direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C6×Dic9, C62.132D6, C62.21Dic3, (C6×C18)⋊6C4, C18⋊5(C2×C12), (C2×C18)⋊11C12, C9⋊5(C22×C12), (C2×C6).54D18, C23.4(C3×D9), (C2×C62).23S3, C6.21(C6×Dic3), C6.57(C22×D9), C22.11(C6×D9), (C22×C6).12D9, C18.23(C22×C6), (C6×C18).46C22, (C3×C18).46C23, (C22×C18).15C6, C32.3(C22×Dic3), C2.2(C2×C6×D9), C6.43(S3×C2×C6), (C2×C6×C18).6C2, (C3×C18)⋊8(C2×C4), (C3×C9)⋊9(C22×C4), C3.1(Dic3×C2×C6), (C2×C6).61(S3×C6), (C2×C18).32(C2×C6), (C22×C6).27(C3×S3), (C2×C6).21(C3×Dic3), (C3×C6).64(C2×Dic3), (C3×C6).160(C22×S3), SmallGroup(432,372)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C2×C6×Dic9 |
Generators and relations for C2×C6×Dic9
G = < a,b,c,d | a2=b6=c18=1, d2=c9, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 446 in 194 conjugacy classes, 118 normal (20 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, Dic9, C2×C18, C2×C18, C3×Dic3, C62, C22×Dic3, C22×C12, C3×C18, C3×C18, C2×Dic9, C22×C18, C22×C18, C6×Dic3, C2×C62, C3×Dic9, C6×C18, C22×Dic9, Dic3×C2×C6, C6×Dic9, C2×C6×C18, C2×C6×Dic9
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, Dic3, C12, D6, C2×C6, C22×C4, D9, C3×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, Dic9, D18, C3×Dic3, S3×C6, C22×Dic3, C22×C12, C3×D9, C2×Dic9, C22×D9, C6×Dic3, S3×C2×C6, C3×Dic9, C6×D9, C22×Dic9, Dic3×C2×C6, C6×Dic9, C2×C6×D9, C2×C6×Dic9
►On 144 points
Generators in S144
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(73 125)(74 126)(75 109)(76 110)(77 111)(78 112)(79 113)(80 114)(81 115)(82 116)(83 117)(84 118)(85 119)(86 120)(87 121)(88 122)(89 123)(90 124)(91 129)(92 130)(93 131)(94 132)(95 133)(96 134)(97 135)(98 136)(99 137)(100 138)(101 139)(102 140)(103 141)(104 142)(105 143)(106 144)(107 127)(108 128)
(1 34 13 28 7 22)(2 35 14 29 8 23)(3 36 15 30 9 24)(4 19 16 31 10 25)(5 20 17 32 11 26)(6 21 18 33 12 27)(37 70 49 64 43 58)(38 71 50 65 44 59)(39 72 51 66 45 60)(40 55 52 67 46 61)(41 56 53 68 47 62)(42 57 54 69 48 63)(73 103 79 91 85 97)(74 104 80 92 86 98)(75 105 81 93 87 99)(76 106 82 94 88 100)(77 107 83 95 89 101)(78 108 84 96 90 102)(109 143 115 131 121 137)(110 144 116 132 122 138)(111 127 117 133 123 139)(112 128 118 134 124 140)(113 129 119 135 125 141)(114 130 120 136 126 142)
(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 88 10 79)(2 87 11 78)(3 86 12 77)(4 85 13 76)(5 84 14 75)(6 83 15 74)(7 82 16 73)(8 81 17 90)(9 80 18 89)(19 97 28 106)(20 96 29 105)(21 95 30 104)(22 94 31 103)(23 93 32 102)(24 92 33 101)(25 91 34 100)(26 108 35 99)(27 107 36 98)(37 113 46 122)(38 112 47 121)(39 111 48 120)(40 110 49 119)(41 109 50 118)(42 126 51 117)(43 125 52 116)(44 124 53 115)(45 123 54 114)(55 144 64 135)(56 143 65 134)(57 142 66 133)(58 141 67 132)(59 140 68 131)(60 139 69 130)(61 138 70 129)(62 137 71 128)(63 136 72 127)
G:=sub<Sym(144)| (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(73,125)(74,126)(75,109)(76,110)(77,111)(78,112)(79,113)(80,114)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,121)(88,122)(89,123)(90,124)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,127)(108,128), (1,34,13,28,7,22)(2,35,14,29,8,23)(3,36,15,30,9,24)(4,19,16,31,10,25)(5,20,17,32,11,26)(6,21,18,33,12,27)(37,70,49,64,43,58)(38,71,50,65,44,59)(39,72,51,66,45,60)(40,55,52,67,46,61)(41,56,53,68,47,62)(42,57,54,69,48,63)(73,103,79,91,85,97)(74,104,80,92,86,98)(75,105,81,93,87,99)(76,106,82,94,88,100)(77,107,83,95,89,101)(78,108,84,96,90,102)(109,143,115,131,121,137)(110,144,116,132,122,138)(111,127,117,133,123,139)(112,128,118,134,124,140)(113,129,119,135,125,141)(114,130,120,136,126,142), (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), (1,88,10,79)(2,87,11,78)(3,86,12,77)(4,85,13,76)(5,84,14,75)(6,83,15,74)(7,82,16,73)(8,81,17,90)(9,80,18,89)(19,97,28,106)(20,96,29,105)(21,95,30,104)(22,94,31,103)(23,93,32,102)(24,92,33,101)(25,91,34,100)(26,108,35,99)(27,107,36,98)(37,113,46,122)(38,112,47,121)(39,111,48,120)(40,110,49,119)(41,109,50,118)(42,126,51,117)(43,125,52,116)(44,124,53,115)(45,123,54,114)(55,144,64,135)(56,143,65,134)(57,142,66,133)(58,141,67,132)(59,140,68,131)(60,139,69,130)(61,138,70,129)(62,137,71,128)(63,136,72,127)>;
G:=Group( (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,64)(20,65)(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(73,125)(74,126)(75,109)(76,110)(77,111)(78,112)(79,113)(80,114)(81,115)(82,116)(83,117)(84,118)(85,119)(86,120)(87,121)(88,122)(89,123)(90,124)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,127)(108,128), (1,34,13,28,7,22)(2,35,14,29,8,23)(3,36,15,30,9,24)(4,19,16,31,10,25)(5,20,17,32,11,26)(6,21,18,33,12,27)(37,70,49,64,43,58)(38,71,50,65,44,59)(39,72,51,66,45,60)(40,55,52,67,46,61)(41,56,53,68,47,62)(42,57,54,69,48,63)(73,103,79,91,85,97)(74,104,80,92,86,98)(75,105,81,93,87,99)(76,106,82,94,88,100)(77,107,83,95,89,101)(78,108,84,96,90,102)(109,143,115,131,121,137)(110,144,116,132,122,138)(111,127,117,133,123,139)(112,128,118,134,124,140)(113,129,119,135,125,141)(114,130,120,136,126,142), (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), (1,88,10,79)(2,87,11,78)(3,86,12,77)(4,85,13,76)(5,84,14,75)(6,83,15,74)(7,82,16,73)(8,81,17,90)(9,80,18,89)(19,97,28,106)(20,96,29,105)(21,95,30,104)(22,94,31,103)(23,93,32,102)(24,92,33,101)(25,91,34,100)(26,108,35,99)(27,107,36,98)(37,113,46,122)(38,112,47,121)(39,111,48,120)(40,110,49,119)(41,109,50,118)(42,126,51,117)(43,125,52,116)(44,124,53,115)(45,123,54,114)(55,144,64,135)(56,143,65,134)(57,142,66,133)(58,141,67,132)(59,140,68,131)(60,139,69,130)(61,138,70,129)(62,137,71,128)(63,136,72,127) );
G=PermutationGroup([[(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,64),(20,65),(21,66),(22,67),(23,68),(24,69),(25,70),(26,71),(27,72),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,61),(35,62),(36,63),(73,125),(74,126),(75,109),(76,110),(77,111),(78,112),(79,113),(80,114),(81,115),(82,116),(83,117),(84,118),(85,119),(86,120),(87,121),(88,122),(89,123),(90,124),(91,129),(92,130),(93,131),(94,132),(95,133),(96,134),(97,135),(98,136),(99,137),(100,138),(101,139),(102,140),(103,141),(104,142),(105,143),(106,144),(107,127),(108,128)], [(1,34,13,28,7,22),(2,35,14,29,8,23),(3,36,15,30,9,24),(4,19,16,31,10,25),(5,20,17,32,11,26),(6,21,18,33,12,27),(37,70,49,64,43,58),(38,71,50,65,44,59),(39,72,51,66,45,60),(40,55,52,67,46,61),(41,56,53,68,47,62),(42,57,54,69,48,63),(73,103,79,91,85,97),(74,104,80,92,86,98),(75,105,81,93,87,99),(76,106,82,94,88,100),(77,107,83,95,89,101),(78,108,84,96,90,102),(109,143,115,131,121,137),(110,144,116,132,122,138),(111,127,117,133,123,139),(112,128,118,134,124,140),(113,129,119,135,125,141),(114,130,120,136,126,142)], [(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)], [(1,88,10,79),(2,87,11,78),(3,86,12,77),(4,85,13,76),(5,84,14,75),(6,83,15,74),(7,82,16,73),(8,81,17,90),(9,80,18,89),(19,97,28,106),(20,96,29,105),(21,95,30,104),(22,94,31,103),(23,93,32,102),(24,92,33,101),(25,91,34,100),(26,108,35,99),(27,107,36,98),(37,113,46,122),(38,112,47,121),(39,111,48,120),(40,110,49,119),(41,109,50,118),(42,126,51,117),(43,125,52,116),(44,124,53,115),(45,123,54,114),(55,144,64,135),(56,143,65,134),(57,142,66,133),(58,141,67,132),(59,140,68,131),(60,139,69,130),(61,138,70,129),(62,137,71,128),(63,136,72,127)]])
144 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4H | 6A | ··· | 6N | 6O | ··· | 6AI | 9A | ··· | 9I | 12A | ··· | 12P | 18A | ··· | 18BK |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 2 | 2 | 2 | 9 | ··· | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 9 | ··· | 9 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | |||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | Dic3 | D6 | D9 | C3×S3 | Dic9 | D18 | C3×Dic3 | S3×C6 | C3×D9 | C3×Dic9 | C6×D9 |
| kernel | C2×C6×Dic9 | C6×Dic9 | C2×C6×C18 | C22×Dic9 | C6×C18 | C2×Dic9 | C22×C18 | C2×C18 | C2×C62 | C62 | C62 | C22×C6 | C22×C6 | C2×C6 | C2×C6 | C2×C6 | C2×C6 | C23 | C22 | C22 |
| # reps | 1 | 6 | 1 | 2 | 8 | 12 | 2 | 16 | 1 | 4 | 3 | 3 | 2 | 12 | 9 | 8 | 6 | 6 | 24 | 18 |
Matrix representation of C2×C6×Dic9 ►in GL4(𝔽37) generated by
| 1 | 0 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 36 |
| 11 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 36 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 14 | 7 |
| 31 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 23 | 9 |
| 0 | 0 | 3 | 14 |
G:=sub<GL(4,GF(37))| [1,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[11,0,0,0,0,27,0,0,0,0,10,0,0,0,0,10],[36,0,0,0,0,1,0,0,0,0,16,14,0,0,0,7],[31,0,0,0,0,1,0,0,0,0,23,3,0,0,9,14] >;
C2×C6×Dic9 in GAP, Magma, Sage, TeX
C_2\times C_6\times {\rm Dic}_9 % in TeX
G:=Group("C2xC6xDic9"); // GroupNames label
G:=SmallGroup(432,372);
// by ID
G=gap.SmallGroup(432,372);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^18=1,d^2=c^9,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