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## G = C14.802- 1+4order 448 = 26·7

### 35th non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C14.802- 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — C2×Dic7⋊C4 — C14.802- 1+4
 Lower central C7 — C2×C14 — C14.802- 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C14.802- 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=e2=a7b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=a7b-1, bd=db, be=eb, cd=dc, ece-1=a7c, ede-1=b2d >

Subgroups: 780 in 214 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic7, Dic7, C28, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C22.D4, C42.C2, C422C2, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C22.46C24, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C22×Dic7, C22×C28, D4×C14, C23.11D14, C22⋊Dic14, C23.D14, Dic73Q8, Dic7.Q8, C28.3Q8, C2×Dic7⋊C4, C23.21D14, D4×Dic7, C23.18D14, C7×C22.D4, C14.802- 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2- 1+4, C22×D7, C22.46C24, D42D7, C23×D7, C2×D42D7, D7×C4○D4, D4.10D14, C14.802- 1+4

Smallest permutation representation of C14.802- 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 33 62 189)(2 32 63 188)(3 31 64 187)(4 30 65 186)(5 29 66 185)(6 42 67 184)(7 41 68 183)(8 40 69 196)(9 39 70 195)(10 38 57 194)(11 37 58 193)(12 36 59 192)(13 35 60 191)(14 34 61 190)(15 161 52 202)(16 160 53 201)(17 159 54 200)(18 158 55 199)(19 157 56 198)(20 156 43 197)(21 155 44 210)(22 168 45 209)(23 167 46 208)(24 166 47 207)(25 165 48 206)(26 164 49 205)(27 163 50 204)(28 162 51 203)(71 175 122 218)(72 174 123 217)(73 173 124 216)(74 172 125 215)(75 171 126 214)(76 170 113 213)(77 169 114 212)(78 182 115 211)(79 181 116 224)(80 180 117 223)(81 179 118 222)(82 178 119 221)(83 177 120 220)(84 176 121 219)(85 110 153 129)(86 109 154 128)(87 108 141 127)(88 107 142 140)(89 106 143 139)(90 105 144 138)(91 104 145 137)(92 103 146 136)(93 102 147 135)(94 101 148 134)(95 100 149 133)(96 99 150 132)(97 112 151 131)(98 111 152 130)
(1 118)(2 119)(3 120)(4 121)(5 122)(6 123)(7 124)(8 125)(9 126)(10 113)(11 114)(12 115)(13 116)(14 117)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 110)(22 111)(23 112)(24 99)(25 100)(26 101)(27 102)(28 103)(29 182)(30 169)(31 170)(32 171)(33 172)(34 173)(35 174)(36 175)(37 176)(38 177)(39 178)(40 179)(41 180)(42 181)(43 128)(44 129)(45 130)(46 131)(47 132)(48 133)(49 134)(50 135)(51 136)(52 137)(53 138)(54 139)(55 140)(56 127)(57 76)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 71)(67 72)(68 73)(69 74)(70 75)(85 162)(86 163)(87 164)(88 165)(89 166)(90 167)(91 168)(92 155)(93 156)(94 157)(95 158)(96 159)(97 160)(98 161)(141 205)(142 206)(143 207)(144 208)(145 209)(146 210)(147 197)(148 198)(149 199)(150 200)(151 201)(152 202)(153 203)(154 204)(183 223)(184 224)(185 211)(186 212)(187 213)(188 214)(189 215)(190 216)(191 217)(192 218)(193 219)(194 220)(195 221)(196 222)
(1 147 69 86)(2 146 70 85)(3 145 57 98)(4 144 58 97)(5 143 59 96)(6 142 60 95)(7 141 61 94)(8 154 62 93)(9 153 63 92)(10 152 64 91)(11 151 65 90)(12 150 66 89)(13 149 67 88)(14 148 68 87)(15 177 45 213)(16 176 46 212)(17 175 47 211)(18 174 48 224)(19 173 49 223)(20 172 50 222)(21 171 51 221)(22 170 52 220)(23 169 53 219)(24 182 54 218)(25 181 55 217)(26 180 56 216)(27 179 43 215)(28 178 44 214)(29 139 192 99)(30 138 193 112)(31 137 194 111)(32 136 195 110)(33 135 196 109)(34 134 183 108)(35 133 184 107)(36 132 185 106)(37 131 186 105)(38 130 187 104)(39 129 188 103)(40 128 189 102)(41 127 190 101)(42 140 191 100)(71 166 115 200)(72 165 116 199)(73 164 117 198)(74 163 118 197)(75 162 119 210)(76 161 120 209)(77 160 121 208)(78 159 122 207)(79 158 123 206)(80 157 124 205)(81 156 125 204)(82 155 126 203)(83 168 113 202)(84 167 114 201)
(1 109 69 135)(2 110 70 136)(3 111 57 137)(4 112 58 138)(5 99 59 139)(6 100 60 140)(7 101 61 127)(8 102 62 128)(9 103 63 129)(10 104 64 130)(11 105 65 131)(12 106 66 132)(13 107 67 133)(14 108 68 134)(15 76 45 120)(16 77 46 121)(17 78 47 122)(18 79 48 123)(19 80 49 124)(20 81 50 125)(21 82 51 126)(22 83 52 113)(23 84 53 114)(24 71 54 115)(25 72 55 116)(26 73 56 117)(27 74 43 118)(28 75 44 119)(29 150 192 89)(30 151 193 90)(31 152 194 91)(32 153 195 92)(33 154 196 93)(34 141 183 94)(35 142 184 95)(36 143 185 96)(37 144 186 97)(38 145 187 98)(39 146 188 85)(40 147 189 86)(41 148 190 87)(42 149 191 88)(155 178 203 214)(156 179 204 215)(157 180 205 216)(158 181 206 217)(159 182 207 218)(160 169 208 219)(161 170 209 220)(162 171 210 221)(163 172 197 222)(164 173 198 223)(165 174 199 224)(166 175 200 211)(167 176 201 212)(168 177 202 213)```

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

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

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

67 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 4P 4Q 4R 7A 7B 7C 14A ··· 14I 14J ··· 14O 14P 14Q 14R 28A ··· 28L 28M ··· 28U order 1 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 2 2 4 2 2 4 4 4 4 14 ··· 14 28 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 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 D7×C4○D4 D4.10D14 kernel C14.802- 1+4 C23.11D14 C22⋊Dic14 C23.D14 Dic7⋊3Q8 Dic7.Q8 C28.3Q8 C2×Dic7⋊C4 C23.21D14 D4×Dic7 C23.18D14 C7×C22.D4 C22.D4 Dic7 C2×C14 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C14 C22 C2 C2 # reps 1 2 2 2 1 2 1 1 1 1 1 1 3 4 4 9 6 3 3 1 6 6 6

Matrix representation of C14.802- 1+4 in GL6(𝔽29)

 28 0 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 0 25 0 0 0 0 22 7
,
 28 4 0 0 0 0 0 1 0 0 0 0 0 0 17 0 0 0 0 0 0 17 0 0 0 0 0 0 4 3 0 0 0 0 24 25
,
 28 0 0 0 0 0 14 1 0 0 0 0 0 0 28 0 0 0 0 0 27 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 28 0 0 0 0 0 27 1 0 0 0 0 0 0 4 3 0 0 0 0 24 25
,
 12 10 0 0 0 0 0 17 0 0 0 0 0 0 12 17 0 0 0 0 24 17 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,22,0,0,0,0,25,7],[28,0,0,0,0,0,4,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,4,24,0,0,0,0,3,25],[28,14,0,0,0,0,0,1,0,0,0,0,0,0,28,27,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,28,27,0,0,0,0,0,1,0,0,0,0,0,0,4,24,0,0,0,0,3,25],[12,0,0,0,0,0,10,17,0,0,0,0,0,0,12,24,0,0,0,0,17,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C14.802- 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{80}2_-^{1+4}`
`% in TeX`

`G:=Group("C14.80ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1103);`
`// by ID`

`G=gap.SmallGroup(448,1103);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,100,346,297,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=e^2=a^7*b^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,a*e=e*a,c*b*c=a^7*b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^7*c,e*d*e^-1=b^2*d>;`
`// generators/relations`

׿
×
𝔽