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## G = C2×C27⋊D4order 432 = 24·33

### Direct product of C2 and C27⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C54 — C2×C27⋊D4
 Chief series C1 — C3 — C9 — C27 — C54 — D54 — C22×D27 — C2×C27⋊D4
 Lower central C27 — C54 — C2×C27⋊D4
 Upper central C1 — C22 — C23

Generators and relations for C2×C27⋊D4
G = < a,b,c,d | a2=b27=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 824 in 108 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C9, Dic3, D6, C2×C6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C18, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C27, Dic9, D18, C2×C18, C2×C18, C2×C18, C2×C3⋊D4, D27, C54, C54, C54, C2×Dic9, C9⋊D4, C22×D9, C22×C18, Dic27, D54, D54, C2×C54, C2×C54, C2×C54, C2×C9⋊D4, C2×Dic27, C27⋊D4, C22×D27, C22×C54, C2×C27⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊D4, C22×S3, D18, C2×C3⋊D4, D27, C9⋊D4, C22×D9, D54, C2×C9⋊D4, C27⋊D4, C22×D27, C2×C27⋊D4

Smallest permutation representation of C2×C27⋊D4
On 216 points
Generators in S216
(1 130)(2 131)(3 132)(4 133)(5 134)(6 135)(7 109)(8 110)(9 111)(10 112)(11 113)(12 114)(13 115)(14 116)(15 117)(16 118)(17 119)(18 120)(19 121)(20 122)(21 123)(22 124)(23 125)(24 126)(25 127)(26 128)(27 129)(28 160)(29 161)(30 162)(31 136)(32 137)(33 138)(34 139)(35 140)(36 141)(37 142)(38 143)(39 144)(40 145)(41 146)(42 147)(43 148)(44 149)(45 150)(46 151)(47 152)(48 153)(49 154)(50 155)(51 156)(52 157)(53 158)(54 159)(55 179)(56 180)(57 181)(58 182)(59 183)(60 184)(61 185)(62 186)(63 187)(64 188)(65 189)(66 163)(67 164)(68 165)(69 166)(70 167)(71 168)(72 169)(73 170)(74 171)(75 172)(76 173)(77 174)(78 175)(79 176)(80 177)(81 178)(82 200)(83 201)(84 202)(85 203)(86 204)(87 205)(88 206)(89 207)(90 208)(91 209)(92 210)(93 211)(94 212)(95 213)(96 214)(97 215)(98 216)(99 190)(100 191)(101 192)(102 193)(103 194)(104 195)(105 196)(106 197)(107 198)(108 199)
(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)
(1 66 47 101)(2 65 48 100)(3 64 49 99)(4 63 50 98)(5 62 51 97)(6 61 52 96)(7 60 53 95)(8 59 54 94)(9 58 28 93)(10 57 29 92)(11 56 30 91)(12 55 31 90)(13 81 32 89)(14 80 33 88)(15 79 34 87)(16 78 35 86)(17 77 36 85)(18 76 37 84)(19 75 38 83)(20 74 39 82)(21 73 40 108)(22 72 41 107)(23 71 42 106)(24 70 43 105)(25 69 44 104)(26 68 45 103)(27 67 46 102)(109 184 158 213)(110 183 159 212)(111 182 160 211)(112 181 161 210)(113 180 162 209)(114 179 136 208)(115 178 137 207)(116 177 138 206)(117 176 139 205)(118 175 140 204)(119 174 141 203)(120 173 142 202)(121 172 143 201)(122 171 144 200)(123 170 145 199)(124 169 146 198)(125 168 147 197)(126 167 148 196)(127 166 149 195)(128 165 150 194)(129 164 151 193)(130 163 152 192)(131 189 153 191)(132 188 154 190)(133 187 155 216)(134 186 156 215)(135 185 157 214)
(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 39)(29 38)(30 37)(31 36)(32 35)(33 34)(40 54)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(55 85)(56 84)(57 83)(58 82)(59 108)(60 107)(61 106)(62 105)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 98)(70 97)(71 96)(72 95)(73 94)(74 93)(75 92)(76 91)(77 90)(78 89)(79 88)(80 87)(81 86)(109 124)(110 123)(111 122)(112 121)(113 120)(114 119)(115 118)(116 117)(125 135)(126 134)(127 133)(128 132)(129 131)(136 141)(137 140)(138 139)(142 162)(143 161)(144 160)(145 159)(146 158)(147 157)(148 156)(149 155)(150 154)(151 153)(163 192)(164 191)(165 190)(166 216)(167 215)(168 214)(169 213)(170 212)(171 211)(172 210)(173 209)(174 208)(175 207)(176 206)(177 205)(178 204)(179 203)(180 202)(181 201)(182 200)(183 199)(184 198)(185 197)(186 196)(187 195)(188 194)(189 193)

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

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

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

114 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A ··· 6G 9A 9B 9C 18A ··· 18U 27A ··· 27I 54A ··· 54BK order 1 2 2 2 2 2 2 2 3 4 4 6 ··· 6 9 9 9 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 1 1 2 2 54 54 2 54 54 2 ··· 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

114 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D6 D9 C3⋊D4 D18 D27 C9⋊D4 D54 C27⋊D4 kernel C2×C27⋊D4 C2×Dic27 C27⋊D4 C22×D27 C22×C54 C22×C18 C54 C2×C18 C22×C6 C18 C2×C6 C23 C6 C22 C2 # reps 1 1 4 1 1 1 2 3 3 4 9 9 12 27 36

Matrix representation of C2×C27⋊D4 in GL4(𝔽109) generated by

 108 0 0 0 0 108 0 0 0 0 1 0 0 0 0 1
,
 108 108 0 0 1 0 0 0 0 0 7 92 0 0 17 99
,
 108 0 0 0 1 1 0 0 0 0 12 103 0 0 6 97
,
 1 0 0 0 108 108 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(109))| [108,0,0,0,0,108,0,0,0,0,1,0,0,0,0,1],[108,1,0,0,108,0,0,0,0,0,7,17,0,0,92,99],[108,1,0,0,0,1,0,0,0,0,12,6,0,0,103,97],[1,108,0,0,0,108,0,0,0,0,0,1,0,0,1,0] >;

C2×C27⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_{27}\rtimes D_4
% in TeX

G:=Group("C2xC27:D4");
// GroupNames label

G:=SmallGroup(432,52);
// by ID

G=gap.SmallGroup(432,52);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,2804,557,10085,292,14118]);
// Polycyclic

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

׿
×
𝔽