direct product, metabelian, supersoluble, monomial
Aliases: C18×D12, C36⋊9D6, C6⋊1(D4×C9), C4⋊2(S3×C18), (C2×C36)⋊6S3, C3⋊1(D4×C18), (C3×C18)⋊4D4, (C6×D12).C3, C12⋊2(C2×C18), (C6×C36)⋊12C2, (C2×C12)⋊3C18, D6⋊1(C2×C18), C3.4(C6×D12), (C6×C12).36C6, (C2×C18).53D6, C6.36(C3×D12), C12.110(S3×C6), C32.2(C6×D4), (S3×C18)⋊5C22, (C22×S3)⋊2C18, (C3×C36)⋊13C22, (C3×D12).11C6, C62.55(C2×C6), C6.3(C22×C18), (C3×C18).30C23, C22.10(S3×C18), C18.51(C22×S3), (C6×C18).28C22, (S3×C2×C18)⋊1C2, (C2×C4)⋊2(S3×C9), (C3×C9)⋊10(C2×D4), (S3×C2×C6).3C6, C6.64(S3×C2×C6), C2.4(S3×C2×C18), (S3×C6).5(C2×C6), (C2×C6).87(S3×C6), (C3×C6).43(C3×D4), (C2×C12).43(C3×S3), (C2×C6).13(C2×C18), (C3×C12).81(C2×C6), (C3×C6).40(C22×C6), SmallGroup(432,346)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C18×D12
G = < a,b,c | a18=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 404 in 178 conjugacy classes, 81 normal (27 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C9, C9, C32, C12, C12, D6, D6, C2×C6, C2×C6, C2×D4, C18, C18, C18, C3×S3, C3×C6, C3×C6, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C3×C9, C36, C36, C2×C18, C2×C18, C3×C12, S3×C6, S3×C6, C62, C2×D12, C6×D4, S3×C9, C3×C18, C3×C18, C2×C36, C2×C36, D4×C9, C22×C18, C3×D12, C6×C12, S3×C2×C6, C3×C36, S3×C18, S3×C18, C6×C18, D4×C18, C6×D12, C9×D12, C6×C36, S3×C2×C18, C18×D12
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, C9, D6, C2×C6, C2×D4, C18, C3×S3, D12, C3×D4, C22×S3, C22×C6, C2×C18, S3×C6, C2×D12, C6×D4, S3×C9, D4×C9, C22×C18, C3×D12, S3×C2×C6, S3×C18, D4×C18, C6×D12, C9×D12, S3×C2×C18, C18×D12
►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 75 113 140 7 81 119 128 13 87 125 134)(2 76 114 141 8 82 120 129 14 88 126 135)(3 77 115 142 9 83 121 130 15 89 109 136)(4 78 116 143 10 84 122 131 16 90 110 137)(5 79 117 144 11 85 123 132 17 73 111 138)(6 80 118 127 12 86 124 133 18 74 112 139)(19 61 50 98 31 55 44 92 25 67 38 104)(20 62 51 99 32 56 45 93 26 68 39 105)(21 63 52 100 33 57 46 94 27 69 40 106)(22 64 53 101 34 58 47 95 28 70 41 107)(23 65 54 102 35 59 48 96 29 71 42 108)(24 66 37 103 36 60 49 97 30 72 43 91)
(1 34)(2 35)(3 36)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(37 115)(38 116)(39 117)(40 118)(41 119)(42 120)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 131)(56 132)(57 133)(58 134)(59 135)(60 136)(61 137)(62 138)(63 139)(64 140)(65 141)(66 142)(67 143)(68 144)(69 127)(70 128)(71 129)(72 130)(73 99)(74 100)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 107)(82 108)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)
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), (1,75,113,140,7,81,119,128,13,87,125,134)(2,76,114,141,8,82,120,129,14,88,126,135)(3,77,115,142,9,83,121,130,15,89,109,136)(4,78,116,143,10,84,122,131,16,90,110,137)(5,79,117,144,11,85,123,132,17,73,111,138)(6,80,118,127,12,86,124,133,18,74,112,139)(19,61,50,98,31,55,44,92,25,67,38,104)(20,62,51,99,32,56,45,93,26,68,39,105)(21,63,52,100,33,57,46,94,27,69,40,106)(22,64,53,101,34,58,47,95,28,70,41,107)(23,65,54,102,35,59,48,96,29,71,42,108)(24,66,37,103,36,60,49,97,30,72,43,91), (1,34)(2,35)(3,36)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,131)(56,132)(57,133)(58,134)(59,135)(60,136)(61,137)(62,138)(63,139)(64,140)(65,141)(66,142)(67,143)(68,144)(69,127)(70,128)(71,129)(72,130)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)>;
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), (1,75,113,140,7,81,119,128,13,87,125,134)(2,76,114,141,8,82,120,129,14,88,126,135)(3,77,115,142,9,83,121,130,15,89,109,136)(4,78,116,143,10,84,122,131,16,90,110,137)(5,79,117,144,11,85,123,132,17,73,111,138)(6,80,118,127,12,86,124,133,18,74,112,139)(19,61,50,98,31,55,44,92,25,67,38,104)(20,62,51,99,32,56,45,93,26,68,39,105)(21,63,52,100,33,57,46,94,27,69,40,106)(22,64,53,101,34,58,47,95,28,70,41,107)(23,65,54,102,35,59,48,96,29,71,42,108)(24,66,37,103,36,60,49,97,30,72,43,91), (1,34)(2,35)(3,36)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(37,115)(38,116)(39,117)(40,118)(41,119)(42,120)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,131)(56,132)(57,133)(58,134)(59,135)(60,136)(61,137)(62,138)(63,139)(64,140)(65,141)(66,142)(67,143)(68,144)(69,127)(70,128)(71,129)(72,130)(73,99)(74,100)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,107)(82,108)(83,91)(84,92)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98) );
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)], [(1,75,113,140,7,81,119,128,13,87,125,134),(2,76,114,141,8,82,120,129,14,88,126,135),(3,77,115,142,9,83,121,130,15,89,109,136),(4,78,116,143,10,84,122,131,16,90,110,137),(5,79,117,144,11,85,123,132,17,73,111,138),(6,80,118,127,12,86,124,133,18,74,112,139),(19,61,50,98,31,55,44,92,25,67,38,104),(20,62,51,99,32,56,45,93,26,68,39,105),(21,63,52,100,33,57,46,94,27,69,40,106),(22,64,53,101,34,58,47,95,28,70,41,107),(23,65,54,102,35,59,48,96,29,71,42,108),(24,66,37,103,36,60,49,97,30,72,43,91)], [(1,34),(2,35),(3,36),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(37,115),(38,116),(39,117),(40,118),(41,119),(42,120),(43,121),(44,122),(45,123),(46,124),(47,125),(48,126),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,131),(56,132),(57,133),(58,134),(59,135),(60,136),(61,137),(62,138),(63,139),(64,140),(65,141),(66,142),(67,143),(68,144),(69,127),(70,128),(71,129),(72,130),(73,99),(74,100),(75,101),(76,102),(77,103),(78,104),(79,105),(80,106),(81,107),(82,108),(83,91),(84,92),(85,93),(86,94),(87,95),(88,96),(89,97),(90,98)]])
162 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6W | 9A | ··· | 9F | 9G | ··· | 9L | 12A | ··· | 12P | 18A | ··· | 18R | 18S | ··· | 18AJ | 18AK | ··· | 18BH | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
162 irreducible representations
| dim | 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 |
| type | + | + | + | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C9 | C18 | C18 | C18 | S3 | D4 | D6 | D6 | C3×S3 | D12 | C3×D4 | S3×C6 | S3×C6 | S3×C9 | D4×C9 | C3×D12 | S3×C18 | S3×C18 | C9×D12 |
| kernel | C18×D12 | C9×D12 | C6×C36 | S3×C2×C18 | C6×D12 | C3×D12 | C6×C12 | S3×C2×C6 | C2×D12 | D12 | C2×C12 | C22×S3 | C2×C36 | C3×C18 | C36 | C2×C18 | C2×C12 | C18 | C3×C6 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 8 | 2 | 4 | 6 | 24 | 6 | 12 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 6 | 12 | 8 | 12 | 6 | 24 |
Matrix representation of C18×D12 ►in GL4(𝔽37) generated by
| 25 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 33 | 0 |
| 0 | 0 | 0 | 33 |
| 10 | 0 | 0 | 0 |
| 0 | 26 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 22 | 14 |
| 0 | 26 | 0 | 0 |
| 10 | 0 | 0 | 0 |
| 0 | 0 | 18 | 15 |
| 0 | 0 | 13 | 19 |
G:=sub<GL(4,GF(37))| [25,0,0,0,0,25,0,0,0,0,33,0,0,0,0,33],[10,0,0,0,0,26,0,0,0,0,8,22,0,0,0,14],[0,10,0,0,26,0,0,0,0,0,18,13,0,0,15,19] >;
C18×D12 in GAP, Magma, Sage, TeX
C_{18}\times D_{12} % in TeX
G:=Group("C18xD12"); // GroupNames label
G:=SmallGroup(432,346);
// by ID
G=gap.SmallGroup(432,346);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,142,192,14118]);
// Polycyclic
G:=Group<a,b,c|a^18=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations