direct product, metabelian, supersoluble, monomial
Aliases: C6×Dic18, C12.74D18, C62.128D6, C9⋊4(C6×Q8), C18⋊3(C3×Q8), (C3×C18)⋊3Q8, (C6×C36).9C2, C4.11(C6×D9), C12.73(S3×C6), (C6×C12).42S3, (C2×C36).17C6, C36.34(C2×C6), (C2×C6).51D18, (C2×C12).20D9, C3.1(C6×Dic6), C6.9(C3×Dic6), C22.8(C6×D9), (C3×C12).205D6, (C2×Dic9).7C6, (C6×Dic9).8C2, Dic9.6(C2×C6), (C3×C6).26Dic6, C6.49(C22×D9), C18.15(C22×C6), (C6×C18).42C22, (C3×C18).38C23, (C3×C36).61C22, C32.4(C2×Dic6), (C3×Dic9).13C22, (C3×C9)⋊6(C2×Q8), C2.3(C2×C6×D9), C6.19(S3×C2×C6), (C2×C4).4(C3×D9), (C2×C6).49(S3×C6), (C2×C12).14(C3×S3), (C2×C18).28(C2×C6), (C3×C6).152(C22×S3), SmallGroup(432,340)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×Dic18
G = < a,b,c | a6=b36=1, c2=b18, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 366 in 130 conjugacy classes, 70 normal (26 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C9, C9, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×Q8, C18, C18, C18, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×C9, Dic9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, C2×Dic6, C6×Q8, C3×C18, C3×C18, Dic18, C2×Dic9, C2×C36, C2×C36, C3×Dic6, C6×Dic3, C6×C12, C3×Dic9, C3×C36, C6×C18, C2×Dic18, C6×Dic6, C3×Dic18, C6×Dic9, C6×C36, C6×Dic18
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, D9, C3×S3, Dic6, C3×Q8, C22×S3, C22×C6, D18, S3×C6, C2×Dic6, C6×Q8, C3×D9, Dic18, C22×D9, C3×Dic6, S3×C2×C6, C6×D9, C2×Dic18, C6×Dic6, C3×Dic18, C2×C6×D9, C6×Dic18
►On 144 points
Generators in S144
(1 47 13 59 25 71)(2 48 14 60 26 72)(3 49 15 61 27 37)(4 50 16 62 28 38)(5 51 17 63 29 39)(6 52 18 64 30 40)(7 53 19 65 31 41)(8 54 20 66 32 42)(9 55 21 67 33 43)(10 56 22 68 34 44)(11 57 23 69 35 45)(12 58 24 70 36 46)(73 129 97 117 85 141)(74 130 98 118 86 142)(75 131 99 119 87 143)(76 132 100 120 88 144)(77 133 101 121 89 109)(78 134 102 122 90 110)(79 135 103 123 91 111)(80 136 104 124 92 112)(81 137 105 125 93 113)(82 138 106 126 94 114)(83 139 107 127 95 115)(84 140 108 128 96 116)
(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 77 19 95)(2 76 20 94)(3 75 21 93)(4 74 22 92)(5 73 23 91)(6 108 24 90)(7 107 25 89)(8 106 26 88)(9 105 27 87)(10 104 28 86)(11 103 29 85)(12 102 30 84)(13 101 31 83)(14 100 32 82)(15 99 33 81)(16 98 34 80)(17 97 35 79)(18 96 36 78)(37 143 55 125)(38 142 56 124)(39 141 57 123)(40 140 58 122)(41 139 59 121)(42 138 60 120)(43 137 61 119)(44 136 62 118)(45 135 63 117)(46 134 64 116)(47 133 65 115)(48 132 66 114)(49 131 67 113)(50 130 68 112)(51 129 69 111)(52 128 70 110)(53 127 71 109)(54 126 72 144)
G:=sub<Sym(144)| (1,47,13,59,25,71)(2,48,14,60,26,72)(3,49,15,61,27,37)(4,50,16,62,28,38)(5,51,17,63,29,39)(6,52,18,64,30,40)(7,53,19,65,31,41)(8,54,20,66,32,42)(9,55,21,67,33,43)(10,56,22,68,34,44)(11,57,23,69,35,45)(12,58,24,70,36,46)(73,129,97,117,85,141)(74,130,98,118,86,142)(75,131,99,119,87,143)(76,132,100,120,88,144)(77,133,101,121,89,109)(78,134,102,122,90,110)(79,135,103,123,91,111)(80,136,104,124,92,112)(81,137,105,125,93,113)(82,138,106,126,94,114)(83,139,107,127,95,115)(84,140,108,128,96,116), (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), (1,77,19,95)(2,76,20,94)(3,75,21,93)(4,74,22,92)(5,73,23,91)(6,108,24,90)(7,107,25,89)(8,106,26,88)(9,105,27,87)(10,104,28,86)(11,103,29,85)(12,102,30,84)(13,101,31,83)(14,100,32,82)(15,99,33,81)(16,98,34,80)(17,97,35,79)(18,96,36,78)(37,143,55,125)(38,142,56,124)(39,141,57,123)(40,140,58,122)(41,139,59,121)(42,138,60,120)(43,137,61,119)(44,136,62,118)(45,135,63,117)(46,134,64,116)(47,133,65,115)(48,132,66,114)(49,131,67,113)(50,130,68,112)(51,129,69,111)(52,128,70,110)(53,127,71,109)(54,126,72,144)>;
G:=Group( (1,47,13,59,25,71)(2,48,14,60,26,72)(3,49,15,61,27,37)(4,50,16,62,28,38)(5,51,17,63,29,39)(6,52,18,64,30,40)(7,53,19,65,31,41)(8,54,20,66,32,42)(9,55,21,67,33,43)(10,56,22,68,34,44)(11,57,23,69,35,45)(12,58,24,70,36,46)(73,129,97,117,85,141)(74,130,98,118,86,142)(75,131,99,119,87,143)(76,132,100,120,88,144)(77,133,101,121,89,109)(78,134,102,122,90,110)(79,135,103,123,91,111)(80,136,104,124,92,112)(81,137,105,125,93,113)(82,138,106,126,94,114)(83,139,107,127,95,115)(84,140,108,128,96,116), (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), (1,77,19,95)(2,76,20,94)(3,75,21,93)(4,74,22,92)(5,73,23,91)(6,108,24,90)(7,107,25,89)(8,106,26,88)(9,105,27,87)(10,104,28,86)(11,103,29,85)(12,102,30,84)(13,101,31,83)(14,100,32,82)(15,99,33,81)(16,98,34,80)(17,97,35,79)(18,96,36,78)(37,143,55,125)(38,142,56,124)(39,141,57,123)(40,140,58,122)(41,139,59,121)(42,138,60,120)(43,137,61,119)(44,136,62,118)(45,135,63,117)(46,134,64,116)(47,133,65,115)(48,132,66,114)(49,131,67,113)(50,130,68,112)(51,129,69,111)(52,128,70,110)(53,127,71,109)(54,126,72,144) );
G=PermutationGroup([[(1,47,13,59,25,71),(2,48,14,60,26,72),(3,49,15,61,27,37),(4,50,16,62,28,38),(5,51,17,63,29,39),(6,52,18,64,30,40),(7,53,19,65,31,41),(8,54,20,66,32,42),(9,55,21,67,33,43),(10,56,22,68,34,44),(11,57,23,69,35,45),(12,58,24,70,36,46),(73,129,97,117,85,141),(74,130,98,118,86,142),(75,131,99,119,87,143),(76,132,100,120,88,144),(77,133,101,121,89,109),(78,134,102,122,90,110),(79,135,103,123,91,111),(80,136,104,124,92,112),(81,137,105,125,93,113),(82,138,106,126,94,114),(83,139,107,127,95,115),(84,140,108,128,96,116)], [(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)], [(1,77,19,95),(2,76,20,94),(3,75,21,93),(4,74,22,92),(5,73,23,91),(6,108,24,90),(7,107,25,89),(8,106,26,88),(9,105,27,87),(10,104,28,86),(11,103,29,85),(12,102,30,84),(13,101,31,83),(14,100,32,82),(15,99,33,81),(16,98,34,80),(17,97,35,79),(18,96,36,78),(37,143,55,125),(38,142,56,124),(39,141,57,123),(40,140,58,122),(41,139,59,121),(42,138,60,120),(43,137,61,119),(44,136,62,118),(45,135,63,117),(46,134,64,116),(47,133,65,115),(48,132,66,114),(49,131,67,113),(50,130,68,112),(51,129,69,111),(52,128,70,110),(53,127,71,109),(54,126,72,144)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6O | 9A | ··· | 9I | 12A | ··· | 12P | 12Q | ··· | 12X | 18A | ··· | 18AA | 36A | ··· | 36AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 18 | ··· | 18 | 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 | Q8 | D6 | D6 | D9 | C3×S3 | C3×Q8 | Dic6 | D18 | S3×C6 | D18 | S3×C6 | C3×D9 | Dic18 | C3×Dic6 | C6×D9 | C6×D9 | C3×Dic18 |
| kernel | C6×Dic18 | C3×Dic18 | C6×Dic9 | C6×C36 | C2×Dic18 | Dic18 | C2×Dic9 | C2×C36 | 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 | 2 | 1 | 2 | 8 | 4 | 2 | 1 | 2 | 2 | 1 | 3 | 2 | 4 | 4 | 6 | 4 | 3 | 2 | 6 | 12 | 8 | 12 | 6 | 24 |
Matrix representation of C6×Dic18 ►in GL3(𝔽37) generated by
| 27 | 0 | 0 |
| 0 | 26 | 0 |
| 0 | 0 | 26 |
| 1 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 30 | 22 |
| 1 | 0 | 0 |
| 0 | 6 | 35 |
| 0 | 0 | 31 |
G:=sub<GL(3,GF(37))| [27,0,0,0,26,0,0,0,26],[1,0,0,0,32,30,0,0,22],[1,0,0,0,6,0,0,35,31] >;
C6×Dic18 in GAP, Magma, Sage, TeX
C_6\times {\rm Dic}_{18} % in TeX
G:=Group("C6xDic18"); // GroupNames label
G:=SmallGroup(432,340);
// by ID
G=gap.SmallGroup(432,340);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,590,142,10085,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^6=b^36=1,c^2=b^18,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations