Copied to
clipboard

## G = C14.2+ 1+4order 448 = 26·7

### 9th 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 — C14.2+ 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — C2×C4○D28 — C14.2+ 1+4
 Lower central C7 — C2×C14 — C14.2+ 1+4
 Upper central C1 — C22 — C2×C4⋊C4

Generators and relations for C14.2+ 1+4
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=b2, bab-1=dad-1=eae=a-1, ac=ca, cbc-1=a7b-1, dbd-1=a7b, be=eb, dcd-1=ece=a7c, ede=a7b2d >

Subgroups: 1540 in 294 conjugacy classes, 111 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C22.31C24, C4⋊Dic7, D14⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, C22×C28, C4⋊D28, D142Q8, C287D4, C14×C4⋊C4, C2×C4○D28, C14.2+ 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, 2- 1+4, D28, C22×D7, C22.31C24, C2×D28, C23×D7, C22×D28, D46D14, Q8.10D14, C14.2+ 1+4

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

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 7A 7B 7C 14A ··· 14U 28A ··· 28AJ order 1 2 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 28 ··· 28 size 1 1 1 1 2 2 28 28 28 28 2 2 2 2 4 4 4 4 28 28 28 28 2 2 2 2 ··· 2 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D7 D14 D14 D28 2+ 1+4 2- 1+4 D4⋊6D14 Q8.10D14 kernel C14.2+ 1+4 C4⋊D28 D14⋊2Q8 C28⋊7D4 C14×C4⋊C4 C2×C4○D28 C2×C28 C2×C4⋊C4 C4⋊C4 C22×C4 C2×C4 C14 C14 C2 C2 # reps 1 4 4 4 1 2 4 3 12 9 24 1 1 6 6

Matrix representation of C14.2+ 1+4 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 25 0 0 0 0 0 0 25
,
 18 25 0 0 0 0 1 11 0 0 0 0 0 0 0 0 0 17 0 0 0 0 12 0 0 0 0 12 0 0 0 0 17 0 0 0
,
 22 14 0 0 0 0 13 7 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 25 21 0 0 0 0 20 4 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 1 0 0 0 0 0 0 1 0 0
,
 18 25 0 0 0 0 1 11 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0

`G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,25,0,0,0,0,0,0,25],[18,1,0,0,0,0,25,11,0,0,0,0,0,0,0,0,0,17,0,0,0,0,12,0,0,0,0,12,0,0,0,0,17,0,0,0],[22,13,0,0,0,0,14,7,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[25,20,0,0,0,0,21,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,28,0,0,0,0,0,0,28,0,0],[18,1,0,0,0,0,25,11,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;`

C14.2+ 1+4 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(448,963);`
`// by ID`

`G=gap.SmallGroup(448,963);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,675,570,80,18822]);`
`// Polycyclic`

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