Copied to
clipboard

## G = C14.172- 1+4order 448 = 26·7

### 17th non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C14.172- 1+4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — D7×C4⋊C4 — C14.172- 1+4
 Lower central C7 — C2×C14 — C14.172- 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C14.172- 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=a7b2, e2=b2, ab=ba, cac=dad-1=a-1, ae=ea, cbc=b-1, dbd-1=a7b, be=eb, dcd-1=a7c, ce=ec, ede-1=b2d >

Subgroups: 1564 in 294 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C22⋊Q8, C2×C4○D4, Dic14, C4×D7, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×Q8, C22×D7, C22×D7, C22×C14, C22.31C24, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, C4○D28, Q82D7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, Q8×C14, D14⋊D4, D7×C4⋊C4, C4⋊D28, C4⋊D28, D14⋊Q8, C287D4, D143Q8, C7×C22⋊Q8, C2×C4○D28, C2×Q82D7, C14.172- 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2+ 1+4, 2- 1+4, C22×D7, C22.31C24, D4×D7, C23×D7, C2×D4×D7, Q8.10D14, D48D14, C14.172- 1+4

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

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C ··· 4G 4H 4I 4J 4K 4L 7A 7B 7C 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28X order 1 2 2 2 2 2 2 2 2 2 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 4 14 14 28 28 28 2 2 4 ··· 4 14 14 28 28 28 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D7 D14 D14 D14 D14 2+ 1+4 2- 1+4 D4×D7 Q8.10D14 D4⋊8D14 kernel C14.172- 1+4 D14⋊D4 D7×C4⋊C4 C4⋊D28 D14⋊Q8 C28⋊7D4 D14⋊3Q8 C7×C22⋊Q8 C2×C4○D28 C2×Q8⋊2D7 C4×D7 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C14 C14 C4 C2 C2 # reps 1 4 1 3 2 1 1 1 1 1 4 3 6 9 3 3 1 1 6 6 6

Matrix representation of C14.172- 1+4 in GL8(𝔽29)

 19 21 0 0 0 0 0 0 17 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 27 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 7 16 0 11 0 0 0 0 16 8 8 8 0 0 0 0 11 13 1 15 0 0 0 0 17 23 20 13
,
 0 26 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 28 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 16 0 11 0 0 0 0 10 10 21 8 0 0 0 0 3 6 28 15 0 0 0 0 18 28 9 13
,
 0 3 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 22 3 0 0 0 0 0 0 22 7 0 0 0 0 0 0 0 0 0 7 2 0 0 0 0 0 24 4 12 12 0 0 0 0 18 15 16 16 0 0 0 0 3 6 5 9
,
 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 14 5 18 0 0 0 0 0 13 21 21 21 0 0 0 0 8 8 14 28 0 0 0 0 9 1 16 9

`G:=sub<GL(8,GF(29))| [19,17,0,0,0,0,0,0,21,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,28,0,0,0,0,0,0,0,0,7,16,11,17,0,0,0,0,16,8,13,23,0,0,0,0,0,8,1,20,0,0,0,0,11,8,15,13],[0,19,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,7,10,3,18,0,0,0,0,16,10,6,28,0,0,0,0,0,21,28,9,0,0,0,0,11,8,15,13],[0,10,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,22,22,0,0,0,0,0,0,3,7,0,0,0,0,0,0,0,0,0,24,18,3,0,0,0,0,7,4,15,6,0,0,0,0,2,12,16,5,0,0,0,0,0,12,16,9],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,14,13,8,9,0,0,0,0,5,21,8,1,0,0,0,0,18,21,14,16,0,0,0,0,0,21,28,9] >;`

C14.172- 1+4 in GAP, Magma, Sage, TeX

`C_{14}._{17}2_-^{1+4}`
`% in TeX`

`G:=Group("C14.17ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(448,1082);`
`// by ID`

`G=gap.SmallGroup(448,1082);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,268,675,297,18822]);`
`// Polycyclic`

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

׿
×
𝔽