 160 53 167)(47 161 54 168)(48 162 55 155)(49 163 56 156)(57 173 64 180)(58 174 65 181)(59 175 66 182)(60 176 67 169)(61 177 68 170)(62 178 69 171)(63 179 70 172)(71 185 78 192)(72 186 79 193)(73 187 80 194)(74 188 81 195)(75 189 82 196)(76 190 83 183)(77 191 84 184)(85 207 92 200)(86 208 93 201)(87 209 94 202)(88 210 95 203)(89 197 96 204)(90 198 97 205)(91 199 98 206)(99 212 106 219)(100 213 107 220)(101 214 108 221)(102 215 109 222)(103 216 110 223)(104 217 111 224)(105 218 112 211)
(1 50 17 41)(2 49 18 40)(3 48 19 39)(4 47 20 38)(5 46 21 37)(6 45 22 36)(7 44 23 35)(8 43 24 34)(9 56 25 33)(10 55 26 32)(11 54 27 31)(12 53 28 30)(13 52 15 29)(14 51 16 42)(57 110 75 94)(58 109 76 93)(59 108 77 92)(60 107 78 91)(61 106 79 90)(62 105 80 89)(63 104 81 88)(64 103 82 87)(65 102 83 86)(66 101 84 85)(67 100 71 98)(68 99 72 97)(69 112 73 96)(70 111 74 95)(113 158 139 142)(114 157 140 141)(115 156 127 154)(116 155 128 153)(117 168 129 152)(118 167 130 151)(119 166 131 150)(120 165 132 149)(121 164 133 148)(122 163 134 147)(123 162 135 146)(124 161 136 145)(125 160 137 144)(126 159 138 143)(169 213 185 206)(170 212 186 205)(171 211 187 204)(172 224 188 203)(173 223 189 202)(174 222 190 201)(175 221 191 200)(176 220 192 199)(177 219 193 198)(178 218 194 197)(179 217 195 210)(180 216 196 209)(181 215 183 208)(182 214 184 207)
(1 34 8 41)(2 35 9 42)(3 36 10 29)(4 37 11 30)(5 38 12 31)(6 39 13 32)(7 40 14 33)(15 55 22 48)(16 56 23 49)(17 43 24 50)(18 44 25 51)(19 45 26 52)(20 46 27 53)(21 47 28 54)(57 88 64 95)(58 89 65 96)(59 90 66 97)(60 91 67 98)(61 92 68 85)(62 93 69 86)(63 94 70 87)(71 100 78 107)(72 101 79 108)(73 102 80 109)(74 103 81 110)(75 104 82 111)(76 105 83 112)(77 106 84 99)(113 147 120 154)(114 148 121 141)(115 149 122 142)(116 150 123 143)(117 151 124 144)(118 152 125 145)(119 153 126 146)(127 165 134 158)(128 166 135 159)(129 167 136 160)(130 168 137 161)(131 155 138 162)(132 156 139 163)(133 157 140 164)(169 206 176 199)(170 207 177 200)(171 208 178 201)(172 209 179 202)(173 210 180 203)(174 197 181 204)(175 198 182 205)(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)
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), (1,185,17,169)(2,186,18,170)(3,187,19,171)(4,188,20,172)(5,189,21,173)(6,190,22,174)(7,191,23,175)(8,192,24,176)(9,193,25,177)(10,194,26,178)(11,195,27,179)(12,196,28,180)(13,183,15,181)(14,184,16,182)(29,215,52,208)(30,216,53,209)(31,217,54,210)(32,218,55,197)(33,219,56,198)(34,220,43,199)(35,221,44,200)(36,222,45,201)(37,223,46,202)(38,224,47,203)(39,211,48,204)(40,212,49,205)(41,213,50,206)(42,214,51,207)(57,137,75,125)(58,138,76,126)(59,139,77,113)(60,140,78,114)(61,127,79,115)(62,128,80,116)(63,129,81,117)(64,130,82,118)(65,131,83,119)(66,132,84,120)(67,133,71,121)(68,134,72,122)(69,135,73,123)(70,136,74,124)(85,165,101,149)(86,166,102,150)(87,167,103,151)(88,168,104,152)(89,155,105,153)(90,156,106,154)(91,157,107,141)(92,158,108,142)(93,159,109,143)(94,160,110,144)(95,161,111,145)(96,162,112,146)(97,163,99,147)(98,164,100,148), (1,121,8,114)(2,122,9,115)(3,123,10,116)(4,124,11,117)(5,125,12,118)(6,126,13,119)(7,113,14,120)(15,131,22,138)(16,132,23,139)(17,133,24,140)(18,134,25,127)(19,135,26,128)(20,136,27,129)(21,137,28,130)(29,150,36,143)(30,151,37,144)(31,152,38,145)(32,153,39,146)(33,154,40,147)(34,141,41,148)(35,142,42,149)(43,157,50,164)(44,158,51,165)(45,159,52,166)(46,160,53,167)(47,161,54,168)(48,162,55,155)(49,163,56,156)(57,173,64,180)(58,174,65,181)(59,175,66,182)(60,176,67,169)(61,177,68,170)(62,178,69,171)(63,179,70,172)(71,185,78,192)(72,186,79,193)(73,187,80,194)(74,188,81,195)(75,189,82,196)(76,190,83,183)(77,191,84,184)(85,207,92,200)(86,208,93,201)(87,209,94,202)(88,210,95,203)(89,197,96,204)(90,198,97,205)(91,199,98,206)(99,212,106,219)(100,213,107,220)(101,214,108,221)(102,215,109,222)(103,216,110,223)(104,217,111,224)(105,218,112,211), (1,50,17,41)(2,49,18,40)(3,48,19,39)(4,47,20,38)(5,46,21,37)(6,45,22,36)(7,44,23,35)(8,43,24,34)(9,56,25,33)(10,55,26,32)(11,54,27,31)(12,53,28,30)(13,52,15,29)(14,51,16,42)(57,110,75,94)(58,109,76,93)(59,108,77,92)(60,107,78,91)(61,106,79,90)(62,105,80,89)(63,104,81,88)(64,103,82,87)(65,102,83,86)(66,101,84,85)(67,100,71,98)(68,99,72,97)(69,112,73,96)(70,111,74,95)(113,158,139,142)(114,157,140,141)(115,156,127,154)(116,155,128,153)(117,168,129,152)(118,167,130,151)(119,166,131,150)(120,165,132,149)(121,164,133,148)(122,163,134,147)(123,162,135,146)(124,161,136,145)(125,160,137,144)(126,159,138,143)(169,213,185,206)(170,212,186,205)(171,211,187,204)(172,224,188,203)(173,223,189,202)(174,222,190,201)(175,221,191,200)(176,220,192,199)(177,219,193,198)(178,218,194,197)(179,217,195,210)(180,216,196,209)(181,215,183,208)(182,214,184,207), (1,34,8,41)(2,35,9,42)(3,36,10,29)(4,37,11,30)(5,38,12,31)(6,39,13,32)(7,40,14,33)(15,55,22,48)(16,56,23,49)(17,43,24,50)(18,44,25,51)(19,45,26,52)(20,46,27,53)(21,47,28,54)(57,88,64,95)(58,89,65,96)(59,90,66,97)(60,91,67,98)(61,92,68,85)(62,93,69,86)(63,94,70,87)(71,100,78,107)(72,101,79,108)(73,102,80,109)(74,103,81,110)(75,104,82,111)(76,105,83,112)(77,106,84,99)(113,147,120,154)(114,148,121,141)(115,149,122,142)(116,150,123,143)(117,151,124,144)(118,152,125,145)(119,153,126,146)(127,165,134,158)(128,166,135,159)(129,167,136,160)(130,168,137,161)(131,155,138,162)(132,156,139,163)(133,157,140,164)(169,206,176,199)(170,207,177,200)(171,208,178,201)(172,209,179,202)(173,210,180,203)(174,197,181,204)(175,198,182,205)(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)>;
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), (1,185,17,169)(2,186,18,170)(3,187,19,171)(4,188,20,172)(5,189,21,173)(6,190,22,174)(7,191,23,175)(8,192,24,176)(9,193,25,177)(10,194,26,178)(11,195,27,179)(12,196,28,180)(13,183,15,181)(14,184,16,182)(29,215,52,208)(30,216,53,209)(31,217,54,210)(32,218,55,197)(33,219,56,198)(34,220,43,199)(35,221,44,200)(36,222,45,201)(37,223,46,202)(38,224,47,203)(39,211,48,204)(40,212,49,205)(41,213,50,206)(42,214,51,207)(57,137,75,125)(58,138,76,126)(59,139,77,113)(60,140,78,114)(61,127,79,115)(62,128,80,116)(63,129,81,117)(64,130,82,118)(65,131,83,119)(66,132,84,120)(67,133,71,121)(68,134,72,122)(69,135,73,123)(70,136,74,124)(85,165,101,149)(86,166,102,150)(87,167,103,151)(88,168,104,152)(89,155,105,153)(90,156,106,154)(91,157,107,141)(92,158,108,142)(93,159,109,143)(94,160,110,144)(95,161,111,145)(96,162,112,146)(97,163,99,147)(98,164,100,148), (1,121,8,114)(2,122,9,115)(3,123,10,116)(4,124,11,117)(5,125,12,118)(6,126,13,119)(7,113,14,120)(15,131,22,138)(16,132,23,139)(17,133,24,140)(18,134,25,127)(19,135,26,128)(20,136,27,129)(21,137,28,130)(29,150,36,143)(30,151,37,144)(31,152,38,145)(32,153,39,146)(33,154,40,147)(34,141,41,148)(35,142,42,149)(43,157,50,164)(44,158,51,165)(45,159,52,166)(46,160,53,167)(47,161,54,168)(48,162,55,155)(49,163,56,156)(57,173,64,180)(58,174,65,181)(59,175,66,182)(60,176,67,169)(61,177,68,170)(62,178,69,171)(63,179,70,172)(71,185,78,192)(72,186,79,193)(73,187,80,194)(74,188,81,195)(75,189,82,196)(76,190,83,183)(77,191,84,184)(85,207,92,200)(86,208,93,201)(87,209,94,202)(88,210,95,203)(89,197,96,204)(90,198,97,205)(91,199,98,206)(99,212,106,219)(100,213,107,220)(101,214,108,221)(102,215,109,222)(103,216,110,223)(104,217,111,224)(105,218,112,211), (1,50,17,41)(2,49,18,40)(3,48,19,39)(4,47,20,38)(5,46,21,37)(6,45,22,36)(7,44,23,35)(8,43,24,34)(9,56,25,33)(10,55,26,32)(11,54,27,31)(12,53,28,30)(13,52,15,29)(14,51,16,42)(57,110,75,94)(58,109,76,93)(59,108,77,92)(60,107,78,91)(61,106,79,90)(62,105,80,89)(63,104,81,88)(64,103,82,87)(65,102,83,86)(66,101,84,85)(67,100,71,98)(68,99,72,97)(69,112,73,96)(70,111,74,95)(113,158,139,142)(114,157,140,141)(115,156,127,154)(116,155,128,153)(117,168,129,152)(118,167,130,151)(119,166,131,150)(120,165,132,149)(121,164,133,148)(122,163,134,147)(123,162,135,146)(124,161,136,145)(125,160,137,144)(126,159,138,143)(169,213,185,206)(170,212,186,205)(171,211,187,204)(172,224,188,203)(173,223,189,202)(174,222,190,201)(175,221,191,200)(176,220,192,199)(177,219,193,198)(178,218,194,197)(179,217,195,210)(180,216,196,209)(181,215,183,208)(182,214,184,207), (1,34,8,41)(2,35,9,42)(3,36,10,29)(4,37,11,30)(5,38,12,31)(6,39,13,32)(7,40,14,33)(15,55,22,48)(16,56,23,49)(17,43,24,50)(18,44,25,51)(19,45,26,52)(20,46,27,53)(21,47,28,54)(57,88,64,95)(58,89,65,96)(59,90,66,97)(60,91,67,98)(61,92,68,85)(62,93,69,86)(63,94,70,87)(71,100,78,107)(72,101,79,108)(73,102,80,109)(74,103,81,110)(75,104,82,111)(76,105,83,112)(77,106,84,99)(113,147,120,154)(114,148,121,141)(115,149,122,142)(116,150,123,143)(117,151,124,144)(118,152,125,145)(119,153,126,146)(127,165,134,158)(128,166,135,159)(129,167,136,160)(130,168,137,161)(131,155,138,162)(132,156,139,163)(133,157,140,164)(169,206,176,199)(170,207,177,200)(171,208,178,201)(172,209,179,202)(173,210,180,203)(174,197,181,204)(175,198,182,205)(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) );
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)], [(1,185,17,169),(2,186,18,170),(3,187,19,171),(4,188,20,172),(5,189,21,173),(6,190,22,174),(7,191,23,175),(8,192,24,176),(9,193,25,177),(10,194,26,178),(11,195,27,179),(12,196,28,180),(13,183,15,181),(14,184,16,182),(29,215,52,208),(30,216,53,209),(31,217,54,210),(32,218,55,197),(33,219,56,198),(34,220,43,199),(35,221,44,200),(36,222,45,201),(37,223,46,202),(38,224,47,203),(39,211,48,204),(40,212,49,205),(41,213,50,206),(42,214,51,207),(57,137,75,125),(58,138,76,126),(59,139,77,113),(60,140,78,114),(61,127,79,115),(62,128,80,116),(63,129,81,117),(64,130,82,118),(65,131,83,119),(66,132,84,120),(67,133,71,121),(68,134,72,122),(69,135,73,123),(70,136,74,124),(85,165,101,149),(86,166,102,150),(87,167,103,151),(88,168,104,152),(89,155,105,153),(90,156,106,154),(91,157,107,141),(92,158,108,142),(93,159,109,143),(94,160,110,144),(95,161,111,145),(96,162,112,146),(97,163,99,147),(98,164,100,148)], [(1,121,8,114),(2,122,9,115),(3,123,10,116),(4,124,11,117),(5,125,12,118),(6,126,13,119),(7,113,14,120),(15,131,22,138),(16,132,23,139),(17,133,24,140),(18,134,25,127),(19,135,26,128),(20,136,27,129),(21,137,28,130),(29,150,36,143),(30,151,37,144),(31,152,38,145),(32,153,39,146),(33,154,40,147),(34,141,41,148),(35,142,42,149),(43,157,50,164),(44,158,51,165),(45,159,52,166),(46,160,53,167),(47,161,54,168),(48,162,55,155),(49,163,56,156),(57,173,64,180),(58,174,65,181),(59,175,66,182),(60,176,67,169),(61,177,68,170),(62,178,69,171),(63,179,70,172),(71,185,78,192),(72,186,79,193),(73,187,80,194),(74,188,81,195),(75,189,82,196),(76,190,83,183),(77,191,84,184),(85,207,92,200),(86,208,93,201),(87,209,94,202),(88,210,95,203),(89,197,96,204),(90,198,97,205),(91,199,98,206),(99,212,106,219),(100,213,107,220),(101,214,108,221),(102,215,109,222),(103,216,110,223),(104,217,111,224),(105,218,112,211)], [(1,50,17,41),(2,49,18,40),(3,48,19,39),(4,47,20,38),(5,46,21,37),(6,45,22,36),(7,44,23,35),(8,43,24,34),(9,56,25,33),(10,55,26,32),(11,54,27,31),(12,53,28,30),(13,52,15,29),(14,51,16,42),(57,110,75,94),(58,109,76,93),(59,108,77,92),(60,107,78,91),(61,106,79,90),(62,105,80,89),(63,104,81,88),(64,103,82,87),(65,102,83,86),(66,101,84,85),(67,100,71,98),(68,99,72,97),(69,112,73,96),(70,111,74,95),(113,158,139,142),(114,157,140,141),(115,156,127,154),(116,155,128,153),(117,168,129,152),(118,167,130,151),(119,166,131,150),(120,165,132,149),(121,164,133,148),(122,163,134,147),(123,162,135,146),(124,161,136,145),(125,160,137,144),(126,159,138,143),(169,213,185,206),(170,212,186,205),(171,211,187,204),(172,224,188,203),(173,223,189,202),(174,222,190,201),(175,221,191,200),(176,220,192,199),(177,219,193,198),(178,218,194,197),(179,217,195,210),(180,216,196,209),(181,215,183,208),(182,214,184,207)], [(1,34,8,41),(2,35,9,42),(3,36,10,29),(4,37,11,30),(5,38,12,31),(6,39,13,32),(7,40,14,33),(15,55,22,48),(16,56,23,49),(17,43,24,50),(18,44,25,51),(19,45,26,52),(20,46,27,53),(21,47,28,54),(57,88,64,95),(58,89,65,96),(59,90,66,97),(60,91,67,98),(61,92,68,85),(62,93,69,86),(63,94,70,87),(71,100,78,107),(72,101,79,108),(73,102,80,109),(74,103,81,110),(75,104,82,111),(76,105,83,112),(77,106,84,99),(113,147,120,154),(114,148,121,141),(115,149,122,142),(116,150,123,143),(117,151,124,144),(118,152,125,145),(119,153,126,146),(127,165,134,158),(128,166,135,159),(129,167,136,160),(130,168,137,161),(131,155,138,162),(132,156,139,163),(133,157,140,164),(169,206,176,199),(170,207,177,200),(171,208,178,201),(172,209,179,202),(173,210,180,203),(174,197,181,204),(175,198,182,205),(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)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 28 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 19 | 12 | 27 |
| 0 | 0 | 0 | 0 | 4 | 25 | 28 | 17 |
| 28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 19 | 12 | 27 |
| 0 | 0 | 0 | 0 | 0 | 19 | 24 | 27 |
| 0 | 0 | 0 | 0 | 17 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 6 | 25 | 10 |
| 1 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 19 | 12 | 27 |
| 0 | 0 | 0 | 0 | 25 | 0 | 28 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 | 17 | 2 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 12 |
G:=sub<GL(8,GF(29))| [28,14,0,0,0,0,0,0,10,4,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,17,5,0,4,0,0,0,0,12,12,19,25,0,0,0,0,0,0,12,28,0,0,0,0,0,0,27,17],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,0,0,17,1,0,0,0,0,19,19,12,6,0,0,0,0,12,24,0,25,0,0,0,0,27,27,0,10],[1,0,0,0,0,0,0,0,19,28,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,24,0,25,0,0,0,0,17,17,19,0,0,0,0,0,0,0,12,28,0,0,0,0,0,0,27,17],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,24,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,17,17,10,4,0,0,0,0,0,0,17,0,0,0,0,0,0,0,2,12] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 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 | 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 | 28 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4⋊6D14 | D7×C4○D4 |
| kernel | C14.442+ (1+4) | C23.11D14 | Dic7⋊4D4 | D14.D4 | D14⋊D4 | Dic7.Q8 | D14.5D4 | C4×C7⋊D4 | C23.23D14 | C23.18D14 | C28⋊2D4 | Dic7⋊D4 | C28⋊D4 | C7×C4⋊D4 | C4⋊D4 | Dic7 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 3 | 4 | 6 | 3 | 3 | 9 | 2 | 12 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{44}2_+^{(1+4)} % in TeX
G:=Group("C14.44ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1068);
// by ID
G=gap.SmallGroup(448,1068);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,675,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=e^2=a^7,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^7*b^-1,b*d=d*b,e*b*e^-1=a^7*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations
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