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## G = C22⋊C4×C27order 432 = 24·33

### Direct product of C27 and C22⋊C4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C22⋊C4×C27
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×C54 — C2×C108 — C22⋊C4×C27
 Lower central C1 — C2 — C22⋊C4×C27
 Upper central C1 — C2×C54 — C22⋊C4×C27

Generators and relations for C22⋊C4×C27
G = < a,b,c,d | a27=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 92 in 68 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C9, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C18, C18, C18, C2×C12, C22×C6, C27, C36, C2×C18, C2×C18, C2×C18, C3×C22⋊C4, C54, C54, C54, C2×C36, C22×C18, C108, C2×C54, C2×C54, C2×C54, C9×C22⋊C4, C2×C108, C22×C54, C22⋊C4×C27
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C9, C12, C2×C6, C22⋊C4, C18, C2×C12, C3×D4, C27, C36, C2×C18, C3×C22⋊C4, C54, C2×C36, D4×C9, C108, C2×C54, C9×C22⋊C4, C2×C108, D4×C27, C22⋊C4×C27

Smallest permutation representation of C22⋊C4×C27
On 216 points
Generators in S216
(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)
(1 107)(2 108)(3 82)(4 83)(5 84)(6 85)(7 86)(8 87)(9 88)(10 89)(11 90)(12 91)(13 92)(14 93)(15 94)(16 95)(17 96)(18 97)(19 98)(20 99)(21 100)(22 101)(23 102)(24 103)(25 104)(26 105)(27 106)(28 159)(29 160)(30 161)(31 162)(32 136)(33 137)(34 138)(35 139)(36 140)(37 141)(38 142)(39 143)(40 144)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 164)(56 165)(57 166)(58 167)(59 168)(60 169)(61 170)(62 171)(63 172)(64 173)(65 174)(66 175)(67 176)(68 177)(69 178)(70 179)(71 180)(72 181)(73 182)(74 183)(75 184)(76 185)(77 186)(78 187)(79 188)(80 189)(81 163)(109 212)(110 213)(111 214)(112 215)(113 216)(114 190)(115 191)(116 192)(117 193)(118 194)(119 195)(120 196)(121 197)(122 198)(123 199)(124 200)(125 201)(126 202)(127 203)(128 204)(129 205)(130 206)(131 207)(132 208)(133 209)(134 210)(135 211)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 81)(24 55)(25 56)(26 57)(27 58)(28 135)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(82 170)(83 171)(84 172)(85 173)(86 174)(87 175)(88 176)(89 177)(90 178)(91 179)(92 180)(93 181)(94 182)(95 183)(96 184)(97 185)(98 186)(99 187)(100 188)(101 189)(102 163)(103 164)(104 165)(105 166)(106 167)(107 168)(108 169)(136 215)(137 216)(138 190)(139 191)(140 192)(141 193)(142 194)(143 195)(144 196)(145 197)(146 198)(147 199)(148 200)(149 201)(150 202)(151 203)(152 204)(153 205)(154 206)(155 207)(156 208)(157 209)(158 210)(159 211)(160 212)(161 213)(162 214)
(1 120 107 144)(2 121 108 145)(3 122 82 146)(4 123 83 147)(5 124 84 148)(6 125 85 149)(7 126 86 150)(8 127 87 151)(9 128 88 152)(10 129 89 153)(11 130 90 154)(12 131 91 155)(13 132 92 156)(14 133 93 157)(15 134 94 158)(16 135 95 159)(17 109 96 160)(18 110 97 161)(19 111 98 162)(20 112 99 136)(21 113 100 137)(22 114 101 138)(23 115 102 139)(24 116 103 140)(25 117 104 141)(26 118 105 142)(27 119 106 143)(28 183 211 74)(29 184 212 75)(30 185 213 76)(31 186 214 77)(32 187 215 78)(33 188 216 79)(34 189 190 80)(35 163 191 81)(36 164 192 55)(37 165 193 56)(38 166 194 57)(39 167 195 58)(40 168 196 59)(41 169 197 60)(42 170 198 61)(43 171 199 62)(44 172 200 63)(45 173 201 64)(46 174 202 65)(47 175 203 66)(48 176 204 67)(49 177 205 68)(50 178 206 69)(51 179 207 70)(52 180 208 71)(53 181 209 72)(54 182 210 73)

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

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

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

270 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 6A ··· 6F 6G 6H 6I 6J 9A ··· 9F 12A ··· 12H 18A ··· 18R 18S ··· 18AD 27A ··· 27R 36A ··· 36X 54A ··· 54BB 54BC ··· 54CL 108A ··· 108BT order 1 2 2 2 2 2 3 3 4 4 4 4 6 ··· 6 6 6 6 6 9 ··· 9 12 ··· 12 18 ··· 18 18 ··· 18 27 ··· 27 36 ··· 36 54 ··· 54 54 ··· 54 108 ··· 108 size 1 1 1 1 2 2 1 1 2 2 2 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

270 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C2 C3 C4 C6 C6 C9 C12 C18 C18 C27 C36 C54 C54 C108 D4 C3×D4 D4×C9 D4×C27 kernel C22⋊C4×C27 C2×C108 C22×C54 C9×C22⋊C4 C2×C54 C2×C36 C22×C18 C3×C22⋊C4 C2×C18 C2×C12 C22×C6 C22⋊C4 C2×C6 C2×C4 C23 C22 C54 C18 C6 C2 # reps 1 2 1 2 4 4 2 6 8 12 6 18 24 36 18 72 2 4 12 36

Matrix representation of C22⋊C4×C27 in GL3(𝔽109) generated by

 45 0 0 0 81 0 0 0 81
,
 1 0 0 0 108 0 0 18 1
,
 1 0 0 0 108 0 0 0 108
,
 33 0 0 0 108 12 0 18 1
G:=sub<GL(3,GF(109))| [45,0,0,0,81,0,0,0,81],[1,0,0,0,108,18,0,0,1],[1,0,0,0,108,0,0,0,108],[33,0,0,0,108,18,0,12,1] >;

C22⋊C4×C27 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times C_{27}
% in TeX

G:=Group("C2^2:C4xC27");
// GroupNames label

G:=SmallGroup(432,21);
// by ID

G=gap.SmallGroup(432,21);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,268,166]);
// Polycyclic

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

׿
×
𝔽