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## G = C2×C4×D27order 432 = 24·33

### Direct product of C2×C4 and D27

Series: Derived Chief Lower central Upper central

 Derived series C1 — C27 — C2×C4×D27
 Chief series C1 — C3 — C9 — C27 — C54 — D54 — C22×D27 — C2×C4×D27
 Lower central C27 — C2×C4×D27
 Upper central C1 — C2×C4

Generators and relations for C2×C4×D27
G = < a,b,c,d | a2=b4=c27=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 792 in 108 conjugacy classes, 51 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C9, Dic3, C12, D6, C2×C6, C22×C4, D9, C18, C18, C4×S3, C2×Dic3, C2×C12, C22×S3, C27, Dic9, C36, D18, C2×C18, S3×C2×C4, D27, C54, C54, C4×D9, C2×Dic9, C2×C36, C22×D9, Dic27, C108, D54, C2×C54, C2×C4×D9, C4×D27, C2×Dic27, C2×C108, C22×D27, C2×C4×D27
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, D9, C4×S3, C22×S3, D18, S3×C2×C4, D27, C4×D9, C22×D9, D54, C2×C4×D9, C4×D27, C22×D27, C2×C4×D27

Smallest permutation representation of C2×C4×D27
On 216 points
Generators in S216
(1 146)(2 147)(3 148)(4 149)(5 150)(6 151)(7 152)(8 153)(9 154)(10 155)(11 156)(12 157)(13 158)(14 159)(15 160)(16 161)(17 162)(18 136)(19 137)(20 138)(21 139)(22 140)(23 141)(24 142)(25 143)(26 144)(27 145)(28 113)(29 114)(30 115)(31 116)(32 117)(33 118)(34 119)(35 120)(36 121)(37 122)(38 123)(39 124)(40 125)(41 126)(42 127)(43 128)(44 129)(45 130)(46 131)(47 132)(48 133)(49 134)(50 135)(51 109)(52 110)(53 111)(54 112)(55 197)(56 198)(57 199)(58 200)(59 201)(60 202)(61 203)(62 204)(63 205)(64 206)(65 207)(66 208)(67 209)(68 210)(69 211)(70 212)(71 213)(72 214)(73 215)(74 216)(75 190)(76 191)(77 192)(78 193)(79 194)(80 195)(81 196)(82 184)(83 185)(84 186)(85 187)(86 188)(87 189)(88 163)(89 164)(90 165)(91 166)(92 167)(93 168)(94 169)(95 170)(96 171)(97 172)(98 173)(99 174)(100 175)(101 176)(102 177)(103 178)(104 179)(105 180)(106 181)(107 182)(108 183)
(1 81 51 105)(2 55 52 106)(3 56 53 107)(4 57 54 108)(5 58 28 82)(6 59 29 83)(7 60 30 84)(8 61 31 85)(9 62 32 86)(10 63 33 87)(11 64 34 88)(12 65 35 89)(13 66 36 90)(14 67 37 91)(15 68 38 92)(16 69 39 93)(17 70 40 94)(18 71 41 95)(19 72 42 96)(20 73 43 97)(21 74 44 98)(22 75 45 99)(23 76 46 100)(24 77 47 101)(25 78 48 102)(26 79 49 103)(27 80 50 104)(109 180 146 196)(110 181 147 197)(111 182 148 198)(112 183 149 199)(113 184 150 200)(114 185 151 201)(115 186 152 202)(116 187 153 203)(117 188 154 204)(118 189 155 205)(119 163 156 206)(120 164 157 207)(121 165 158 208)(122 166 159 209)(123 167 160 210)(124 168 161 211)(125 169 162 212)(126 170 136 213)(127 171 137 214)(128 172 138 215)(129 173 139 216)(130 174 140 190)(131 175 141 191)(132 176 142 192)(133 177 143 193)(134 178 144 194)(135 179 145 195)
(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)
(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(28 47)(29 46)(30 45)(31 44)(32 43)(33 42)(34 41)(35 40)(36 39)(37 38)(48 54)(49 53)(50 52)(55 80)(56 79)(57 78)(58 77)(59 76)(60 75)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(82 101)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(102 108)(103 107)(104 106)(110 135)(111 134)(112 133)(113 132)(114 131)(115 130)(116 129)(117 128)(118 127)(119 126)(120 125)(121 124)(122 123)(136 156)(137 155)(138 154)(139 153)(140 152)(141 151)(142 150)(143 149)(144 148)(145 147)(157 162)(158 161)(159 160)(163 170)(164 169)(165 168)(166 167)(171 189)(172 188)(173 187)(174 186)(175 185)(176 184)(177 183)(178 182)(179 181)(190 202)(191 201)(192 200)(193 199)(194 198)(195 197)(203 216)(204 215)(205 214)(206 213)(207 212)(208 211)(209 210)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 9A 9B 9C 12A 12B 12C 12D 18A ··· 18I 27A ··· 27I 36A ··· 36L 54A ··· 54AA 108A ··· 108AJ order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 9 9 9 12 12 12 12 18 ··· 18 27 ··· 27 36 ··· 36 54 ··· 54 108 ··· 108 size 1 1 1 1 27 27 27 27 2 1 1 1 1 27 27 27 27 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D6 D6 D9 C4×S3 D18 D18 D27 C4×D9 D54 D54 C4×D27 kernel C2×C4×D27 C4×D27 C2×Dic27 C2×C108 C22×D27 D54 C2×C36 C36 C2×C18 C2×C12 C18 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 8 1 2 1 3 4 6 3 9 12 18 9 36

Matrix representation of C2×C4×D27 in GL3(𝔽109) generated by

 1 0 0 0 108 0 0 0 108
,
 33 0 0 0 108 0 0 0 108
,
 1 0 0 0 30 93 0 16 46
,
 108 0 0 0 1 0 0 108 108
G:=sub<GL(3,GF(109))| [1,0,0,0,108,0,0,0,108],[33,0,0,0,108,0,0,0,108],[1,0,0,0,30,16,0,93,46],[108,0,0,0,1,108,0,0,108] >;

C2×C4×D27 in GAP, Magma, Sage, TeX

C_2\times C_4\times D_{27}
% in TeX

G:=Group("C2xC4xD27");
// GroupNames label

G:=SmallGroup(432,44);
// by ID

G=gap.SmallGroup(432,44);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,58,2804,557,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^27=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽