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## G = C14.1042- 1+4order 448 = 26·7

### 59th 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 — C14.1042- 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C22×Dic7 — C2×Dic7⋊C4 — C14.1042- 1+4
 Lower central C7 — C2×C14 — C14.1042- 1+4
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C14.1042- 1+4
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=e2=b2, bab-1=cac-1=eae-1=a-1, ad=da, cbc-1=a7b-1, bd=db, be=eb, cd=dc, ece-1=a7c, ede-1=b2d >

Subgroups: 1140 in 292 conjugacy classes, 113 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, C2×C4○D4, Dic14, C4×D7, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×C14, C22×C14, D46D4, C4×Dic7, Dic7⋊C4, Dic7⋊C4, C4⋊Dic7, D14⋊C4, D14⋊C4, C23.D7, C23.D7, C2×Dic14, C2×C4×D7, D42D7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C2×Dic7⋊C4, C28.48D4, C4×C7⋊D4, C23.23D14, D4×Dic7, C23.18D14, Dic7⋊D4, Dic7⋊Q8, D143Q8, C2×D42D7, C14×C4○D4, C14.1042- 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, 2- 1+4, C7⋊D4, C22×D7, D46D4, C2×C7⋊D4, C23×D7, D7×C4○D4, D4.10D14, C22×C7⋊D4, C14.1042- 1+4

Smallest permutation representation of C14.1042- 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)
(1 127 224 110)(2 140 211 109)(3 139 212 108)(4 138 213 107)(5 137 214 106)(6 136 215 105)(7 135 216 104)(8 134 217 103)(9 133 218 102)(10 132 219 101)(11 131 220 100)(12 130 221 99)(13 129 222 112)(14 128 223 111)(15 80 52 32)(16 79 53 31)(17 78 54 30)(18 77 55 29)(19 76 56 42)(20 75 43 41)(21 74 44 40)(22 73 45 39)(23 72 46 38)(24 71 47 37)(25 84 48 36)(26 83 49 35)(27 82 50 34)(28 81 51 33)(57 181 126 90)(58 180 113 89)(59 179 114 88)(60 178 115 87)(61 177 116 86)(62 176 117 85)(63 175 118 98)(64 174 119 97)(65 173 120 96)(66 172 121 95)(67 171 122 94)(68 170 123 93)(69 169 124 92)(70 182 125 91)(141 193 163 203)(142 192 164 202)(143 191 165 201)(144 190 166 200)(145 189 167 199)(146 188 168 198)(147 187 155 197)(148 186 156 210)(149 185 157 209)(150 184 158 208)(151 183 159 207)(152 196 160 206)(153 195 161 205)(154 194 162 204)
(1 69 8 62)(2 68 9 61)(3 67 10 60)(4 66 11 59)(5 65 12 58)(6 64 13 57)(7 63 14 70)(15 183 22 190)(16 196 23 189)(17 195 24 188)(18 194 25 187)(19 193 26 186)(20 192 27 185)(21 191 28 184)(29 155 36 162)(30 168 37 161)(31 167 38 160)(32 166 39 159)(33 165 40 158)(34 164 41 157)(35 163 42 156)(43 202 50 209)(44 201 51 208)(45 200 52 207)(46 199 53 206)(47 198 54 205)(48 197 55 204)(49 210 56 203)(71 153 78 146)(72 152 79 145)(73 151 80 144)(74 150 81 143)(75 149 82 142)(76 148 83 141)(77 147 84 154)(85 134 92 127)(86 133 93 140)(87 132 94 139)(88 131 95 138)(89 130 96 137)(90 129 97 136)(91 128 98 135)(99 173 106 180)(100 172 107 179)(101 171 108 178)(102 170 109 177)(103 169 110 176)(104 182 111 175)(105 181 112 174)(113 214 120 221)(114 213 121 220)(115 212 122 219)(116 211 123 218)(117 224 124 217)(118 223 125 216)(119 222 126 215)
(1 197 224 187)(2 198 211 188)(3 199 212 189)(4 200 213 190)(5 201 214 191)(6 202 215 192)(7 203 216 193)(8 204 217 194)(9 205 218 195)(10 206 219 196)(11 207 220 183)(12 208 221 184)(13 209 222 185)(14 210 223 186)(15 66 52 121)(16 67 53 122)(17 68 54 123)(18 69 55 124)(19 70 56 125)(20 57 43 126)(21 58 44 113)(22 59 45 114)(23 60 46 115)(24 61 47 116)(25 62 48 117)(26 63 49 118)(27 64 50 119)(28 65 51 120)(29 92 77 169)(30 93 78 170)(31 94 79 171)(32 95 80 172)(33 96 81 173)(34 97 82 174)(35 98 83 175)(36 85 84 176)(37 86 71 177)(38 87 72 178)(39 88 73 179)(40 89 74 180)(41 90 75 181)(42 91 76 182)(99 158 130 150)(100 159 131 151)(101 160 132 152)(102 161 133 153)(103 162 134 154)(104 163 135 141)(105 164 136 142)(106 165 137 143)(107 166 138 144)(108 167 139 145)(109 168 140 146)(110 155 127 147)(111 156 128 148)(112 157 129 149)
(1 147 224 155)(2 146 211 168)(3 145 212 167)(4 144 213 166)(5 143 214 165)(6 142 215 164)(7 141 216 163)(8 154 217 162)(9 153 218 161)(10 152 219 160)(11 151 220 159)(12 150 221 158)(13 149 222 157)(14 148 223 156)(15 172 52 95)(16 171 53 94)(17 170 54 93)(18 169 55 92)(19 182 56 91)(20 181 43 90)(21 180 44 89)(22 179 45 88)(23 178 46 87)(24 177 47 86)(25 176 48 85)(26 175 49 98)(27 174 50 97)(28 173 51 96)(29 69 77 124)(30 68 78 123)(31 67 79 122)(32 66 80 121)(33 65 81 120)(34 64 82 119)(35 63 83 118)(36 62 84 117)(37 61 71 116)(38 60 72 115)(39 59 73 114)(40 58 74 113)(41 57 75 126)(42 70 76 125)(99 208 130 184)(100 207 131 183)(101 206 132 196)(102 205 133 195)(103 204 134 194)(104 203 135 193)(105 202 136 192)(106 201 137 191)(107 200 138 190)(108 199 139 189)(109 198 140 188)(110 197 127 187)(111 210 128 186)(112 209 129 185)```

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

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

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

85 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K ··· 4O 7A 7B 7C 14A ··· 14I 14J ··· 14AA 28A ··· 28L 28M ··· 28AD 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 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 2 2 4 28 2 2 2 2 4 4 14 14 14 14 28 ··· 28 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

85 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 type + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D7 C4○D4 D14 D14 D14 C7⋊D4 2- 1+4 D7×C4○D4 D4.10D14 kernel C14.1042- 1+4 C2×Dic7⋊C4 C28.48D4 C4×C7⋊D4 C23.23D14 D4×Dic7 C23.18D14 Dic7⋊D4 Dic7⋊Q8 D14⋊3Q8 C2×D4⋊2D7 C14×C4○D4 C7×D4 C2×C4○D4 Dic7 C22×C4 C2×D4 C2×Q8 D4 C14 C2 C2 # reps 1 2 1 1 2 1 2 2 1 1 1 1 4 3 4 9 9 3 24 1 6 6

Matrix representation of C14.1042- 1+4 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 0 21 0 0 0 0 11 18 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 21 11 0 0 0 0 18 8 0 0 0 0 0 0 7 17 0 0 0 0 4 22 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 11 21 0 0 0 0 8 18 0 0 0 0 0 0 7 17 0 0 0 0 4 22 0 0 0 0 0 0 1 27 0 0 0 0 1 28
,
 28 0 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 27 0 0 0 0 1 28
,
 8 18 0 0 0 0 11 21 0 0 0 0 0 0 7 17 0 0 0 0 4 22 0 0 0 0 0 0 17 24 0 0 0 0 0 12

`G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,11,0,0,0,0,21,18,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[21,18,0,0,0,0,11,8,0,0,0,0,0,0,7,4,0,0,0,0,17,22,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[11,8,0,0,0,0,21,18,0,0,0,0,0,0,7,4,0,0,0,0,17,22,0,0,0,0,0,0,1,1,0,0,0,0,27,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,27,28],[8,11,0,0,0,0,18,21,0,0,0,0,0,0,7,4,0,0,0,0,17,22,0,0,0,0,0,0,17,0,0,0,0,0,24,12] >;`

C14.1042- 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{104}2_-^{1+4}`
`% in TeX`

`G:=Group("C14.104ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1277);`
`// by ID`

`G=gap.SmallGroup(448,1277);`
`# by ID`

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

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