metabelian, supersoluble, monomial
Aliases: C18.2Dic6, C6.2Dic18, C62.55D6, C6.3(C4×D9), C18.4(C4×S3), C9⋊Dic3⋊2C4, (C3×C18).2Q8, (C2×C6).10D18, (C3×C18).14D4, (C2×C18).10D6, C22.6(S3×D9), C3⋊2(Dic9⋊C4), C9⋊1(Dic3⋊C4), C6.12(C9⋊D4), (C6×C18).4C22, (C6×Dic9).3C2, (C2×Dic3).2D9, (C6×Dic3).2S3, (C2×Dic9).2S3, (C3×C6).13Dic6, C2.1(D6⋊D9), C18.11(C3⋊D4), (Dic3×C18).2C2, C6.4(C6.D6), C6.2(C32⋊2Q8), C6.16(D6⋊S3), C2.2(C9⋊Dic6), C2.4(C18.D6), C32.3(Dic3⋊C4), C3.1(C62.C22), (C3×C9)⋊2(C4⋊C4), (C2×C6).16S32, (C3×C6).39(C4×S3), (C3×C18).9(C2×C4), (C2×C9⋊Dic3).3C2, (C3×C6).50(C3⋊D4), SmallGroup(432,89)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C18.Dic6
G = < a,b,c | a6=b36=1, c2=b18, bab-1=a-1, ac=ca, cbc-1=a3b-1 >
Subgroups: 444 in 94 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C18, C18, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C3×C9, Dic9, C36, C2×C18, C2×C18, C3×Dic3, C3⋊Dic3, C62, Dic3⋊C4, C3×C18, C2×Dic9, C2×Dic9, C2×C36, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C3×Dic9, C9×Dic3, C9⋊Dic3, C6×C18, Dic9⋊C4, C62.C22, C6×Dic9, Dic3×C18, C2×C9⋊Dic3, C18.Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, D9, Dic6, C4×S3, C3⋊D4, D18, S32, Dic3⋊C4, Dic18, C4×D9, C9⋊D4, C6.D6, D6⋊S3, C32⋊2Q8, S3×D9, Dic9⋊C4, C62.C22, C9⋊Dic6, C18.D6, D6⋊D9, C18.Dic6
►On 144 points
Generators in S144
(1 47 13 59 25 71)(2 72 26 60 14 48)(3 49 15 61 27 37)(4 38 28 62 16 50)(5 51 17 63 29 39)(6 40 30 64 18 52)(7 53 19 65 31 41)(8 42 32 66 20 54)(9 55 21 67 33 43)(10 44 34 68 22 56)(11 57 23 69 35 45)(12 46 36 70 24 58)(73 119 85 131 97 143)(74 144 98 132 86 120)(75 121 87 133 99 109)(76 110 100 134 88 122)(77 123 89 135 101 111)(78 112 102 136 90 124)(79 125 91 137 103 113)(80 114 104 138 92 126)(81 127 93 139 105 115)(82 116 106 140 94 128)(83 129 95 141 107 117)(84 118 108 142 96 130)
(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)
(1 132 19 114)(2 73 20 91)(3 130 21 112)(4 107 22 89)(5 128 23 110)(6 105 24 87)(7 126 25 144)(8 103 26 85)(9 124 27 142)(10 101 28 83)(11 122 29 140)(12 99 30 81)(13 120 31 138)(14 97 32 79)(15 118 33 136)(16 95 34 77)(17 116 35 134)(18 93 36 75)(37 96 55 78)(38 117 56 135)(39 94 57 76)(40 115 58 133)(41 92 59 74)(42 113 60 131)(43 90 61 108)(44 111 62 129)(45 88 63 106)(46 109 64 127)(47 86 65 104)(48 143 66 125)(49 84 67 102)(50 141 68 123)(51 82 69 100)(52 139 70 121)(53 80 71 98)(54 137 72 119)
G:=sub<Sym(144)| (1,47,13,59,25,71)(2,72,26,60,14,48)(3,49,15,61,27,37)(4,38,28,62,16,50)(5,51,17,63,29,39)(6,40,30,64,18,52)(7,53,19,65,31,41)(8,42,32,66,20,54)(9,55,21,67,33,43)(10,44,34,68,22,56)(11,57,23,69,35,45)(12,46,36,70,24,58)(73,119,85,131,97,143)(74,144,98,132,86,120)(75,121,87,133,99,109)(76,110,100,134,88,122)(77,123,89,135,101,111)(78,112,102,136,90,124)(79,125,91,137,103,113)(80,114,104,138,92,126)(81,127,93,139,105,115)(82,116,106,140,94,128)(83,129,95,141,107,117)(84,118,108,142,96,130), (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), (1,132,19,114)(2,73,20,91)(3,130,21,112)(4,107,22,89)(5,128,23,110)(6,105,24,87)(7,126,25,144)(8,103,26,85)(9,124,27,142)(10,101,28,83)(11,122,29,140)(12,99,30,81)(13,120,31,138)(14,97,32,79)(15,118,33,136)(16,95,34,77)(17,116,35,134)(18,93,36,75)(37,96,55,78)(38,117,56,135)(39,94,57,76)(40,115,58,133)(41,92,59,74)(42,113,60,131)(43,90,61,108)(44,111,62,129)(45,88,63,106)(46,109,64,127)(47,86,65,104)(48,143,66,125)(49,84,67,102)(50,141,68,123)(51,82,69,100)(52,139,70,121)(53,80,71,98)(54,137,72,119)>;
G:=Group( (1,47,13,59,25,71)(2,72,26,60,14,48)(3,49,15,61,27,37)(4,38,28,62,16,50)(5,51,17,63,29,39)(6,40,30,64,18,52)(7,53,19,65,31,41)(8,42,32,66,20,54)(9,55,21,67,33,43)(10,44,34,68,22,56)(11,57,23,69,35,45)(12,46,36,70,24,58)(73,119,85,131,97,143)(74,144,98,132,86,120)(75,121,87,133,99,109)(76,110,100,134,88,122)(77,123,89,135,101,111)(78,112,102,136,90,124)(79,125,91,137,103,113)(80,114,104,138,92,126)(81,127,93,139,105,115)(82,116,106,140,94,128)(83,129,95,141,107,117)(84,118,108,142,96,130), (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), (1,132,19,114)(2,73,20,91)(3,130,21,112)(4,107,22,89)(5,128,23,110)(6,105,24,87)(7,126,25,144)(8,103,26,85)(9,124,27,142)(10,101,28,83)(11,122,29,140)(12,99,30,81)(13,120,31,138)(14,97,32,79)(15,118,33,136)(16,95,34,77)(17,116,35,134)(18,93,36,75)(37,96,55,78)(38,117,56,135)(39,94,57,76)(40,115,58,133)(41,92,59,74)(42,113,60,131)(43,90,61,108)(44,111,62,129)(45,88,63,106)(46,109,64,127)(47,86,65,104)(48,143,66,125)(49,84,67,102)(50,141,68,123)(51,82,69,100)(52,139,70,121)(53,80,71,98)(54,137,72,119) );
G=PermutationGroup([[(1,47,13,59,25,71),(2,72,26,60,14,48),(3,49,15,61,27,37),(4,38,28,62,16,50),(5,51,17,63,29,39),(6,40,30,64,18,52),(7,53,19,65,31,41),(8,42,32,66,20,54),(9,55,21,67,33,43),(10,44,34,68,22,56),(11,57,23,69,35,45),(12,46,36,70,24,58),(73,119,85,131,97,143),(74,144,98,132,86,120),(75,121,87,133,99,109),(76,110,100,134,88,122),(77,123,89,135,101,111),(78,112,102,136,90,124),(79,125,91,137,103,113),(80,114,104,138,92,126),(81,127,93,139,105,115),(82,116,106,140,94,128),(83,129,95,141,107,117),(84,118,108,142,96,130)], [(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)], [(1,132,19,114),(2,73,20,91),(3,130,21,112),(4,107,22,89),(5,128,23,110),(6,105,24,87),(7,126,25,144),(8,103,26,85),(9,124,27,142),(10,101,28,83),(11,122,29,140),(12,99,30,81),(13,120,31,138),(14,97,32,79),(15,118,33,136),(16,95,34,77),(17,116,35,134),(18,93,36,75),(37,96,55,78),(38,117,56,135),(39,94,57,76),(40,115,58,133),(41,92,59,74),(42,113,60,131),(43,90,61,108),(44,111,62,129),(45,88,63,106),(46,109,64,127),(47,86,65,104),(48,143,66,125),(49,84,67,102),(50,141,68,123),(51,82,69,100),(52,139,70,121),(53,80,71,98),(54,137,72,119)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | ··· | 18I | 18J | ··· | 18R | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 6 | 6 | 18 | 18 | 54 | 54 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | - | + | - | + | + | - | - | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | S3 | D4 | Q8 | D6 | D6 | D9 | Dic6 | C4×S3 | C3⋊D4 | Dic6 | C4×S3 | C3⋊D4 | D18 | Dic18 | C4×D9 | C9⋊D4 | S32 | C6.D6 | D6⋊S3 | C32⋊2Q8 | S3×D9 | C9⋊Dic6 | C18.D6 | D6⋊D9 |
| kernel | C18.Dic6 | C6×Dic9 | Dic3×C18 | C2×C9⋊Dic3 | C9⋊Dic3 | C2×Dic9 | C6×Dic3 | C3×C18 | C3×C18 | C2×C18 | C62 | C2×Dic3 | C18 | C18 | C18 | C3×C6 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C6 | C2×C6 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
Matrix representation of C18.Dic6 ►in GL6(𝔽37)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 36 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 24 | 26 | 0 | 0 | 0 | 0 |
| 2 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 8 | 0 | 0 |
| 0 | 0 | 12 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 6 |
| 0 | 0 | 0 | 0 | 31 | 17 |
| 32 | 27 | 0 | 0 | 0 | 0 |
| 10 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 23 | 0 | 0 |
| 0 | 0 | 14 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 36 |
G:=sub<GL(6,GF(37))| [0,36,0,0,0,0,1,36,0,0,0,0,0,0,0,1,0,0,0,0,36,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[24,2,0,0,0,0,26,13,0,0,0,0,0,0,33,12,0,0,0,0,8,4,0,0,0,0,0,0,11,31,0,0,0,0,6,17],[32,10,0,0,0,0,27,5,0,0,0,0,0,0,30,14,0,0,0,0,23,7,0,0,0,0,0,0,1,1,0,0,0,0,0,36] >;
C18.Dic6 in GAP, Magma, Sage, TeX
C_{18}.{\rm Dic}_6 % in TeX
G:=Group("C18.Dic6"); // GroupNames label
G:=SmallGroup(432,89);
// by ID
G=gap.SmallGroup(432,89);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,36,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c|a^6=b^36=1,c^2=b^18,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b^-1>;
// generators/relations