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## G = C42.282D14order 448 = 26·7

### 5th central extension by C42 of D14

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C42.282D14
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C4×D7 — D7×C42 — C42.282D14
 Lower central C7 — C14 — C42.282D14
 Upper central C1 — C42 — C4×C8

Generators and relations for C42.282D14
G = < a,b,c,d | a4=b4=1, c14=b-1, d2=a2b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2b2c13 >

Subgroups: 452 in 118 conjugacy classes, 63 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, D7, C14, C42, C42, C2×C8, C2×C8, C22×C4, Dic7, Dic7, C28, D14, D14, C2×C14, C4×C8, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C7⋊C8, C56, C4×D7, C4×D7, C2×Dic7, C2×C28, C22×D7, C42.12C4, C2×C7⋊C8, C4×Dic7, C4×C28, C2×C56, C2×C4×D7, C4×C7⋊C8, Dic7⋊C8, D14⋊C8, C4×C56, D7×C42, C42.282D14
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C2×C8, M4(2), C22×C4, C4○D4, D14, C42⋊C2, C22×C8, C2×M4(2), C4×D7, C22×D7, C42.12C4, C8×D7, C8⋊D7, C2×C4×D7, C4○D28, C42⋊D7, D7×C2×C8, C2×C8⋊D7, C42.282D14

Smallest permutation representation of C42.282D14
On 224 points
Generators in S224
```(1 157 100 202)(2 158 101 203)(3 159 102 204)(4 160 103 205)(5 161 104 206)(6 162 105 207)(7 163 106 208)(8 164 107 209)(9 165 108 210)(10 166 109 211)(11 167 110 212)(12 168 111 213)(13 113 112 214)(14 114 57 215)(15 115 58 216)(16 116 59 217)(17 117 60 218)(18 118 61 219)(19 119 62 220)(20 120 63 221)(21 121 64 222)(22 122 65 223)(23 123 66 224)(24 124 67 169)(25 125 68 170)(26 126 69 171)(27 127 70 172)(28 128 71 173)(29 129 72 174)(30 130 73 175)(31 131 74 176)(32 132 75 177)(33 133 76 178)(34 134 77 179)(35 135 78 180)(36 136 79 181)(37 137 80 182)(38 138 81 183)(39 139 82 184)(40 140 83 185)(41 141 84 186)(42 142 85 187)(43 143 86 188)(44 144 87 189)(45 145 88 190)(46 146 89 191)(47 147 90 192)(48 148 91 193)(49 149 92 194)(50 150 93 195)(51 151 94 196)(52 152 95 197)(53 153 96 198)(54 154 97 199)(55 155 98 200)(56 156 99 201)
(1 43 29 15)(2 44 30 16)(3 45 31 17)(4 46 32 18)(5 47 33 19)(6 48 34 20)(7 49 35 21)(8 50 36 22)(9 51 37 23)(10 52 38 24)(11 53 39 25)(12 54 40 26)(13 55 41 27)(14 56 42 28)(57 99 85 71)(58 100 86 72)(59 101 87 73)(60 102 88 74)(61 103 89 75)(62 104 90 76)(63 105 91 77)(64 106 92 78)(65 107 93 79)(66 108 94 80)(67 109 95 81)(68 110 96 82)(69 111 97 83)(70 112 98 84)(113 155 141 127)(114 156 142 128)(115 157 143 129)(116 158 144 130)(117 159 145 131)(118 160 146 132)(119 161 147 133)(120 162 148 134)(121 163 149 135)(122 164 150 136)(123 165 151 137)(124 166 152 138)(125 167 153 139)(126 168 154 140)(169 211 197 183)(170 212 198 184)(171 213 199 185)(172 214 200 186)(173 215 201 187)(174 216 202 188)(175 217 203 189)(176 218 204 190)(177 219 205 191)(178 220 206 192)(179 221 207 193)(180 222 208 194)(181 223 209 195)(182 224 210 196)
(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 14 86 99 29 42 58 71)(2 98 87 41 30 70 59 13)(3 40 88 69 31 12 60 97)(4 68 89 11 32 96 61 39)(5 10 90 95 33 38 62 67)(6 94 91 37 34 66 63 9)(7 36 92 65 35 8 64 93)(15 28 100 57 43 56 72 85)(16 112 101 55 44 84 73 27)(17 54 102 83 45 26 74 111)(18 82 103 25 46 110 75 53)(19 24 104 109 47 52 76 81)(20 108 105 51 48 80 77 23)(21 50 106 79 49 22 78 107)(113 158 200 189 141 130 172 217)(114 188 201 129 142 216 173 157)(115 128 202 215 143 156 174 187)(116 214 203 155 144 186 175 127)(117 154 204 185 145 126 176 213)(118 184 205 125 146 212 177 153)(119 124 206 211 147 152 178 183)(120 210 207 151 148 182 179 123)(121 150 208 181 149 122 180 209)(131 168 218 199 159 140 190 171)(132 198 219 139 160 170 191 167)(133 138 220 169 161 166 192 197)(134 224 221 165 162 196 193 137)(135 164 222 195 163 136 194 223)```

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

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

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

136 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4L 4M ··· 4R 7A 7B 7C 8A ··· 8H 8I ··· 8P 14A ··· 14I 28A ··· 28AJ 56A ··· 56AV order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 7 7 7 8 ··· 8 8 ··· 8 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 1 1 14 14 1 ··· 1 14 ··· 14 2 2 2 2 ··· 2 14 ··· 14 2 ··· 2 2 ··· 2 2 ··· 2

136 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C8 D7 M4(2) C4○D4 D14 D14 C4×D7 C8×D7 C8⋊D7 C4○D28 kernel C42.282D14 C4×C7⋊C8 Dic7⋊C8 D14⋊C8 C4×C56 D7×C42 C4×Dic7 C2×C4×D7 C4×D7 C4×C8 C28 C28 C42 C2×C8 C2×C4 C4 C4 C4 # reps 1 1 2 2 1 1 4 4 16 3 4 4 3 6 12 24 24 24

Matrix representation of C42.282D14 in GL3(𝔽113) generated by

 112 0 0 0 15 0 0 0 15
,
 98 0 0 0 112 0 0 0 112
,
 18 0 0 0 13 19 0 94 94
,
 18 0 0 0 19 13 0 94 94
`G:=sub<GL(3,GF(113))| [112,0,0,0,15,0,0,0,15],[98,0,0,0,112,0,0,0,112],[18,0,0,0,13,94,0,19,94],[18,0,0,0,19,94,0,13,94] >;`

C42.282D14 in GAP, Magma, Sage, TeX

`C_4^2._{282}D_{14}`
`% in TeX`

`G:=Group("C4^2.282D14");`
`// GroupNames label`

`G:=SmallGroup(448,219);`
`// by ID`

`G=gap.SmallGroup(448,219);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,422,58,136,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^14=b^-1,d^2=a^2*b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c^13>;`
`// generators/relations`

׿
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