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## G = C14.C42order 224 = 25·7

### 5th non-split extension by C14 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C14.C42
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C22×Dic7 — C14.C42
 Lower central C7 — C14 — C14.C42
 Upper central C1 — C23 — C22×C4

Generators and relations for C14.C42
G = < a,b,c | a14=b4=c4=1, bab-1=a-1, ac=ca, cbc-1=a7b >

Subgroups: 238 in 76 conjugacy classes, 45 normal (19 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C22×C4, C22×C4, Dic7, C28, C2×C14, C2×C14, C2.C42, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C22×Dic7, C22×C28, C14.C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C14.C42

Smallest permutation representation of C14.C42
Regular action 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)
(1 150 29 126)(2 149 30 125)(3 148 31 124)(4 147 32 123)(5 146 33 122)(6 145 34 121)(7 144 35 120)(8 143 36 119)(9 142 37 118)(10 141 38 117)(11 154 39 116)(12 153 40 115)(13 152 41 114)(14 151 42 113)(15 167 46 135)(16 166 47 134)(17 165 48 133)(18 164 49 132)(19 163 50 131)(20 162 51 130)(21 161 52 129)(22 160 53 128)(23 159 54 127)(24 158 55 140)(25 157 56 139)(26 156 43 138)(27 155 44 137)(28 168 45 136)(57 201 85 175)(58 200 86 174)(59 199 87 173)(60 198 88 172)(61 197 89 171)(62 210 90 170)(63 209 91 169)(64 208 92 182)(65 207 93 181)(66 206 94 180)(67 205 95 179)(68 204 96 178)(69 203 97 177)(70 202 98 176)(71 222 104 194)(72 221 105 193)(73 220 106 192)(74 219 107 191)(75 218 108 190)(76 217 109 189)(77 216 110 188)(78 215 111 187)(79 214 112 186)(80 213 99 185)(81 212 100 184)(82 211 101 183)(83 224 102 196)(84 223 103 195)
(1 72 27 63)(2 73 28 64)(3 74 15 65)(4 75 16 66)(5 76 17 67)(6 77 18 68)(7 78 19 69)(8 79 20 70)(9 80 21 57)(10 81 22 58)(11 82 23 59)(12 83 24 60)(13 84 25 61)(14 71 26 62)(29 105 44 91)(30 106 45 92)(31 107 46 93)(32 108 47 94)(33 109 48 95)(34 110 49 96)(35 111 50 97)(36 112 51 98)(37 99 52 85)(38 100 53 86)(39 101 54 87)(40 102 55 88)(41 103 56 89)(42 104 43 90)(113 187 138 177)(114 188 139 178)(115 189 140 179)(116 190 127 180)(117 191 128 181)(118 192 129 182)(119 193 130 169)(120 194 131 170)(121 195 132 171)(122 196 133 172)(123 183 134 173)(124 184 135 174)(125 185 136 175)(126 186 137 176)(141 219 160 207)(142 220 161 208)(143 221 162 209)(144 222 163 210)(145 223 164 197)(146 224 165 198)(147 211 166 199)(148 212 167 200)(149 213 168 201)(150 214 155 202)(151 215 156 203)(152 216 157 204)(153 217 158 205)(154 218 159 206)```

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4L 7A 7B 7C 14A ··· 14U 28A ··· 28X order 1 2 ··· 2 4 4 4 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 2 2 2 14 ··· 14 2 2 2 2 ··· 2 2 ··· 2

68 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + - + - + - + image C1 C2 C2 C4 C4 D4 Q8 D7 Dic7 D14 Dic14 C4×D7 D28 C7⋊D4 kernel C14.C42 C22×Dic7 C22×C28 C2×Dic7 C2×C28 C2×C14 C2×C14 C22×C4 C2×C4 C23 C22 C22 C22 C22 # reps 1 2 1 8 4 3 1 3 6 3 6 12 6 12

Matrix representation of C14.C42 in GL6(𝔽29)

 25 25 0 0 0 0 4 11 0 0 0 0 0 0 10 10 0 0 0 0 19 22 0 0 0 0 0 0 3 8 0 0 0 0 21 8
,
 17 13 0 0 0 0 0 12 0 0 0 0 0 0 9 10 0 0 0 0 15 20 0 0 0 0 0 0 3 1 0 0 0 0 21 26
,
 17 0 0 0 0 0 0 17 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 27 18 0 0 0 0 11 2

`G:=sub<GL(6,GF(29))| [25,4,0,0,0,0,25,11,0,0,0,0,0,0,10,19,0,0,0,0,10,22,0,0,0,0,0,0,3,21,0,0,0,0,8,8],[17,0,0,0,0,0,13,12,0,0,0,0,0,0,9,15,0,0,0,0,10,20,0,0,0,0,0,0,3,21,0,0,0,0,1,26],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,11,0,0,0,0,18,2] >;`

C14.C42 in GAP, Magma, Sage, TeX

`C_{14}.C_4^2`
`% in TeX`

`G:=Group("C14.C4^2");`
`// GroupNames label`

`G:=SmallGroup(224,37);`
`// by ID`

`G=gap.SmallGroup(224,37);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,217,55,6917]);`
`// Polycyclic`

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

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