Copied to
clipboard

## G = C14.472+ 1+4order 448 = 26·7

### 47th 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 — C14.472+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — D14.D4 — C14.472+ 1+4
 Lower central C7 — C2×C14 — C14.472+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C14.472+ 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=a7b-1, dbd-1=a7b, be=eb, dcd-1=ece=a7c, ede=a7b2d >

Subgroups: 1164 in 240 conjugacy classes, 95 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C22.D4, C42.C2, C41D4, C4×D7, D28, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C22×C14, C22.34C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, D14⋊C4, C23.D7, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C2×D28, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, D14.D4, C22.D28, C28.3Q8, D28⋊C4, C23.21D14, C287D4, D4×Dic7, C282D4, Dic7⋊D4, C28⋊D4, C7×C4⋊D4, C14.472+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.34C24, D42D7, C23×D7, C2×D42D7, D46D14, D48D14, C14.472+ 1+4

Smallest permutation representation of C14.472+ 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 23 181)(2 191 24 182)(3 192 25 169)(4 193 26 170)(5 194 27 171)(6 195 28 172)(7 196 15 173)(8 183 16 174)(9 184 17 175)(10 185 18 176)(11 186 19 177)(12 187 20 178)(13 188 21 179)(14 189 22 180)(29 221 48 205)(30 222 49 206)(31 223 50 207)(32 224 51 208)(33 211 52 209)(34 212 53 210)(35 213 54 197)(36 214 55 198)(37 215 56 199)(38 216 43 200)(39 217 44 201)(40 218 45 202)(41 219 46 203)(42 220 47 204)(57 135 81 122)(58 136 82 123)(59 137 83 124)(60 138 84 125)(61 139 71 126)(62 140 72 113)(63 127 73 114)(64 128 74 115)(65 129 75 116)(66 130 76 117)(67 131 77 118)(68 132 78 119)(69 133 79 120)(70 134 80 121)(85 164 103 149)(86 165 104 150)(87 166 105 151)(88 167 106 152)(89 168 107 153)(90 155 108 154)(91 156 109 141)(92 157 110 142)(93 158 111 143)(94 159 112 144)(95 160 99 145)(96 161 100 146)(97 162 101 147)(98 163 102 148)
(1 118 8 125)(2 119 9 126)(3 120 10 113)(4 121 11 114)(5 122 12 115)(6 123 13 116)(7 124 14 117)(15 137 22 130)(16 138 23 131)(17 139 24 132)(18 140 25 133)(19 127 26 134)(20 128 27 135)(21 129 28 136)(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)(43 158 50 165)(44 159 51 166)(45 160 52 167)(46 161 53 168)(47 162 54 155)(48 163 55 156)(49 164 56 157)(57 171 64 178)(58 172 65 179)(59 173 66 180)(60 174 67 181)(61 175 68 182)(62 176 69 169)(63 177 70 170)(71 184 78 191)(72 185 79 192)(73 186 80 193)(74 187 81 194)(75 188 82 195)(76 189 83 196)(77 190 84 183)(85 206 92 199)(86 207 93 200)(87 208 94 201)(88 209 95 202)(89 210 96 203)(90 197 97 204)(91 198 98 205)(99 218 106 211)(100 219 107 212)(101 220 108 213)(102 221 109 214)(103 222 110 215)(104 223 111 216)(105 224 112 217)
(1 47 16 35)(2 46 17 34)(3 45 18 33)(4 44 19 32)(5 43 20 31)(6 56 21 30)(7 55 22 29)(8 54 23 42)(9 53 24 41)(10 52 25 40)(11 51 26 39)(12 50 27 38)(13 49 28 37)(14 48 15 36)(57 111 74 86)(58 110 75 85)(59 109 76 98)(60 108 77 97)(61 107 78 96)(62 106 79 95)(63 105 80 94)(64 104 81 93)(65 103 82 92)(66 102 83 91)(67 101 84 90)(68 100 71 89)(69 99 72 88)(70 112 73 87)(113 160 133 152)(114 159 134 151)(115 158 135 150)(116 157 136 149)(117 156 137 148)(118 155 138 147)(119 168 139 146)(120 167 140 145)(121 166 127 144)(122 165 128 143)(123 164 129 142)(124 163 130 141)(125 162 131 154)(126 161 132 153)(169 211 185 202)(170 224 186 201)(171 223 187 200)(172 222 188 199)(173 221 189 198)(174 220 190 197)(175 219 191 210)(176 218 192 209)(177 217 193 208)(178 216 194 207)(179 215 195 206)(180 214 196 205)(181 213 183 204)(182 212 184 203)
(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 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 43)(27 44)(28 45)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 85)(70 86)(71 109)(72 110)(73 111)(74 112)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 105)(82 106)(83 107)(84 108)(113 142)(114 143)(115 144)(116 145)(117 146)(118 147)(119 148)(120 149)(121 150)(122 151)(123 152)(124 153)(125 154)(126 141)(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)(169 199)(170 200)(171 201)(172 202)(173 203)(174 204)(175 205)(176 206)(177 207)(178 208)(179 209)(180 210)(181 197)(182 198)(183 220)(184 221)(185 222)(186 223)(187 224)(188 211)(189 212)(190 213)(191 214)(192 215)(193 216)(194 217)(195 218)(196 219)```

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

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

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

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 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 D14 D14 D14 D14 2+ 1+4 D4⋊2D7 D4⋊6D14 D4⋊8D14 kernel C14.472+ 1+4 D14.D4 C22.D28 C28.3Q8 D28⋊C4 C23.21D14 C28⋊7D4 D4×Dic7 C28⋊2D4 Dic7⋊D4 C28⋊D4 C7×C4⋊D4 C4⋊D4 C28 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C14 C4 C2 C2 # reps 1 2 2 1 1 1 1 1 2 2 1 1 3 4 6 3 3 9 2 6 6 6

Matrix representation of C14.472+ 1+4 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 8 8 0 0 0 0 21 3 0 0 0 0 0 0 8 8 0 0 0 0 21 3
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 2 18 0 0 0 0 11 27 0 0 0 0 0 0 2 18 0 0 0 0 11 27
,
 12 0 0 0 0 0 0 17 0 0 0 0 0 0 14 0 17 0 0 0 0 14 0 17 0 0 14 0 15 0 0 0 0 14 0 15
,
 0 28 0 0 0 0 1 0 0 0 0 0 0 0 1 0 27 0 0 0 3 28 23 2 0 0 0 0 28 0 0 0 0 0 26 1
,
 0 28 0 0 0 0 28 0 0 0 0 0 0 0 1 0 27 0 0 0 0 1 0 27 0 0 0 0 28 0 0 0 0 0 0 28

`G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,8,21,0,0,0,0,8,3,0,0,0,0,0,0,8,21,0,0,0,0,8,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,11,0,0,0,0,18,27,0,0,0,0,0,0,2,11,0,0,0,0,18,27],[12,0,0,0,0,0,0,17,0,0,0,0,0,0,14,0,14,0,0,0,0,14,0,14,0,0,17,0,15,0,0,0,0,17,0,15],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,1,3,0,0,0,0,0,28,0,0,0,0,27,23,28,26,0,0,0,2,0,1],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,27,0,28,0,0,0,0,27,0,28] >;`

C14.472+ 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{47}2_+^{1+4}`
`% in TeX`

`G:=Group("C14.47ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1072);`
`// by ID`

`G=gap.SmallGroup(448,1072);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,675,570,297,18822]);`
`// 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=a^7*b^-1,d*b*d^-1=a^7*b,b*e=e*b,d*c*d^-1=e*c*e=a^7*c,e*d*e=a^7*b^2*d>;`
`// generators/relations`

׿
×
𝔽