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## G = C14.642+ 1+4order 448 = 26·7

### 64th non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C14.642+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — D7×C4⋊C4 — C14.642+ 1+4
 Lower central C7 — C2×C14 — C14.642+ 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C14.642+ 1+4
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=a7b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, dbd-1=ebe=a7b, cd=dc, ce=ec, ede=a7b2d >

Subgroups: 1100 in 238 conjugacy classes, 95 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C22.D4, C42.C2, C422C2, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C22.47C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23.11D14, C23.D14, Dic74D4, D14.D4, D14⋊D4, Dic7.Q8, D7×C4⋊C4, D28⋊C4, C4⋊C4⋊D7, C4×C7⋊D4, C282D4, Dic7⋊D4, C7×C22.D4, C14.642+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.47C24, C23×D7, D46D14, D7×C4○D4, C14.642+ 1+4

Smallest permutation representation of C14.642+ 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)
(1 190 21 182)(2 191 22 169)(3 192 23 170)(4 193 24 171)(5 194 25 172)(6 195 26 173)(7 196 27 174)(8 183 28 175)(9 184 15 176)(10 185 16 177)(11 186 17 178)(12 187 18 179)(13 188 19 180)(14 189 20 181)(29 214 43 206)(30 215 44 207)(31 216 45 208)(32 217 46 209)(33 218 47 210)(34 219 48 197)(35 220 49 198)(36 221 50 199)(37 222 51 200)(38 223 52 201)(39 224 53 202)(40 211 54 203)(41 212 55 204)(42 213 56 205)(57 127 79 116)(58 128 80 117)(59 129 81 118)(60 130 82 119)(61 131 83 120)(62 132 84 121)(63 133 71 122)(64 134 72 123)(65 135 73 124)(66 136 74 125)(67 137 75 126)(68 138 76 113)(69 139 77 114)(70 140 78 115)(85 155 99 144)(86 156 100 145)(87 157 101 146)(88 158 102 147)(89 159 103 148)(90 160 104 149)(91 161 105 150)(92 162 106 151)(93 163 107 152)(94 164 108 153)(95 165 109 154)(96 166 110 141)(97 167 111 142)(98 168 112 143)
(1 124 8 117)(2 125 9 118)(3 126 10 119)(4 113 11 120)(5 114 12 121)(6 115 13 122)(7 116 14 123)(15 129 22 136)(16 130 23 137)(17 131 24 138)(18 132 25 139)(19 133 26 140)(20 134 27 127)(21 135 28 128)(29 148 36 141)(30 149 37 142)(31 150 38 143)(32 151 39 144)(33 152 40 145)(34 153 41 146)(35 154 42 147)(43 159 50 166)(44 160 51 167)(45 161 52 168)(46 162 53 155)(47 163 54 156)(48 164 55 157)(49 165 56 158)(57 181 64 174)(58 182 65 175)(59 169 66 176)(60 170 67 177)(61 171 68 178)(62 172 69 179)(63 173 70 180)(71 195 78 188)(72 196 79 189)(73 183 80 190)(74 184 81 191)(75 185 82 192)(76 186 83 193)(77 187 84 194)(85 209 92 202)(86 210 93 203)(87 197 94 204)(88 198 95 205)(89 199 96 206)(90 200 97 207)(91 201 98 208)(99 217 106 224)(100 218 107 211)(101 219 108 212)(102 220 109 213)(103 221 110 214)(104 222 111 215)(105 223 112 216)
(1 47 28 40)(2 46 15 39)(3 45 16 38)(4 44 17 37)(5 43 18 36)(6 56 19 35)(7 55 20 34)(8 54 21 33)(9 53 22 32)(10 52 23 31)(11 51 24 30)(12 50 25 29)(13 49 26 42)(14 48 27 41)(57 108 72 87)(58 107 73 86)(59 106 74 85)(60 105 75 98)(61 104 76 97)(62 103 77 96)(63 102 78 95)(64 101 79 94)(65 100 80 93)(66 99 81 92)(67 112 82 91)(68 111 83 90)(69 110 84 89)(70 109 71 88)(113 160 131 142)(114 159 132 141)(115 158 133 154)(116 157 134 153)(117 156 135 152)(118 155 136 151)(119 168 137 150)(120 167 138 149)(121 166 139 148)(122 165 140 147)(123 164 127 146)(124 163 128 145)(125 162 129 144)(126 161 130 143)(169 224 184 209)(170 223 185 208)(171 222 186 207)(172 221 187 206)(173 220 188 205)(174 219 189 204)(175 218 190 203)(176 217 191 202)(177 216 192 201)(178 215 193 200)(179 214 194 199)(180 213 195 198)(181 212 196 197)(182 211 183 210)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 29)(12 30)(13 31)(14 32)(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 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 112)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)(82 109)(83 110)(84 111)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(169 204)(170 205)(171 206)(172 207)(173 208)(174 209)(175 210)(176 197)(177 198)(178 199)(179 200)(180 201)(181 202)(182 203)(183 218)(184 219)(185 220)(186 221)(187 222)(188 223)(189 224)(190 211)(191 212)(192 213)(193 214)(194 215)(195 216)(196 217)```

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

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

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

67 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H ··· 4M 4N 4O 4P 7A 7B 7C 14A ··· 14I 14J ··· 14O 14P 14Q 14R 28A ··· 28L 28M ··· 28U order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 ··· 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 14 14 14 28 ··· 28 28 ··· 28 size 1 1 1 1 4 4 14 14 28 2 2 2 2 4 4 4 14 ··· 14 28 28 28 2 2 2 2 ··· 2 4 ··· 4 8 8 8 4 ··· 4 8 ··· 8

67 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 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 C4○D4 D14 D14 D14 D14 2+ 1+4 D4⋊6D14 D7×C4○D4 kernel C14.642+ 1+4 C23.11D14 C23.D14 Dic7⋊4D4 D14.D4 D14⋊D4 Dic7.Q8 D7×C4⋊C4 D28⋊C4 C4⋊C4⋊D7 C4×C7⋊D4 C28⋊2D4 Dic7⋊D4 C7×C22.D4 C22.D4 Dic7 D14 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C14 C2 C2 # reps 1 1 1 1 1 2 1 1 1 1 2 1 1 1 3 4 4 9 6 3 3 1 6 12

Matrix representation of C14.642+ 1+4 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 25 25 0 0 0 0 4 11 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 1 28 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 15 4 0 0 0 0 16 14
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 19 3 0 0 0 0 5 10
,
 28 2 0 0 0 0 28 1 0 0 0 0 0 0 28 0 0 0 0 0 11 1 0 0 0 0 0 0 25 7 0 0 0 0 2 4
,
 1 27 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 4 22 0 0 0 0 27 25

`G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,25,4,0,0,0,0,25,11,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,1,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,16,0,0,0,0,4,14],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,5,0,0,0,0,3,10],[28,28,0,0,0,0,2,1,0,0,0,0,0,0,28,11,0,0,0,0,0,1,0,0,0,0,0,0,25,2,0,0,0,0,7,4],[1,0,0,0,0,0,27,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,4,27,0,0,0,0,22,25] >;`

C14.642+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{64}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.64ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1114);`
`// by ID`

`G=gap.SmallGroup(448,1114);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,219,184,1571,297,18822]);`
`// 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,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,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`

׿
×
𝔽