direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×S3×Dic9, D6.11D18, C62.65D6, C18⋊3(C4×S3), (S3×C18)⋊3C4, C6⋊1(C2×Dic9), (C6×Dic9)⋊6C2, (S3×C6).31D6, (C2×C6).19D18, (C2×C18).20D6, C9⋊Dic3⋊6C22, (S3×C6).4Dic3, C6.27(S3×Dic3), (C22×S3).2D9, C3⋊1(C22×Dic9), C22.15(S3×D9), C6.20(C22×D9), C18.20(C22×S3), (C6×C18).14C22, (C3×C18).20C23, (C3×Dic9)⋊6C22, (S3×C18).14C22, C32.2(C22×Dic3), C9⋊4(S3×C2×C4), C2.4(C2×S3×D9), C6.39(C2×S32), (C2×C6).26S32, (S3×C2×C6).3S3, (S3×C9)⋊2(C2×C4), (C3×C18)⋊3(C2×C4), (S3×C2×C18).3C2, (C3×C9)⋊4(C22×C4), C3.3(C2×S3×Dic3), (C2×C9⋊Dic3)⋊6C2, (C3×S3).(C2×Dic3), (C3×C6).88(C22×S3), (C3×C6).34(C2×Dic3), SmallGroup(432,308)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C9 — C2×S3×Dic9 |
Generators and relations for C2×S3×Dic9
G = < a,b,c,d,e | a2=b3=c2=d18=1, e2=d9, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 748 in 178 conjugacy classes, 77 normal (25 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C18, C18, C18, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C3×C9, Dic9, Dic9, C2×C18, C2×C18, C3×Dic3, C3⋊Dic3, S3×C6, C62, S3×C2×C4, C22×Dic3, S3×C9, C3×C18, C3×C18, C2×Dic9, C2×Dic9, C22×C18, S3×Dic3, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C3×Dic9, C9⋊Dic3, S3×C18, C6×C18, C22×Dic9, C2×S3×Dic3, S3×Dic9, C6×Dic9, C2×C9⋊Dic3, S3×C2×C18, C2×S3×Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, D9, C4×S3, C2×Dic3, C22×S3, Dic9, D18, S32, S3×C2×C4, C22×Dic3, C2×Dic9, C22×D9, S3×Dic3, C2×S32, S3×D9, C22×Dic9, C2×S3×Dic3, S3×Dic9, C2×S3×D9, C2×S3×Dic9
►On 144 points
Generators in S144
(1 32)(2 33)(3 34)(4 35)(5 36)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)(85 97)(86 98)(87 99)(88 100)(89 101)(90 102)(109 133)(110 134)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 141)(118 142)(119 143)(120 144)(121 127)(122 128)(123 129)(124 130)(125 131)(126 132)
(1 7 13)(2 8 14)(3 9 15)(4 10 16)(5 11 17)(6 12 18)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 67 61)(56 68 62)(57 69 63)(58 70 64)(59 71 65)(60 72 66)(73 85 79)(74 86 80)(75 87 81)(76 88 82)(77 89 83)(78 90 84)(91 103 97)(92 104 98)(93 105 99)(94 106 100)(95 107 101)(96 108 102)(109 115 121)(110 116 122)(111 117 123)(112 118 124)(113 119 125)(114 120 126)(127 133 139)(128 134 140)(129 135 141)(130 136 142)(131 137 143)(132 138 144)
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 96)(20 97)(21 98)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 106)(30 107)(31 108)(32 91)(33 92)(34 93)(35 94)(36 95)(37 110)(38 111)(39 112)(40 113)(41 114)(42 115)(43 116)(44 117)(45 118)(46 119)(47 120)(48 121)(49 122)(50 123)(51 124)(52 125)(53 126)(54 109)(55 142)(56 143)(57 144)(58 127)(59 128)(60 129)(61 130)(62 131)(63 132)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(71 140)(72 141)
(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)
(1 47 10 38)(2 46 11 37)(3 45 12 54)(4 44 13 53)(5 43 14 52)(6 42 15 51)(7 41 16 50)(8 40 17 49)(9 39 18 48)(19 70 28 61)(20 69 29 60)(21 68 30 59)(22 67 31 58)(23 66 32 57)(24 65 33 56)(25 64 34 55)(26 63 35 72)(27 62 36 71)(73 126 82 117)(74 125 83 116)(75 124 84 115)(76 123 85 114)(77 122 86 113)(78 121 87 112)(79 120 88 111)(80 119 89 110)(81 118 90 109)(91 144 100 135)(92 143 101 134)(93 142 102 133)(94 141 103 132)(95 140 104 131)(96 139 105 130)(97 138 106 129)(98 137 107 128)(99 136 108 127)
G:=sub<Sym(144)| (1,32)(2,33)(3,34)(4,35)(5,36)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(109,133)(110,134)(111,135)(112,136)(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)(120,144)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,67,61)(56,68,62)(57,69,63)(58,70,64)(59,71,65)(60,72,66)(73,85,79)(74,86,80)(75,87,81)(76,88,82)(77,89,83)(78,90,84)(91,103,97)(92,104,98)(93,105,99)(94,106,100)(95,107,101)(96,108,102)(109,115,121)(110,116,122)(111,117,123)(112,118,124)(113,119,125)(114,120,126)(127,133,139)(128,134,140)(129,135,141)(130,136,142)(131,137,143)(132,138,144), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,96)(20,97)(21,98)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,106)(30,107)(31,108)(32,91)(33,92)(34,93)(35,94)(36,95)(37,110)(38,111)(39,112)(40,113)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,109)(55,142)(56,143)(57,144)(58,127)(59,128)(60,129)(61,130)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,140)(72,141), (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), (1,47,10,38)(2,46,11,37)(3,45,12,54)(4,44,13,53)(5,43,14,52)(6,42,15,51)(7,41,16,50)(8,40,17,49)(9,39,18,48)(19,70,28,61)(20,69,29,60)(21,68,30,59)(22,67,31,58)(23,66,32,57)(24,65,33,56)(25,64,34,55)(26,63,35,72)(27,62,36,71)(73,126,82,117)(74,125,83,116)(75,124,84,115)(76,123,85,114)(77,122,86,113)(78,121,87,112)(79,120,88,111)(80,119,89,110)(81,118,90,109)(91,144,100,135)(92,143,101,134)(93,142,102,133)(94,141,103,132)(95,140,104,131)(96,139,105,130)(97,138,106,129)(98,137,107,128)(99,136,108,127)>;
G:=Group( (1,32)(2,33)(3,34)(4,35)(5,36)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(109,133)(110,134)(111,135)(112,136)(113,137)(114,138)(115,139)(116,140)(117,141)(118,142)(119,143)(120,144)(121,127)(122,128)(123,129)(124,130)(125,131)(126,132), (1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,67,61)(56,68,62)(57,69,63)(58,70,64)(59,71,65)(60,72,66)(73,85,79)(74,86,80)(75,87,81)(76,88,82)(77,89,83)(78,90,84)(91,103,97)(92,104,98)(93,105,99)(94,106,100)(95,107,101)(96,108,102)(109,115,121)(110,116,122)(111,117,123)(112,118,124)(113,119,125)(114,120,126)(127,133,139)(128,134,140)(129,135,141)(130,136,142)(131,137,143)(132,138,144), (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,96)(20,97)(21,98)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,106)(30,107)(31,108)(32,91)(33,92)(34,93)(35,94)(36,95)(37,110)(38,111)(39,112)(40,113)(41,114)(42,115)(43,116)(44,117)(45,118)(46,119)(47,120)(48,121)(49,122)(50,123)(51,124)(52,125)(53,126)(54,109)(55,142)(56,143)(57,144)(58,127)(59,128)(60,129)(61,130)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,140)(72,141), (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), (1,47,10,38)(2,46,11,37)(3,45,12,54)(4,44,13,53)(5,43,14,52)(6,42,15,51)(7,41,16,50)(8,40,17,49)(9,39,18,48)(19,70,28,61)(20,69,29,60)(21,68,30,59)(22,67,31,58)(23,66,32,57)(24,65,33,56)(25,64,34,55)(26,63,35,72)(27,62,36,71)(73,126,82,117)(74,125,83,116)(75,124,84,115)(76,123,85,114)(77,122,86,113)(78,121,87,112)(79,120,88,111)(80,119,89,110)(81,118,90,109)(91,144,100,135)(92,143,101,134)(93,142,102,133)(94,141,103,132)(95,140,104,131)(96,139,105,130)(97,138,106,129)(98,137,107,128)(99,136,108,127) );
G=PermutationGroup([[(1,32),(2,33),(3,34),(4,35),(5,36),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,28),(16,29),(17,30),(18,31),(37,65),(38,66),(39,67),(40,68),(41,69),(42,70),(43,71),(44,72),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96),(85,97),(86,98),(87,99),(88,100),(89,101),(90,102),(109,133),(110,134),(111,135),(112,136),(113,137),(114,138),(115,139),(116,140),(117,141),(118,142),(119,143),(120,144),(121,127),(122,128),(123,129),(124,130),(125,131),(126,132)], [(1,7,13),(2,8,14),(3,9,15),(4,10,16),(5,11,17),(6,12,18),(19,25,31),(20,26,32),(21,27,33),(22,28,34),(23,29,35),(24,30,36),(37,49,43),(38,50,44),(39,51,45),(40,52,46),(41,53,47),(42,54,48),(55,67,61),(56,68,62),(57,69,63),(58,70,64),(59,71,65),(60,72,66),(73,85,79),(74,86,80),(75,87,81),(76,88,82),(77,89,83),(78,90,84),(91,103,97),(92,104,98),(93,105,99),(94,106,100),(95,107,101),(96,108,102),(109,115,121),(110,116,122),(111,117,123),(112,118,124),(113,119,125),(114,120,126),(127,133,139),(128,134,140),(129,135,141),(130,136,142),(131,137,143),(132,138,144)], [(1,79),(2,80),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,96),(20,97),(21,98),(22,99),(23,100),(24,101),(25,102),(26,103),(27,104),(28,105),(29,106),(30,107),(31,108),(32,91),(33,92),(34,93),(35,94),(36,95),(37,110),(38,111),(39,112),(40,113),(41,114),(42,115),(43,116),(44,117),(45,118),(46,119),(47,120),(48,121),(49,122),(50,123),(51,124),(52,125),(53,126),(54,109),(55,142),(56,143),(57,144),(58,127),(59,128),(60,129),(61,130),(62,131),(63,132),(64,133),(65,134),(66,135),(67,136),(68,137),(69,138),(70,139),(71,140),(72,141)], [(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)], [(1,47,10,38),(2,46,11,37),(3,45,12,54),(4,44,13,53),(5,43,14,52),(6,42,15,51),(7,41,16,50),(8,40,17,49),(9,39,18,48),(19,70,28,61),(20,69,29,60),(21,68,30,59),(22,67,31,58),(23,66,32,57),(24,65,33,56),(25,64,34,55),(26,63,35,72),(27,62,36,71),(73,126,82,117),(74,125,83,116),(75,124,84,115),(76,123,85,114),(77,122,86,113),(78,121,87,112),(79,120,88,111),(80,119,89,110),(81,118,90,109),(91,144,100,135),(92,143,101,134),(93,142,102,133),(94,141,103,132),(95,140,104,131),(96,139,105,130),(97,138,106,129),(98,137,107,128),(99,136,108,127)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 18A | ··· | 18I | 18J | ··· | 18R | 18S | ··· | 18AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 4 | 9 | 9 | 9 | 9 | 27 | 27 | 27 | 27 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | + | + | - | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | S3 | D6 | D6 | Dic3 | D6 | D6 | D9 | C4×S3 | Dic9 | D18 | D18 | S32 | S3×Dic3 | C2×S32 | S3×D9 | S3×Dic9 | C2×S3×D9 |
| kernel | C2×S3×Dic9 | S3×Dic9 | C6×Dic9 | C2×C9⋊Dic3 | S3×C2×C18 | S3×C18 | C2×Dic9 | S3×C2×C6 | Dic9 | C2×C18 | S3×C6 | S3×C6 | C62 | C22×S3 | C18 | D6 | D6 | C2×C6 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 1 | 4 | 2 | 1 | 3 | 4 | 12 | 6 | 3 | 1 | 2 | 1 | 3 | 6 | 3 |
Matrix representation of C2×S3×Dic9 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 36 | 1 | 0 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 36 | 0 | 0 |
| 36 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 11 | 20 |
| 0 | 0 | 17 | 31 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 36 | 1 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[36,36,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,36,0,0,36,0,0,0,0,0,36,0,0,0,0,36],[36,0,0,0,0,36,0,0,0,0,11,17,0,0,20,31],[6,0,0,0,0,6,0,0,0,0,36,36,0,0,0,1] >;
C2×S3×Dic9 in GAP, Magma, Sage, TeX
C_2\times S_3\times {\rm Dic}_9 % in TeX
G:=Group("C2xS3xDic9"); // GroupNames label
G:=SmallGroup(432,308);
// by ID
G=gap.SmallGroup(432,308);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^18=1,e^2=d^9,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations