Copied to
clipboard

## G = C42.243D14order 448 = 26·7

### 2nd non-split extension by C42 of D14 acting via D14/C14=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C42.243D14
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C4×D7 — C42⋊D7 — C42.243D14
 Lower central C7 — C2×C14 — C42.243D14
 Upper central C1 — C2×C4 — C4×C8

Generators and relations for C42.243D14
G = < a,b,c,d | a4=b4=1, c14=b-1, d2=a2b, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=a2b2c13 >

Subgroups: 388 in 96 conjugacy classes, 47 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, D7, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic7, C28, C28, D14, C2×C14, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C42⋊C2, C7⋊C8, C56, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C42.7C22, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, D14⋊C4, C4×C28, C2×C56, C2×C4×D7, C42.D7, Dic7⋊C8, D14⋊C8, C4×C56, C42⋊D7, C42.243D14
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, D14, C42⋊C2, C8○D4, C4×D7, C22×D7, C42.7C22, C2×C4×D7, C4○D28, C42⋊D7, D28.2C4, C42.243D14

Smallest permutation representation of C42.243D14
On 224 points
Generators in S224
```(1 159 182 91)(2 160 183 92)(3 161 184 93)(4 162 185 94)(5 163 186 95)(6 164 187 96)(7 165 188 97)(8 166 189 98)(9 167 190 99)(10 168 191 100)(11 113 192 101)(12 114 193 102)(13 115 194 103)(14 116 195 104)(15 117 196 105)(16 118 197 106)(17 119 198 107)(18 120 199 108)(19 121 200 109)(20 122 201 110)(21 123 202 111)(22 124 203 112)(23 125 204 57)(24 126 205 58)(25 127 206 59)(26 128 207 60)(27 129 208 61)(28 130 209 62)(29 131 210 63)(30 132 211 64)(31 133 212 65)(32 134 213 66)(33 135 214 67)(34 136 215 68)(35 137 216 69)(36 138 217 70)(37 139 218 71)(38 140 219 72)(39 141 220 73)(40 142 221 74)(41 143 222 75)(42 144 223 76)(43 145 224 77)(44 146 169 78)(45 147 170 79)(46 148 171 80)(47 149 172 81)(48 150 173 82)(49 151 174 83)(50 152 175 84)(51 153 176 85)(52 154 177 86)(53 155 178 87)(54 156 179 88)(55 157 180 89)(56 158 181 90)
(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 42 224 209 29 14 196 181)(2 208 169 13 30 180 197 41)(3 12 170 179 31 40 198 207)(4 178 171 39 32 206 199 11)(5 38 172 205 33 10 200 177)(6 204 173 9 34 176 201 37)(7 8 174 175 35 36 202 203)(15 56 182 223 43 28 210 195)(16 222 183 27 44 194 211 55)(17 26 184 193 45 54 212 221)(18 192 185 53 46 220 213 25)(19 52 186 219 47 24 214 191)(20 218 187 23 48 190 215 51)(21 22 188 189 49 50 216 217)(57 110 167 164 85 82 139 136)(58 163 168 81 86 135 140 109)(59 80 113 134 87 108 141 162)(60 133 114 107 88 161 142 79)(61 106 115 160 89 78 143 132)(62 159 116 77 90 131 144 105)(63 76 117 130 91 104 145 158)(64 129 118 103 92 157 146 75)(65 102 119 156 93 74 147 128)(66 155 120 73 94 127 148 101)(67 72 121 126 95 100 149 154)(68 125 122 99 96 153 150 71)(69 98 123 152 97 70 151 124)(83 112 137 166 111 84 165 138)```

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

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

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

124 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 7A 7B 7C 8A ··· 8H 8I 8J 8K 8L 14A ··· 14I 28A ··· 28AJ 56A ··· 56AV order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 7 7 7 8 ··· 8 8 8 8 8 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 1 1 28 1 1 1 1 2 2 2 2 28 28 28 2 2 2 2 ··· 2 28 28 28 28 2 ··· 2 2 ··· 2 2 ··· 2

124 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 D7 C4○D4 D14 D14 C8○D4 C4×D7 C4○D28 D28.2C4 kernel C42.243D14 C42.D7 Dic7⋊C8 D14⋊C8 C4×C56 C42⋊D7 Dic7⋊C4 D14⋊C4 C4×C8 C28 C42 C2×C8 C14 C2×C4 C4 C2 # reps 1 1 2 2 1 1 4 4 3 4 3 6 8 12 24 48

Matrix representation of C42.243D14 in GL4(𝔽113) generated by

 111 112 0 0 3 2 0 0 0 0 15 0 0 0 0 15
,
 98 0 0 0 0 98 0 0 0 0 112 0 0 0 0 112
,
 18 0 0 0 0 18 0 0 0 0 19 94 0 0 19 100
,
 18 0 0 0 41 95 0 0 0 0 94 19 0 0 100 19
`G:=sub<GL(4,GF(113))| [111,3,0,0,112,2,0,0,0,0,15,0,0,0,0,15],[98,0,0,0,0,98,0,0,0,0,112,0,0,0,0,112],[18,0,0,0,0,18,0,0,0,0,19,19,0,0,94,100],[18,41,0,0,0,95,0,0,0,0,94,100,0,0,19,19] >;`

C42.243D14 in GAP, Magma, Sage, TeX

`C_4^2._{243}D_{14}`
`% in TeX`

`G:=Group("C4^2.243D14");`
`// GroupNames label`

`G:=SmallGroup(448,224);`
`// by ID`

`G=gap.SmallGroup(448,224);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,120,422,58,136,18822]);`
`// 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,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c^13>;`
`// generators/relations`

׿
×
𝔽