direct product, metabelian, supersoluble, monomial
Aliases: C6×D36, C12⋊9D18, C62.130D6, C4⋊2(C6×D9), C9⋊5(C6×D4), C36⋊7(C2×C6), (C6×C36)⋊6C2, (C2×C12)⋊6D9, C18⋊4(C3×D4), (C3×C18)⋊5D4, (C2×C36)⋊11C6, D18⋊5(C2×C6), C3.1(C6×D12), C12.74(S3×C6), (C6×C12).43S3, (C3×C36)⋊9C22, C6.16(C3×D12), (C3×C6).59D12, (C2×C6).53D18, (C22×D9)⋊7C6, (C6×D9)⋊8C22, (C3×C12).206D6, C6.51(C22×D9), C32.4(C2×D12), C22.10(C6×D9), C18.17(C22×C6), (C6×C18).44C22, (C3×C18).40C23, (C2×C6×D9)⋊4C2, C2.4(C2×C6×D9), (C2×C4)⋊2(C3×D9), (C3×C9)⋊11(C2×D4), C6.21(S3×C2×C6), (C2×C6).51(S3×C6), (C2×C12).15(C3×S3), (C2×C18).30(C2×C6), (C3×C6).154(C22×S3), SmallGroup(432,343)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×D36
G = < a,b,c | a6=b36=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 782 in 178 conjugacy classes, 70 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C9, C9, C32, C12, C12, D6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C18, C3×S3, C3×C6, C3×C6, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C3×C9, C36, C36, D18, D18, C2×C18, C2×C18, C3×C12, S3×C6, C62, C2×D12, C6×D4, C3×D9, C3×C18, C3×C18, D36, C2×C36, C2×C36, C22×D9, C3×D12, C6×C12, S3×C2×C6, C3×C36, C6×D9, C6×D9, C6×C18, C2×D36, C6×D12, C3×D36, C6×C36, C2×C6×D9, C6×D36
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, D9, C3×S3, D12, C3×D4, C22×S3, C22×C6, D18, S3×C6, C2×D12, C6×D4, C3×D9, D36, C22×D9, C3×D12, S3×C2×C6, C6×D9, C2×D36, C6×D12, C3×D36, C2×C6×D9, C6×D36
►On 144 points
Generators in S144
(1 63 25 51 13 39)(2 64 26 52 14 40)(3 65 27 53 15 41)(4 66 28 54 16 42)(5 67 29 55 17 43)(6 68 30 56 18 44)(7 69 31 57 19 45)(8 70 32 58 20 46)(9 71 33 59 21 47)(10 72 34 60 22 48)(11 37 35 61 23 49)(12 38 36 62 24 50)(73 110 85 122 97 134)(74 111 86 123 98 135)(75 112 87 124 99 136)(76 113 88 125 100 137)(77 114 89 126 101 138)(78 115 90 127 102 139)(79 116 91 128 103 140)(80 117 92 129 104 141)(81 118 93 130 105 142)(82 119 94 131 106 143)(83 120 95 132 107 144)(84 121 96 133 108 109)
(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)
(1 117)(2 116)(3 115)(4 114)(5 113)(6 112)(7 111)(8 110)(9 109)(10 144)(11 143)(12 142)(13 141)(14 140)(15 139)(16 138)(17 137)(18 136)(19 135)(20 134)(21 133)(22 132)(23 131)(24 130)(25 129)(26 128)(27 127)(28 126)(29 125)(30 124)(31 123)(32 122)(33 121)(34 120)(35 119)(36 118)(37 82)(38 81)(39 80)(40 79)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 108)(48 107)(49 106)(50 105)(51 104)(52 103)(53 102)(54 101)(55 100)(56 99)(57 98)(58 97)(59 96)(60 95)(61 94)(62 93)(63 92)(64 91)(65 90)(66 89)(67 88)(68 87)(69 86)(70 85)(71 84)(72 83)
G:=sub<Sym(144)| (1,63,25,51,13,39)(2,64,26,52,14,40)(3,65,27,53,15,41)(4,66,28,54,16,42)(5,67,29,55,17,43)(6,68,30,56,18,44)(7,69,31,57,19,45)(8,70,32,58,20,46)(9,71,33,59,21,47)(10,72,34,60,22,48)(11,37,35,61,23,49)(12,38,36,62,24,50)(73,110,85,122,97,134)(74,111,86,123,98,135)(75,112,87,124,99,136)(76,113,88,125,100,137)(77,114,89,126,101,138)(78,115,90,127,102,139)(79,116,91,128,103,140)(80,117,92,129,104,141)(81,118,93,130,105,142)(82,119,94,131,106,143)(83,120,95,132,107,144)(84,121,96,133,108,109), (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), (1,117)(2,116)(3,115)(4,114)(5,113)(6,112)(7,111)(8,110)(9,109)(10,144)(11,143)(12,142)(13,141)(14,140)(15,139)(16,138)(17,137)(18,136)(19,135)(20,134)(21,133)(22,132)(23,131)(24,130)(25,129)(26,128)(27,127)(28,126)(29,125)(30,124)(31,123)(32,122)(33,121)(34,120)(35,119)(36,118)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)>;
G:=Group( (1,63,25,51,13,39)(2,64,26,52,14,40)(3,65,27,53,15,41)(4,66,28,54,16,42)(5,67,29,55,17,43)(6,68,30,56,18,44)(7,69,31,57,19,45)(8,70,32,58,20,46)(9,71,33,59,21,47)(10,72,34,60,22,48)(11,37,35,61,23,49)(12,38,36,62,24,50)(73,110,85,122,97,134)(74,111,86,123,98,135)(75,112,87,124,99,136)(76,113,88,125,100,137)(77,114,89,126,101,138)(78,115,90,127,102,139)(79,116,91,128,103,140)(80,117,92,129,104,141)(81,118,93,130,105,142)(82,119,94,131,106,143)(83,120,95,132,107,144)(84,121,96,133,108,109), (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), (1,117)(2,116)(3,115)(4,114)(5,113)(6,112)(7,111)(8,110)(9,109)(10,144)(11,143)(12,142)(13,141)(14,140)(15,139)(16,138)(17,137)(18,136)(19,135)(20,134)(21,133)(22,132)(23,131)(24,130)(25,129)(26,128)(27,127)(28,126)(29,125)(30,124)(31,123)(32,122)(33,121)(34,120)(35,119)(36,118)(37,82)(38,81)(39,80)(40,79)(41,78)(42,77)(43,76)(44,75)(45,74)(46,73)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83) );
G=PermutationGroup([[(1,63,25,51,13,39),(2,64,26,52,14,40),(3,65,27,53,15,41),(4,66,28,54,16,42),(5,67,29,55,17,43),(6,68,30,56,18,44),(7,69,31,57,19,45),(8,70,32,58,20,46),(9,71,33,59,21,47),(10,72,34,60,22,48),(11,37,35,61,23,49),(12,38,36,62,24,50),(73,110,85,122,97,134),(74,111,86,123,98,135),(75,112,87,124,99,136),(76,113,88,125,100,137),(77,114,89,126,101,138),(78,115,90,127,102,139),(79,116,91,128,103,140),(80,117,92,129,104,141),(81,118,93,130,105,142),(82,119,94,131,106,143),(83,120,95,132,107,144),(84,121,96,133,108,109)], [(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)], [(1,117),(2,116),(3,115),(4,114),(5,113),(6,112),(7,111),(8,110),(9,109),(10,144),(11,143),(12,142),(13,141),(14,140),(15,139),(16,138),(17,137),(18,136),(19,135),(20,134),(21,133),(22,132),(23,131),(24,130),(25,129),(26,128),(27,127),(28,126),(29,125),(30,124),(31,123),(32,122),(33,121),(34,120),(35,119),(36,118),(37,82),(38,81),(39,80),(40,79),(41,78),(42,77),(43,76),(44,75),(45,74),(46,73),(47,108),(48,107),(49,106),(50,105),(51,104),(52,103),(53,102),(54,101),(55,100),(56,99),(57,98),(58,97),(59,96),(60,95),(61,94),(62,93),(63,92),(64,91),(65,90),(66,89),(67,88),(68,87),(69,86),(70,85),(71,84),(72,83)]])
126 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 | ··· | 9I | 12A | ··· | 12P | 18A | ··· | 18AA | 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 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 18 | 18 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 18 | ··· | 18 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 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 | C6 | C6 | C6 | S3 | D4 | D6 | D6 | D9 | C3×S3 | C3×D4 | D12 | D18 | S3×C6 | D18 | S3×C6 | C3×D9 | D36 | C3×D12 | C6×D9 | C6×D9 | C3×D36 |
| kernel | C6×D36 | C3×D36 | C6×C36 | C2×C6×D9 | C2×D36 | D36 | C2×C36 | C22×D9 | C6×C12 | C3×C18 | C3×C12 | C62 | C2×C12 | C2×C12 | C18 | C3×C6 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 8 | 2 | 4 | 1 | 2 | 2 | 1 | 3 | 2 | 4 | 4 | 6 | 4 | 3 | 2 | 6 | 12 | 8 | 12 | 6 | 24 |
Matrix representation of C6×D36 ►in GL3(𝔽37) generated by
| 11 | 0 | 0 |
| 0 | 27 | 0 |
| 0 | 0 | 27 |
| 1 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 22 |
| 36 | 0 | 0 |
| 0 | 0 | 21 |
| 0 | 30 | 0 |
G:=sub<GL(3,GF(37))| [11,0,0,0,27,0,0,0,27],[1,0,0,0,32,0,0,0,22],[36,0,0,0,0,30,0,21,0] >;
C6×D36 in GAP, Magma, Sage, TeX
C_6\times D_{36} % in TeX
G:=Group("C6xD36"); // GroupNames label
G:=SmallGroup(432,343);
// by ID
G=gap.SmallGroup(432,343);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,142,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^6=b^36=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations