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G = C14.452+ 1+4order 448 = 26·7

45th 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 — C14.452+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C7⋊D4 — C28⋊2D4 — C14.452+ 1+4
 Lower central C7 — C2×C14 — C14.452+ 1+4
 Upper central C1 — C22 — C4⋊D4

Generators and relations for C14.452+ 1+4
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 >

Subgroups: 1004 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, 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, C2×Q8, Dic7, Dic7, C28, C28, D14, C2×C14, C2×C14, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊Q8, Dic14, C4×D7, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C22×C14, C22.49C24, C4×Dic7, C4×Dic7, Dic7⋊C4, Dic7⋊C4, C4⋊Dic7, D14⋊C4, D14⋊C4, C23.D7, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C23.11D14, Dic7.D4, C28⋊Q8, C4⋊C47D7, C23.21D14, C4×C7⋊D4, D4×Dic7, C28.17D4, C282D4, Dic7⋊D4, C7×C4⋊D4, C14.452+ 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2+ 1+4, C22×D7, C22.49C24, D42D7, C23×D7, C2×D42D7, D46D14, D7×C4○D4, C14.452+ 1+4

Smallest permutation representation of C14.452+ 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)
(1 191 19 176)(2 192 20 177)(3 193 21 178)(4 194 22 179)(5 195 23 180)(6 196 24 181)(7 183 25 182)(8 184 26 169)(9 185 27 170)(10 186 28 171)(11 187 15 172)(12 188 16 173)(13 189 17 174)(14 190 18 175)(29 224 47 208)(30 211 48 209)(31 212 49 210)(32 213 50 197)(33 214 51 198)(34 215 52 199)(35 216 53 200)(36 217 54 201)(37 218 55 202)(38 219 56 203)(39 220 43 204)(40 221 44 205)(41 222 45 206)(42 223 46 207)(57 139 76 119)(58 140 77 120)(59 127 78 121)(60 128 79 122)(61 129 80 123)(62 130 81 124)(63 131 82 125)(64 132 83 126)(65 133 84 113)(66 134 71 114)(67 135 72 115)(68 136 73 116)(69 137 74 117)(70 138 75 118)(85 160 108 146)(86 161 109 147)(87 162 110 148)(88 163 111 149)(89 164 112 150)(90 165 99 151)(91 166 100 152)(92 167 101 153)(93 168 102 154)(94 155 103 141)(95 156 104 142)(96 157 105 143)(97 158 106 144)(98 159 107 145)
(1 116)(2 117)(3 118)(4 119)(5 120)(6 121)(7 122)(8 123)(9 124)(10 125)(11 126)(12 113)(13 114)(14 115)(15 132)(16 133)(17 134)(18 135)(19 136)(20 137)(21 138)(22 139)(23 140)(24 127)(25 128)(26 129)(27 130)(28 131)(29 153)(30 154)(31 141)(32 142)(33 143)(34 144)(35 145)(36 146)(37 147)(38 148)(39 149)(40 150)(41 151)(42 152)(43 163)(44 164)(45 165)(46 166)(47 167)(48 168)(49 155)(50 156)(51 157)(52 158)(53 159)(54 160)(55 161)(56 162)(57 179)(58 180)(59 181)(60 182)(61 169)(62 170)(63 171)(64 172)(65 173)(66 174)(67 175)(68 176)(69 177)(70 178)(71 189)(72 190)(73 191)(74 192)(75 193)(76 194)(77 195)(78 196)(79 183)(80 184)(81 185)(82 186)(83 187)(84 188)(85 201)(86 202)(87 203)(88 204)(89 205)(90 206)(91 207)(92 208)(93 209)(94 210)(95 197)(96 198)(97 199)(98 200)(99 222)(100 223)(101 224)(102 211)(103 212)(104 213)(105 214)(106 215)(107 216)(108 217)(109 218)(110 219)(111 220)(112 221)
(1 54 26 29)(2 53 27 42)(3 52 28 41)(4 51 15 40)(5 50 16 39)(6 49 17 38)(7 48 18 37)(8 47 19 36)(9 46 20 35)(10 45 21 34)(11 44 22 33)(12 43 23 32)(13 56 24 31)(14 55 25 30)(57 105 83 89)(58 104 84 88)(59 103 71 87)(60 102 72 86)(61 101 73 85)(62 100 74 98)(63 99 75 97)(64 112 76 96)(65 111 77 95)(66 110 78 94)(67 109 79 93)(68 108 80 92)(69 107 81 91)(70 106 82 90)(113 163 140 142)(114 162 127 141)(115 161 128 154)(116 160 129 153)(117 159 130 152)(118 158 131 151)(119 157 132 150)(120 156 133 149)(121 155 134 148)(122 168 135 147)(123 167 136 146)(124 166 137 145)(125 165 138 144)(126 164 139 143)(169 224 191 201)(170 223 192 200)(171 222 193 199)(172 221 194 198)(173 220 195 197)(174 219 196 210)(175 218 183 209)(176 217 184 208)(177 216 185 207)(178 215 186 206)(179 214 187 205)(180 213 188 204)(181 212 189 203)(182 211 190 202)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 88)(58 89)(59 90)(60 91)(61 92)(62 93)(63 94)(64 95)(65 96)(66 97)(67 98)(68 85)(69 86)(70 87)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(113 143)(114 144)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)(121 151)(122 152)(123 153)(124 154)(125 141)(126 142)(127 165)(128 166)(129 167)(130 168)(131 155)(132 156)(133 157)(134 158)(135 159)(136 160)(137 161)(138 162)(139 163)(140 164)(169 201)(170 202)(171 203)(172 204)(173 205)(174 206)(175 207)(176 208)(177 209)(178 210)(179 197)(180 198)(181 199)(182 200)(183 216)(184 217)(185 218)(186 219)(187 220)(188 221)(189 222)(190 223)(191 224)(192 211)(193 212)(194 213)(195 214)(196 215)```

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

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

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

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

Matrix representation of C14.452+ 1+4 in 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] >;`

C14.452+ 1+4 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`

׿
×
𝔽