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G = C23×D27  order 432 = 24·33

Direct product of C23 and D27

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C23×D27, C27⋊C24, C54⋊C23, C9.(S3×C23), C3.(C23×D9), (C2×C54)⋊4C22, (C22×C54)⋊3C2, (C2×C18).40D6, (C2×C6).40D18, C6.38(C22×D9), (C22×C6).10D9, C18.38(C22×S3), (C22×C18).10S3, SmallGroup(432,227)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C23×D27
 G = < a,b,c,d,e | a2=b2=c2=d27=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 2104 in 268 conjugacy classes, 115 normal (9 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C23, C23, C9, D6, C2×C6, C24, D9, C18, C22×S3, C22×C6, C27, D18, C2×C18, S3×C23, D27, C54, C22×D9, C22×C18, D54, C2×C54, C23×D9, C22×D27, C22×C54, C23×D27
Quotients: C1, C2, C22, S3, C23, D6, C24, D9, C22×S3, D18, S3×C23, D27, C22×D9, D54, C23×D9, C22×D27, C23×D27

Smallest permutation representation of C23×D27
►On 216 points
Generators in S216
(1 215)(2 216)(3 190)(4 191)(5 192)(6 193)(7 194)(8 195)(9 196)(10 197)(11 198)(12 199)(13 200)(14 201)(15 202)(16 203)(17 204)(18 205)(19 206)(20 207)(21 208)(22 209)(23 210)(24 211)(25 212)(26 213)(27 214)(28 173)(29 174)(30 175)(31 176)(32 177)(33 178)(34 179)(35 180)(36 181)(37 182)(38 183)(39 184)(40 185)(41 186)(42 187)(43 188)(44 189)(45 163)(46 164)(47 165)(48 166)(49 167)(50 168)(51 169)(52 170)(53 171)(54 172)(55 137)(56 138)(57 139)(58 140)(59 141)(60 142)(61 143)(62 144)(63 145)(64 146)(65 147)(66 148)(67 149)(68 150)(69 151)(70 152)(71 153)(72 154)(73 155)(74 156)(75 157)(76 158)(77 159)(78 160)(79 161)(80 162)(81 136)(82 125)(83 126)(84 127)(85 128)(86 129)(87 130)(88 131)(89 132)(90 133)(91 134)(92 135)(93 109)(94 110)(95 111)(96 112)(97 113)(98 114)(99 115)(100 116)(101 117)(102 118)(103 119)(104 120)(105 121)(106 122)(107 123)(108 124)

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

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

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

120 conjugacy classes

class 1 2A···2G2H···2O 3 6A···6G9A9B9C18A···18U27A···27I54A···54BK
order12···22···236···699918···1827···2754···54
size11···127···2722···22222···22···22···2

120 irreducible representations

dim111222222
type+++++++++
imageC1C2C2S3D6D9D18D27D54
kernelC23×D27C22×D27C22×C54C22×C18C2×C18C22×C6C2×C6C23C22
# reps114117321963

Matrix representation of C23×D27 ►in GL4(𝔽109) generated by

108000
0100
001080
000108
,
108000
010800
0010
0001
,
108000
0100
0010
0001
,
1000
0100
00717
009299
,
1000
0100
001679
006393
G:=sub<GL(4,GF(109))| [108,0,0,0,0,1,0,0,0,0,108,0,0,0,0,108],[108,0,0,0,0,108,0,0,0,0,1,0,0,0,0,1],[108,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,7,92,0,0,17,99],[1,0,0,0,0,1,0,0,0,0,16,63,0,0,79,93] >;
 

C23×D27 in GAP, Magma, Sage, TeX 

C_2^3\times D_{27}
 
% in TeX
 
G:=Group("C2^3xD27");
 
// GroupNames label
 
G:=SmallGroup(432,227);
 
// by ID
 
G=gap.SmallGroup(432,227);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,2804,557,10085,292,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^27=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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×
⋊
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𝔽
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