direct product, metabelian, supersoluble, monomial
Aliases: C3×D18⋊C4, D18⋊3C12, C6.23D36, C62.121D6, (C6×C36)⋊1C2, (C2×C36)⋊9C6, (C6×D9)⋊2C4, (C2×C12)⋊1D9, C2.2(C3×D36), C6.25(C4×D9), C2.5(C12×D9), C6.13(S3×C12), (C6×C12).24S3, (C6×Dic9)⋊3C2, (C2×Dic9)⋊7C6, C18.21(C3×D4), (C2×C6).49D18, C6.11(C3×D12), (C3×C6).53D12, (C3×C18).28D4, C22.6(C6×D9), C18.19(C2×C12), C6.30(C9⋊D4), (C22×D9).3C6, (C6×C18).35C22, C32.4(D6⋊C4), (C2×C4)⋊1(C3×D9), (C2×C6×D9).2C2, C9⋊4(C3×C22⋊C4), C3.1(C3×D6⋊C4), C2.2(C3×C9⋊D4), (C3×C9)⋊5(C22⋊C4), (C3×C6).67(C4×S3), (C2×C12).3(C3×S3), (C2×C6).37(S3×C6), C6.13(C3×C3⋊D4), (C2×C18).26(C2×C6), (C3×C18).22(C2×C4), (C3×C6).91(C3⋊D4), SmallGroup(432,134)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D18⋊C4
G = < a,b,c,d | a3=b18=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b9c >
Subgroups: 470 in 118 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, D9, C18, C18, C3×S3, C3×C6, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C3×C9, Dic9, C36, D18, D18, C2×C18, C2×C18, C3×Dic3, C3×C12, S3×C6, C62, D6⋊C4, C3×C22⋊C4, C3×D9, C3×C18, C2×Dic9, C2×C36, C2×C36, C22×D9, C6×Dic3, C6×C12, S3×C2×C6, C3×Dic9, C3×C36, C6×D9, C6×D9, C6×C18, D18⋊C4, C3×D6⋊C4, C6×Dic9, C6×C36, C2×C6×D9, C3×D18⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, D9, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, D18, S3×C6, D6⋊C4, C3×C22⋊C4, C3×D9, C4×D9, D36, C9⋊D4, S3×C12, C3×D12, C3×C3⋊D4, C6×D9, D18⋊C4, C3×D6⋊C4, C12×D9, C3×D36, C3×C9⋊D4, C3×D18⋊C4
►On 144 points
Generators in S144
(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 43 49)(38 44 50)(39 45 51)(40 46 52)(41 47 53)(42 48 54)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 72)(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)(109 121 115)(110 122 116)(111 123 117)(112 124 118)(113 125 119)(114 126 120)(127 139 133)(128 140 134)(129 141 135)(130 142 136)(131 143 137)(132 144 138)
(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 80)(2 79)(3 78)(4 77)(5 76)(6 75)(7 74)(8 73)(9 90)(10 89)(11 88)(12 87)(13 86)(14 85)(15 84)(16 83)(17 82)(18 81)(19 92)(20 91)(21 108)(22 107)(23 106)(24 105)(25 104)(26 103)(27 102)(28 101)(29 100)(30 99)(31 98)(32 97)(33 96)(34 95)(35 94)(36 93)(37 113)(38 112)(39 111)(40 110)(41 109)(42 126)(43 125)(44 124)(45 123)(46 122)(47 121)(48 120)(49 119)(50 118)(51 117)(52 116)(53 115)(54 114)(55 135)(56 134)(57 133)(58 132)(59 131)(60 130)(61 129)(62 128)(63 127)(64 144)(65 143)(66 142)(67 141)(68 140)(69 139)(70 138)(71 137)(72 136)
(1 55 31 38)(2 56 32 39)(3 57 33 40)(4 58 34 41)(5 59 35 42)(6 60 36 43)(7 61 19 44)(8 62 20 45)(9 63 21 46)(10 64 22 47)(11 65 23 48)(12 66 24 49)(13 67 25 50)(14 68 26 51)(15 69 27 52)(16 70 28 53)(17 71 29 54)(18 72 30 37)(73 137 91 114)(74 138 92 115)(75 139 93 116)(76 140 94 117)(77 141 95 118)(78 142 96 119)(79 143 97 120)(80 144 98 121)(81 127 99 122)(82 128 100 123)(83 129 101 124)(84 130 102 125)(85 131 103 126)(86 132 104 109)(87 133 105 110)(88 134 106 111)(89 135 107 112)(90 136 108 113)
G:=sub<Sym(144)| (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)(37,43,49)(38,44,50)(39,45,51)(40,46,52)(41,47,53)(42,48,54)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72)(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)(109,121,115)(110,122,116)(111,123,117)(112,124,118)(113,125,119)(114,126,120)(127,139,133)(128,140,134)(129,141,135)(130,142,136)(131,143,137)(132,144,138), (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), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,90)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,92)(20,91)(21,108)(22,107)(23,106)(24,105)(25,104)(26,103)(27,102)(28,101)(29,100)(30,99)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,113)(38,112)(39,111)(40,110)(41,109)(42,126)(43,125)(44,124)(45,123)(46,122)(47,121)(48,120)(49,119)(50,118)(51,117)(52,116)(53,115)(54,114)(55,135)(56,134)(57,133)(58,132)(59,131)(60,130)(61,129)(62,128)(63,127)(64,144)(65,143)(66,142)(67,141)(68,140)(69,139)(70,138)(71,137)(72,136), (1,55,31,38)(2,56,32,39)(3,57,33,40)(4,58,34,41)(5,59,35,42)(6,60,36,43)(7,61,19,44)(8,62,20,45)(9,63,21,46)(10,64,22,47)(11,65,23,48)(12,66,24,49)(13,67,25,50)(14,68,26,51)(15,69,27,52)(16,70,28,53)(17,71,29,54)(18,72,30,37)(73,137,91,114)(74,138,92,115)(75,139,93,116)(76,140,94,117)(77,141,95,118)(78,142,96,119)(79,143,97,120)(80,144,98,121)(81,127,99,122)(82,128,100,123)(83,129,101,124)(84,130,102,125)(85,131,103,126)(86,132,104,109)(87,133,105,110)(88,134,106,111)(89,135,107,112)(90,136,108,113)>;
G:=Group( (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)(37,43,49)(38,44,50)(39,45,51)(40,46,52)(41,47,53)(42,48,54)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72)(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)(109,121,115)(110,122,116)(111,123,117)(112,124,118)(113,125,119)(114,126,120)(127,139,133)(128,140,134)(129,141,135)(130,142,136)(131,143,137)(132,144,138), (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), (1,80)(2,79)(3,78)(4,77)(5,76)(6,75)(7,74)(8,73)(9,90)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,92)(20,91)(21,108)(22,107)(23,106)(24,105)(25,104)(26,103)(27,102)(28,101)(29,100)(30,99)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,113)(38,112)(39,111)(40,110)(41,109)(42,126)(43,125)(44,124)(45,123)(46,122)(47,121)(48,120)(49,119)(50,118)(51,117)(52,116)(53,115)(54,114)(55,135)(56,134)(57,133)(58,132)(59,131)(60,130)(61,129)(62,128)(63,127)(64,144)(65,143)(66,142)(67,141)(68,140)(69,139)(70,138)(71,137)(72,136), (1,55,31,38)(2,56,32,39)(3,57,33,40)(4,58,34,41)(5,59,35,42)(6,60,36,43)(7,61,19,44)(8,62,20,45)(9,63,21,46)(10,64,22,47)(11,65,23,48)(12,66,24,49)(13,67,25,50)(14,68,26,51)(15,69,27,52)(16,70,28,53)(17,71,29,54)(18,72,30,37)(73,137,91,114)(74,138,92,115)(75,139,93,116)(76,140,94,117)(77,141,95,118)(78,142,96,119)(79,143,97,120)(80,144,98,121)(81,127,99,122)(82,128,100,123)(83,129,101,124)(84,130,102,125)(85,131,103,126)(86,132,104,109)(87,133,105,110)(88,134,106,111)(89,135,107,112)(90,136,108,113) );
G=PermutationGroup([[(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),(37,43,49),(38,44,50),(39,45,51),(40,46,52),(41,47,53),(42,48,54),(55,61,67),(56,62,68),(57,63,69),(58,64,70),(59,65,71),(60,66,72),(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),(109,121,115),(110,122,116),(111,123,117),(112,124,118),(113,125,119),(114,126,120),(127,139,133),(128,140,134),(129,141,135),(130,142,136),(131,143,137),(132,144,138)], [(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)], [(1,80),(2,79),(3,78),(4,77),(5,76),(6,75),(7,74),(8,73),(9,90),(10,89),(11,88),(12,87),(13,86),(14,85),(15,84),(16,83),(17,82),(18,81),(19,92),(20,91),(21,108),(22,107),(23,106),(24,105),(25,104),(26,103),(27,102),(28,101),(29,100),(30,99),(31,98),(32,97),(33,96),(34,95),(35,94),(36,93),(37,113),(38,112),(39,111),(40,110),(41,109),(42,126),(43,125),(44,124),(45,123),(46,122),(47,121),(48,120),(49,119),(50,118),(51,117),(52,116),(53,115),(54,114),(55,135),(56,134),(57,133),(58,132),(59,131),(60,130),(61,129),(62,128),(63,127),(64,144),(65,143),(66,142),(67,141),(68,140),(69,139),(70,138),(71,137),(72,136)], [(1,55,31,38),(2,56,32,39),(3,57,33,40),(4,58,34,41),(5,59,35,42),(6,60,36,43),(7,61,19,44),(8,62,20,45),(9,63,21,46),(10,64,22,47),(11,65,23,48),(12,66,24,49),(13,67,25,50),(14,68,26,51),(15,69,27,52),(16,70,28,53),(17,71,29,54),(18,72,30,37),(73,137,91,114),(74,138,92,115),(75,139,93,116),(76,140,94,117),(77,141,95,118),(78,142,96,119),(79,143,97,120),(80,144,98,121),(81,127,99,122),(82,128,100,123),(83,129,101,124),(84,130,102,125),(85,131,103,126),(86,132,104,109),(87,133,105,110),(88,134,106,111),(89,135,107,112),(90,136,108,113)]])
126 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 | ··· | 9I | 12A | ··· | 12P | 12Q | 12R | 12S | 12T | 18A | ··· | 18AA | 36A | ··· | 36AJ |
| 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 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 1 | ··· | 1 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D4 | D6 | D9 | C3×S3 | C3×D4 | C4×S3 | D12 | C3⋊D4 | D18 | S3×C6 | C3×D9 | C4×D9 | D36 | C9⋊D4 | S3×C12 | C3×D12 | C3×C3⋊D4 | C6×D9 | C12×D9 | C3×D36 | C3×C9⋊D4 |
| kernel | C3×D18⋊C4 | C6×Dic9 | C6×C36 | C2×C6×D9 | D18⋊C4 | C6×D9 | C2×Dic9 | C2×C36 | C22×D9 | D18 | C6×C12 | C3×C18 | C62 | C2×C12 | C2×C12 | C18 | C3×C6 | C3×C6 | C3×C6 | C2×C6 | C2×C6 | C2×C4 | C6 | C6 | C6 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 2 | 1 | 3 | 2 | 4 | 2 | 2 | 2 | 3 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 6 | 12 | 12 | 12 |
Matrix representation of C3×D18⋊C4 ►in GL3(𝔽37) generated by
| 26 | 0 | 0 |
| 0 | 26 | 0 |
| 0 | 0 | 26 |
| 1 | 0 | 0 |
| 0 | 30 | 0 |
| 0 | 0 | 21 |
| 36 | 0 | 0 |
| 0 | 0 | 21 |
| 0 | 30 | 0 |
| 6 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(37))| [26,0,0,0,26,0,0,0,26],[1,0,0,0,30,0,0,0,21],[36,0,0,0,0,30,0,21,0],[6,0,0,0,36,0,0,0,1] >;
C3×D18⋊C4 in GAP, Magma, Sage, TeX
C_3\times D_{18}\rtimes C_4 % in TeX
G:=Group("C3xD18:C4"); // GroupNames label
G:=SmallGroup(432,134);
// by ID
G=gap.SmallGroup(432,134);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^18=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^9*c>;
// generators/relations