direct product, metabelian, supersoluble, monomial, A-group
Aliases: Dic3×C2×C18, C62.20C12, (C2×C6)⋊5C36, (C6×C18)⋊3C4, C6⋊2(C2×C36), C3⋊2(C22×C36), (C2×C18).54D6, C23.4(S3×C9), (C2×C62).24C6, (C22×C6).9C18, C62.57(C2×C6), C6.9(C22×C18), C6.34(C6×Dic3), (C22×C18).12S3, C22.11(S3×C18), C18.57(C22×S3), (C6×C18).30C22, (C3×C18).36C23, (C6×Dic3).17C6, C32.3(C22×C12), (C2×C6×C18).3C2, C2.2(S3×C2×C18), C6.70(S3×C2×C6), (C3×C18)⋊7(C2×C4), (C3×C9)⋊8(C22×C4), C3.4(Dic3×C2×C6), (C2×C6).88(S3×C6), (Dic3×C2×C6).2C3, (C2×C6).15(C2×C18), (C3×C6).55(C2×C12), (C3×C6).46(C22×C6), (C22×C6).38(C3×S3), (C2×C6).27(C3×Dic3), (C3×Dic3).21(C2×C6), SmallGroup(432,373)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C2×C18 |
Generators and relations for Dic3×C2×C18
G = < a,b,c,d | a2=b18=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 292 in 194 conjugacy classes, 129 normal (21 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C22×C4, C18, C18, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C3×C9, C36, C2×C18, C2×C18, C3×Dic3, C62, C22×Dic3, C22×C12, C3×C18, C3×C18, C2×C36, C22×C18, C22×C18, C6×Dic3, C2×C62, C9×Dic3, C6×C18, C22×C36, Dic3×C2×C6, Dic3×C18, C2×C6×C18, Dic3×C2×C18
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C9, Dic3, C12, D6, C2×C6, C22×C4, C18, C3×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C36, C2×C18, C3×Dic3, S3×C6, C22×Dic3, C22×C12, S3×C9, C2×C36, C22×C18, C6×Dic3, S3×C2×C6, C9×Dic3, S3×C18, C22×C36, Dic3×C2×C6, Dic3×C18, S3×C2×C18, Dic3×C2×C18
►On 144 points
Generators in S144
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 121)(9 122)(10 123)(11 124)(12 125)(13 126)(14 109)(15 110)(16 111)(17 112)(18 113)(19 50)(20 51)(21 52)(22 53)(23 54)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(55 144)(56 127)(57 128)(58 129)(59 130)(60 131)(61 132)(62 133)(63 134)(64 135)(65 136)(66 137)(67 138)(68 139)(69 140)(70 141)(71 142)(72 143)(73 107)(74 108)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 97)(82 98)(83 99)(84 100)(85 101)(86 102)(87 103)(88 104)(89 105)(90 106)
(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)
(1 102 7 108 13 96)(2 103 8 91 14 97)(3 104 9 92 15 98)(4 105 10 93 16 99)(5 106 11 94 17 100)(6 107 12 95 18 101)(19 63 31 57 25 69)(20 64 32 58 26 70)(21 65 33 59 27 71)(22 66 34 60 28 72)(23 67 35 61 29 55)(24 68 36 62 30 56)(37 139 49 133 43 127)(38 140 50 134 44 128)(39 141 51 135 45 129)(40 142 52 136 46 130)(41 143 53 137 47 131)(42 144 54 138 48 132)(73 125 79 113 85 119)(74 126 80 114 86 120)(75 109 81 115 87 121)(76 110 82 116 88 122)(77 111 83 117 89 123)(78 112 84 118 90 124)
(1 56 108 36)(2 57 91 19)(3 58 92 20)(4 59 93 21)(5 60 94 22)(6 61 95 23)(7 62 96 24)(8 63 97 25)(9 64 98 26)(10 65 99 27)(11 66 100 28)(12 67 101 29)(13 68 102 30)(14 69 103 31)(15 70 104 32)(16 71 105 33)(17 72 106 34)(18 55 107 35)(37 120 133 80)(38 121 134 81)(39 122 135 82)(40 123 136 83)(41 124 137 84)(42 125 138 85)(43 126 139 86)(44 109 140 87)(45 110 141 88)(46 111 142 89)(47 112 143 90)(48 113 144 73)(49 114 127 74)(50 115 128 75)(51 116 129 76)(52 117 130 77)(53 118 131 78)(54 119 132 79)
G:=sub<Sym(144)| (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,109)(15,110)(16,111)(17,112)(18,113)(19,50)(20,51)(21,52)(22,53)(23,54)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(55,144)(56,127)(57,128)(58,129)(59,130)(60,131)(61,132)(62,133)(63,134)(64,135)(65,136)(66,137)(67,138)(68,139)(69,140)(70,141)(71,142)(72,143)(73,107)(74,108)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(81,97)(82,98)(83,99)(84,100)(85,101)(86,102)(87,103)(88,104)(89,105)(90,106), (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), (1,102,7,108,13,96)(2,103,8,91,14,97)(3,104,9,92,15,98)(4,105,10,93,16,99)(5,106,11,94,17,100)(6,107,12,95,18,101)(19,63,31,57,25,69)(20,64,32,58,26,70)(21,65,33,59,27,71)(22,66,34,60,28,72)(23,67,35,61,29,55)(24,68,36,62,30,56)(37,139,49,133,43,127)(38,140,50,134,44,128)(39,141,51,135,45,129)(40,142,52,136,46,130)(41,143,53,137,47,131)(42,144,54,138,48,132)(73,125,79,113,85,119)(74,126,80,114,86,120)(75,109,81,115,87,121)(76,110,82,116,88,122)(77,111,83,117,89,123)(78,112,84,118,90,124), (1,56,108,36)(2,57,91,19)(3,58,92,20)(4,59,93,21)(5,60,94,22)(6,61,95,23)(7,62,96,24)(8,63,97,25)(9,64,98,26)(10,65,99,27)(11,66,100,28)(12,67,101,29)(13,68,102,30)(14,69,103,31)(15,70,104,32)(16,71,105,33)(17,72,106,34)(18,55,107,35)(37,120,133,80)(38,121,134,81)(39,122,135,82)(40,123,136,83)(41,124,137,84)(42,125,138,85)(43,126,139,86)(44,109,140,87)(45,110,141,88)(46,111,142,89)(47,112,143,90)(48,113,144,73)(49,114,127,74)(50,115,128,75)(51,116,129,76)(52,117,130,77)(53,118,131,78)(54,119,132,79)>;
G:=Group( (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,121)(9,122)(10,123)(11,124)(12,125)(13,126)(14,109)(15,110)(16,111)(17,112)(18,113)(19,50)(20,51)(21,52)(22,53)(23,54)(24,37)(25,38)(26,39)(27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(55,144)(56,127)(57,128)(58,129)(59,130)(60,131)(61,132)(62,133)(63,134)(64,135)(65,136)(66,137)(67,138)(68,139)(69,140)(70,141)(71,142)(72,143)(73,107)(74,108)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(81,97)(82,98)(83,99)(84,100)(85,101)(86,102)(87,103)(88,104)(89,105)(90,106), (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), (1,102,7,108,13,96)(2,103,8,91,14,97)(3,104,9,92,15,98)(4,105,10,93,16,99)(5,106,11,94,17,100)(6,107,12,95,18,101)(19,63,31,57,25,69)(20,64,32,58,26,70)(21,65,33,59,27,71)(22,66,34,60,28,72)(23,67,35,61,29,55)(24,68,36,62,30,56)(37,139,49,133,43,127)(38,140,50,134,44,128)(39,141,51,135,45,129)(40,142,52,136,46,130)(41,143,53,137,47,131)(42,144,54,138,48,132)(73,125,79,113,85,119)(74,126,80,114,86,120)(75,109,81,115,87,121)(76,110,82,116,88,122)(77,111,83,117,89,123)(78,112,84,118,90,124), (1,56,108,36)(2,57,91,19)(3,58,92,20)(4,59,93,21)(5,60,94,22)(6,61,95,23)(7,62,96,24)(8,63,97,25)(9,64,98,26)(10,65,99,27)(11,66,100,28)(12,67,101,29)(13,68,102,30)(14,69,103,31)(15,70,104,32)(16,71,105,33)(17,72,106,34)(18,55,107,35)(37,120,133,80)(38,121,134,81)(39,122,135,82)(40,123,136,83)(41,124,137,84)(42,125,138,85)(43,126,139,86)(44,109,140,87)(45,110,141,88)(46,111,142,89)(47,112,143,90)(48,113,144,73)(49,114,127,74)(50,115,128,75)(51,116,129,76)(52,117,130,77)(53,118,131,78)(54,119,132,79) );
G=PermutationGroup([[(1,114),(2,115),(3,116),(4,117),(5,118),(6,119),(7,120),(8,121),(9,122),(10,123),(11,124),(12,125),(13,126),(14,109),(15,110),(16,111),(17,112),(18,113),(19,50),(20,51),(21,52),(22,53),(23,54),(24,37),(25,38),(26,39),(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(55,144),(56,127),(57,128),(58,129),(59,130),(60,131),(61,132),(62,133),(63,134),(64,135),(65,136),(66,137),(67,138),(68,139),(69,140),(70,141),(71,142),(72,143),(73,107),(74,108),(75,91),(76,92),(77,93),(78,94),(79,95),(80,96),(81,97),(82,98),(83,99),(84,100),(85,101),(86,102),(87,103),(88,104),(89,105),(90,106)], [(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)], [(1,102,7,108,13,96),(2,103,8,91,14,97),(3,104,9,92,15,98),(4,105,10,93,16,99),(5,106,11,94,17,100),(6,107,12,95,18,101),(19,63,31,57,25,69),(20,64,32,58,26,70),(21,65,33,59,27,71),(22,66,34,60,28,72),(23,67,35,61,29,55),(24,68,36,62,30,56),(37,139,49,133,43,127),(38,140,50,134,44,128),(39,141,51,135,45,129),(40,142,52,136,46,130),(41,143,53,137,47,131),(42,144,54,138,48,132),(73,125,79,113,85,119),(74,126,80,114,86,120),(75,109,81,115,87,121),(76,110,82,116,88,122),(77,111,83,117,89,123),(78,112,84,118,90,124)], [(1,56,108,36),(2,57,91,19),(3,58,92,20),(4,59,93,21),(5,60,94,22),(6,61,95,23),(7,62,96,24),(8,63,97,25),(9,64,98,26),(10,65,99,27),(11,66,100,28),(12,67,101,29),(13,68,102,30),(14,69,103,31),(15,70,104,32),(16,71,105,33),(17,72,106,34),(18,55,107,35),(37,120,133,80),(38,121,134,81),(39,122,135,82),(40,123,136,83),(41,124,137,84),(42,125,138,85),(43,126,139,86),(44,109,140,87),(45,110,141,88),(46,111,142,89),(47,112,143,90),(48,113,144,73),(49,114,127,74),(50,115,128,75),(51,116,129,76),(52,117,130,77),(53,118,131,78),(54,119,132,79)]])
216 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4H | 6A | ··· | 6N | 6O | ··· | 6AI | 9A | ··· | 9F | 9G | ··· | 9L | 12A | ··· | 12P | 18A | ··· | 18AP | 18AQ | ··· | 18CF | 36A | ··· | 36AV |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
216 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C9 | C12 | C18 | C18 | C36 | S3 | Dic3 | D6 | C3×S3 | C3×Dic3 | S3×C6 | S3×C9 | C9×Dic3 | S3×C18 |
| kernel | Dic3×C2×C18 | Dic3×C18 | C2×C6×C18 | Dic3×C2×C6 | C6×C18 | C6×Dic3 | C2×C62 | C22×Dic3 | C62 | C2×Dic3 | C22×C6 | C2×C6 | C22×C18 | C2×C18 | C2×C18 | C22×C6 | C2×C6 | C2×C6 | C23 | C22 | C22 |
| # reps | 1 | 6 | 1 | 2 | 8 | 12 | 2 | 6 | 16 | 36 | 6 | 48 | 1 | 4 | 3 | 2 | 8 | 6 | 6 | 24 | 18 |
Matrix representation of Dic3×C2×C18 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 21 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 36 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 27 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 36 | 0 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[21,0,0,0,0,9,0,0,0,0,12,0,0,0,0,12],[36,0,0,0,0,36,0,0,0,0,11,0,0,0,0,27],[6,0,0,0,0,6,0,0,0,0,0,36,0,0,1,0] >;
Dic3×C2×C18 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_2\times C_{18} % in TeX
G:=Group("Dic3xC2xC18"); // GroupNames label
G:=SmallGroup(432,373);
// by ID
G=gap.SmallGroup(432,373);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,192,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^18=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations