direct product, metabelian, supersoluble, monomial
Aliases: C9×D6⋊C4, D6⋊C36, C18.23D12, (C6×C36)⋊2C2, (C2×C36)⋊1S3, C6.6(D4×C9), (S3×C18)⋊1C4, (C2×C12)⋊1C18, C2.5(S3×C36), C6.4(C2×C36), C2.2(C9×D12), (S3×C6).2C12, C18.25(C4×S3), C6.37(S3×C12), (C6×C12).22C6, (C2×C18).49D6, C6.35(C3×D12), (C3×C18).23D4, (C22×S3).C18, (C2×Dic3)⋊1C18, (Dic3×C18)⋊3C2, C22.6(S3×C18), C62.51(C2×C6), (C6×Dic3).3C6, C18.30(C3⋊D4), (C6×C18).24C22, (C2×C4)⋊1(S3×C9), (C3×D6⋊C4).C3, (S3×C2×C6).2C6, C3⋊1(C9×C22⋊C4), (S3×C2×C18).1C2, C3.4(C3×D6⋊C4), C2.2(C9×C3⋊D4), (C3×C9)⋊4(C22⋊C4), (C2×C6).83(S3×C6), (C2×C12).4(C3×S3), (C2×C6).9(C2×C18), (C3×C6).54(C3×D4), C6.44(C3×C3⋊D4), (C3×C18).17(C2×C4), (C3×C6).41(C2×C12), C32.2(C3×C22⋊C4), SmallGroup(432,135)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9×D6⋊C4
G = < a,b,c,d | a9=b6=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b3c >
Subgroups: 260 in 118 conjugacy classes, 51 normal (45 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C9, C9, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C18, C18, C3×S3, C3×C6, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C3×C9, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, D6⋊C4, C3×C22⋊C4, S3×C9, C3×C18, C2×C36, C2×C36, C22×C18, C6×Dic3, C6×C12, S3×C2×C6, C9×Dic3, C3×C36, S3×C18, S3×C18, C6×C18, C9×C22⋊C4, C3×D6⋊C4, Dic3×C18, C6×C36, S3×C2×C18, C9×D6⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C9, C12, D6, C2×C6, C22⋊C4, C18, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C36, C2×C18, S3×C6, D6⋊C4, C3×C22⋊C4, S3×C9, C2×C36, D4×C9, S3×C12, C3×D12, C3×C3⋊D4, S3×C18, C9×C22⋊C4, C3×D6⋊C4, S3×C36, C9×D12, C9×C3⋊D4, C9×D6⋊C4
►On 144 points
Generators in S144
(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 43 4 37 7 40)(2 44 5 38 8 41)(3 45 6 39 9 42)(10 143 16 140 13 137)(11 144 17 141 14 138)(12 136 18 142 15 139)(19 135 25 132 22 129)(20 127 26 133 23 130)(21 128 27 134 24 131)(28 51 31 54 34 48)(29 52 32 46 35 49)(30 53 33 47 36 50)(55 86 58 89 61 83)(56 87 59 90 62 84)(57 88 60 82 63 85)(64 73 67 76 70 79)(65 74 68 77 71 80)(66 75 69 78 72 81)(91 119 97 125 94 122)(92 120 98 126 95 123)(93 121 99 118 96 124)(100 115 106 112 103 109)(101 116 107 113 104 110)(102 117 108 114 105 111)
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 71)(11 72)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 63)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 123)(29 124)(30 125)(31 126)(32 118)(33 119)(34 120)(35 121)(36 122)(37 112)(38 113)(39 114)(40 115)(41 116)(42 117)(43 109)(44 110)(45 111)(46 99)(47 91)(48 92)(49 93)(50 94)(51 95)(52 96)(53 97)(54 98)(73 139)(74 140)(75 141)(76 142)(77 143)(78 144)(79 136)(80 137)(81 138)(82 135)(83 127)(84 128)(85 129)(86 130)(87 131)(88 132)(89 133)(90 134)
(1 63 35 79)(2 55 36 80)(3 56 28 81)(4 57 29 73)(5 58 30 74)(6 59 31 75)(7 60 32 76)(8 61 33 77)(9 62 34 78)(10 104 127 125)(11 105 128 126)(12 106 129 118)(13 107 130 119)(14 108 131 120)(15 100 132 121)(16 101 133 122)(17 102 134 123)(18 103 135 124)(19 96 136 112)(20 97 137 113)(21 98 138 114)(22 99 139 115)(23 91 140 116)(24 92 141 117)(25 93 142 109)(26 94 143 110)(27 95 144 111)(37 88 52 67)(38 89 53 68)(39 90 54 69)(40 82 46 70)(41 83 47 71)(42 84 48 72)(43 85 49 64)(44 86 50 65)(45 87 51 66)
G:=sub<Sym(144)| (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), (1,43,4,37,7,40)(2,44,5,38,8,41)(3,45,6,39,9,42)(10,143,16,140,13,137)(11,144,17,141,14,138)(12,136,18,142,15,139)(19,135,25,132,22,129)(20,127,26,133,23,130)(21,128,27,134,24,131)(28,51,31,54,34,48)(29,52,32,46,35,49)(30,53,33,47,36,50)(55,86,58,89,61,83)(56,87,59,90,62,84)(57,88,60,82,63,85)(64,73,67,76,70,79)(65,74,68,77,71,80)(66,75,69,78,72,81)(91,119,97,125,94,122)(92,120,98,126,95,123)(93,121,99,118,96,124)(100,115,106,112,103,109)(101,116,107,113,104,110)(102,117,108,114,105,111), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,71)(11,72)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,63)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,123)(29,124)(30,125)(31,126)(32,118)(33,119)(34,120)(35,121)(36,122)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,109)(44,110)(45,111)(46,99)(47,91)(48,92)(49,93)(50,94)(51,95)(52,96)(53,97)(54,98)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,136)(80,137)(81,138)(82,135)(83,127)(84,128)(85,129)(86,130)(87,131)(88,132)(89,133)(90,134), (1,63,35,79)(2,55,36,80)(3,56,28,81)(4,57,29,73)(5,58,30,74)(6,59,31,75)(7,60,32,76)(8,61,33,77)(9,62,34,78)(10,104,127,125)(11,105,128,126)(12,106,129,118)(13,107,130,119)(14,108,131,120)(15,100,132,121)(16,101,133,122)(17,102,134,123)(18,103,135,124)(19,96,136,112)(20,97,137,113)(21,98,138,114)(22,99,139,115)(23,91,140,116)(24,92,141,117)(25,93,142,109)(26,94,143,110)(27,95,144,111)(37,88,52,67)(38,89,53,68)(39,90,54,69)(40,82,46,70)(41,83,47,71)(42,84,48,72)(43,85,49,64)(44,86,50,65)(45,87,51,66)>;
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), (1,43,4,37,7,40)(2,44,5,38,8,41)(3,45,6,39,9,42)(10,143,16,140,13,137)(11,144,17,141,14,138)(12,136,18,142,15,139)(19,135,25,132,22,129)(20,127,26,133,23,130)(21,128,27,134,24,131)(28,51,31,54,34,48)(29,52,32,46,35,49)(30,53,33,47,36,50)(55,86,58,89,61,83)(56,87,59,90,62,84)(57,88,60,82,63,85)(64,73,67,76,70,79)(65,74,68,77,71,80)(66,75,69,78,72,81)(91,119,97,125,94,122)(92,120,98,126,95,123)(93,121,99,118,96,124)(100,115,106,112,103,109)(101,116,107,113,104,110)(102,117,108,114,105,111), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,71)(11,72)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,63)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,123)(29,124)(30,125)(31,126)(32,118)(33,119)(34,120)(35,121)(36,122)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,109)(44,110)(45,111)(46,99)(47,91)(48,92)(49,93)(50,94)(51,95)(52,96)(53,97)(54,98)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,136)(80,137)(81,138)(82,135)(83,127)(84,128)(85,129)(86,130)(87,131)(88,132)(89,133)(90,134), (1,63,35,79)(2,55,36,80)(3,56,28,81)(4,57,29,73)(5,58,30,74)(6,59,31,75)(7,60,32,76)(8,61,33,77)(9,62,34,78)(10,104,127,125)(11,105,128,126)(12,106,129,118)(13,107,130,119)(14,108,131,120)(15,100,132,121)(16,101,133,122)(17,102,134,123)(18,103,135,124)(19,96,136,112)(20,97,137,113)(21,98,138,114)(22,99,139,115)(23,91,140,116)(24,92,141,117)(25,93,142,109)(26,94,143,110)(27,95,144,111)(37,88,52,67)(38,89,53,68)(39,90,54,69)(40,82,46,70)(41,83,47,71)(42,84,48,72)(43,85,49,64)(44,86,50,65)(45,87,51,66) );
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)], [(1,43,4,37,7,40),(2,44,5,38,8,41),(3,45,6,39,9,42),(10,143,16,140,13,137),(11,144,17,141,14,138),(12,136,18,142,15,139),(19,135,25,132,22,129),(20,127,26,133,23,130),(21,128,27,134,24,131),(28,51,31,54,34,48),(29,52,32,46,35,49),(30,53,33,47,36,50),(55,86,58,89,61,83),(56,87,59,90,62,84),(57,88,60,82,63,85),(64,73,67,76,70,79),(65,74,68,77,71,80),(66,75,69,78,72,81),(91,119,97,125,94,122),(92,120,98,126,95,123),(93,121,99,118,96,124),(100,115,106,112,103,109),(101,116,107,113,104,110),(102,117,108,114,105,111)], [(1,100),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,71),(11,72),(12,64),(13,65),(14,66),(15,67),(16,68),(17,69),(18,70),(19,63),(20,55),(21,56),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,123),(29,124),(30,125),(31,126),(32,118),(33,119),(34,120),(35,121),(36,122),(37,112),(38,113),(39,114),(40,115),(41,116),(42,117),(43,109),(44,110),(45,111),(46,99),(47,91),(48,92),(49,93),(50,94),(51,95),(52,96),(53,97),(54,98),(73,139),(74,140),(75,141),(76,142),(77,143),(78,144),(79,136),(80,137),(81,138),(82,135),(83,127),(84,128),(85,129),(86,130),(87,131),(88,132),(89,133),(90,134)], [(1,63,35,79),(2,55,36,80),(3,56,28,81),(4,57,29,73),(5,58,30,74),(6,59,31,75),(7,60,32,76),(8,61,33,77),(9,62,34,78),(10,104,127,125),(11,105,128,126),(12,106,129,118),(13,107,130,119),(14,108,131,120),(15,100,132,121),(16,101,133,122),(17,102,134,123),(18,103,135,124),(19,96,136,112),(20,97,137,113),(21,98,138,114),(22,99,139,115),(23,91,140,116),(24,92,141,117),(25,93,142,109),(26,94,143,110),(27,95,144,111),(37,88,52,67),(38,89,53,68),(39,90,54,69),(40,82,46,70),(41,83,47,71),(42,84,48,72),(43,85,49,64),(44,86,50,65),(45,87,51,66)]])
162 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6O | 6P | 6Q | 6R | 6S | 9A | ··· | 9F | 9G | ··· | 9L | 12A | ··· | 12P | 12Q | 12R | 12S | 12T | 18A | ··· | 18R | 18S | ··· | 18AJ | 18AK | ··· | 18AV | 36A | ··· | 36AJ | 36AK | ··· | 36AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 6 | ··· | 6 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C9 | C12 | C18 | C18 | C18 | C36 | S3 | D4 | D6 | C3×S3 | C4×S3 | D12 | C3⋊D4 | C3×D4 | S3×C6 | S3×C9 | D4×C9 | S3×C12 | C3×D12 | C3×C3⋊D4 | S3×C18 | S3×C36 | C9×D12 | C9×C3⋊D4 |
| kernel | C9×D6⋊C4 | Dic3×C18 | C6×C36 | S3×C2×C18 | C3×D6⋊C4 | S3×C18 | C6×Dic3 | C6×C12 | S3×C2×C6 | D6⋊C4 | S3×C6 | C2×Dic3 | C2×C12 | C22×S3 | D6 | C2×C36 | C3×C18 | C2×C18 | C2×C12 | C18 | C18 | C18 | C3×C6 | C2×C6 | C2×C4 | C6 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 6 | 8 | 6 | 6 | 6 | 24 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 6 | 12 | 4 | 4 | 4 | 6 | 12 | 12 | 12 |
Matrix representation of C9×D6⋊C4 ►in GL4(𝔽37) generated by
| 7 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 26 | 0 | 0 | 0 |
| 11 | 10 | 0 | 0 |
| 0 | 0 | 11 | 25 |
| 0 | 0 | 0 | 27 |
| 27 | 28 | 0 | 0 |
| 11 | 10 | 0 | 0 |
| 0 | 0 | 9 | 19 |
| 0 | 0 | 25 | 28 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 9 |
| 0 | 0 | 0 | 31 |
G:=sub<GL(4,GF(37))| [7,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[26,11,0,0,0,10,0,0,0,0,11,0,0,0,25,27],[27,11,0,0,28,10,0,0,0,0,9,25,0,0,19,28],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,9,31] >;
C9×D6⋊C4 in GAP, Magma, Sage, TeX
C_9\times D_6\rtimes C_4
% in TeX
G:=Group("C9xD6:C4"); // GroupNames label
G:=SmallGroup(432,135);
// by ID
G=gap.SmallGroup(432,135);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,268,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^6=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations