metabelian, supersoluble, monomial
Aliases: Dic18.1C6, 3- 1+2⋊2Q16, C9⋊C8.C6, C9⋊C24.C2, C9⋊Q16⋊C3, C9⋊2(C3×Q16), C36.3(C2×C6), C18.9(C3×D4), (Q8×C9).1C6, C12.11(S3×C6), (C3×C12).14D6, Q8.3(C9⋊C6), C32.(C3⋊Q16), C36.C6.1C2, (Q8×C32).2S3, C2.6(Dic9⋊C6), (C2×3- 1+2).9D4, (Q8×3- 1+2).1C2, (C4×3- 1+2).3C22, C4.3(C2×C9⋊C6), C6.28(C3×C3⋊D4), C3.3(C3×C3⋊Q16), (C3×Q8).25(C3×S3), (C3×C6).29(C3⋊D4), SmallGroup(432,162)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic18.C6
G = < a,b,c | a36=1, b2=c6=a18, bab-1=a-1, cac-1=a7, cbc-1=a9b >
Subgroups: 190 in 60 conjugacy classes, 26 normal (all characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, Q8, C9, C9, C32, Dic3, C12, C12, Q16, C18, C18, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C3×Q8, 3- 1+2, Dic9, C36, C36, C3×Dic3, C3×C12, C3×C12, C3⋊Q16, C3×Q16, C2×3- 1+2, C9⋊C8, Dic18, Q8×C9, Q8×C9, C3×C3⋊C8, C3×Dic6, Q8×C32, C9⋊C12, C4×3- 1+2, C4×3- 1+2, C9⋊Q16, C3×C3⋊Q16, C9⋊C24, C36.C6, Q8×3- 1+2, Dic18.C6
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, C3×S3, C3⋊D4, C3×D4, S3×C6, C3⋊Q16, C3×Q16, C9⋊C6, C3×C3⋊D4, C2×C9⋊C6, C3×C3⋊Q16, Dic9⋊C6, Dic18.C6
►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)
(1 101 19 83)(2 100 20 82)(3 99 21 81)(4 98 22 80)(5 97 23 79)(6 96 24 78)(7 95 25 77)(8 94 26 76)(9 93 27 75)(10 92 28 74)(11 91 29 73)(12 90 30 108)(13 89 31 107)(14 88 32 106)(15 87 33 105)(16 86 34 104)(17 85 35 103)(18 84 36 102)(37 135 55 117)(38 134 56 116)(39 133 57 115)(40 132 58 114)(41 131 59 113)(42 130 60 112)(43 129 61 111)(44 128 62 110)(45 127 63 109)(46 126 64 144)(47 125 65 143)(48 124 66 142)(49 123 67 141)(50 122 68 140)(51 121 69 139)(52 120 70 138)(53 119 71 137)(54 118 72 136)
(1 72 19 54)(2 67 8 37 14 43 20 49 26 55 32 61)(3 62 33 56 27 50 21 44 15 38 9 68)(4 57 22 39)(5 52 11 58 17 64 23 70 29 40 35 46)(6 47 36 41 30 71 24 65 18 59 12 53)(7 42 25 60)(10 63 28 45)(13 48 31 66)(16 69 34 51)(73 123 103 117 97 111 91 141 85 135 79 129)(74 118 92 136)(75 113 81 119 87 125 93 131 99 137 105 143)(76 144 106 138 100 132 94 126 88 120 82 114)(77 139 95 121)(78 134 84 140 90 110 96 116 102 122 108 128)(80 124 98 142)(83 109 101 127)(86 130 104 112)(89 115 107 133)
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), (1,101,19,83)(2,100,20,82)(3,99,21,81)(4,98,22,80)(5,97,23,79)(6,96,24,78)(7,95,25,77)(8,94,26,76)(9,93,27,75)(10,92,28,74)(11,91,29,73)(12,90,30,108)(13,89,31,107)(14,88,32,106)(15,87,33,105)(16,86,34,104)(17,85,35,103)(18,84,36,102)(37,135,55,117)(38,134,56,116)(39,133,57,115)(40,132,58,114)(41,131,59,113)(42,130,60,112)(43,129,61,111)(44,128,62,110)(45,127,63,109)(46,126,64,144)(47,125,65,143)(48,124,66,142)(49,123,67,141)(50,122,68,140)(51,121,69,139)(52,120,70,138)(53,119,71,137)(54,118,72,136), (1,72,19,54)(2,67,8,37,14,43,20,49,26,55,32,61)(3,62,33,56,27,50,21,44,15,38,9,68)(4,57,22,39)(5,52,11,58,17,64,23,70,29,40,35,46)(6,47,36,41,30,71,24,65,18,59,12,53)(7,42,25,60)(10,63,28,45)(13,48,31,66)(16,69,34,51)(73,123,103,117,97,111,91,141,85,135,79,129)(74,118,92,136)(75,113,81,119,87,125,93,131,99,137,105,143)(76,144,106,138,100,132,94,126,88,120,82,114)(77,139,95,121)(78,134,84,140,90,110,96,116,102,122,108,128)(80,124,98,142)(83,109,101,127)(86,130,104,112)(89,115,107,133)>;
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), (1,101,19,83)(2,100,20,82)(3,99,21,81)(4,98,22,80)(5,97,23,79)(6,96,24,78)(7,95,25,77)(8,94,26,76)(9,93,27,75)(10,92,28,74)(11,91,29,73)(12,90,30,108)(13,89,31,107)(14,88,32,106)(15,87,33,105)(16,86,34,104)(17,85,35,103)(18,84,36,102)(37,135,55,117)(38,134,56,116)(39,133,57,115)(40,132,58,114)(41,131,59,113)(42,130,60,112)(43,129,61,111)(44,128,62,110)(45,127,63,109)(46,126,64,144)(47,125,65,143)(48,124,66,142)(49,123,67,141)(50,122,68,140)(51,121,69,139)(52,120,70,138)(53,119,71,137)(54,118,72,136), (1,72,19,54)(2,67,8,37,14,43,20,49,26,55,32,61)(3,62,33,56,27,50,21,44,15,38,9,68)(4,57,22,39)(5,52,11,58,17,64,23,70,29,40,35,46)(6,47,36,41,30,71,24,65,18,59,12,53)(7,42,25,60)(10,63,28,45)(13,48,31,66)(16,69,34,51)(73,123,103,117,97,111,91,141,85,135,79,129)(74,118,92,136)(75,113,81,119,87,125,93,131,99,137,105,143)(76,144,106,138,100,132,94,126,88,120,82,114)(77,139,95,121)(78,134,84,140,90,110,96,116,102,122,108,128)(80,124,98,142)(83,109,101,127)(86,130,104,112)(89,115,107,133) );
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)], [(1,101,19,83),(2,100,20,82),(3,99,21,81),(4,98,22,80),(5,97,23,79),(6,96,24,78),(7,95,25,77),(8,94,26,76),(9,93,27,75),(10,92,28,74),(11,91,29,73),(12,90,30,108),(13,89,31,107),(14,88,32,106),(15,87,33,105),(16,86,34,104),(17,85,35,103),(18,84,36,102),(37,135,55,117),(38,134,56,116),(39,133,57,115),(40,132,58,114),(41,131,59,113),(42,130,60,112),(43,129,61,111),(44,128,62,110),(45,127,63,109),(46,126,64,144),(47,125,65,143),(48,124,66,142),(49,123,67,141),(50,122,68,140),(51,121,69,139),(52,120,70,138),(53,119,71,137),(54,118,72,136)], [(1,72,19,54),(2,67,8,37,14,43,20,49,26,55,32,61),(3,62,33,56,27,50,21,44,15,38,9,68),(4,57,22,39),(5,52,11,58,17,64,23,70,29,40,35,46),(6,47,36,41,30,71,24,65,18,59,12,53),(7,42,25,60),(10,63,28,45),(13,48,31,66),(16,69,34,51),(73,123,103,117,97,111,91,141,85,135,79,129),(74,118,92,136),(75,113,81,119,87,125,93,131,99,137,105,143),(76,144,106,138,100,132,94,126,88,120,82,114),(77,139,95,121),(78,134,84,140,90,110,96,116,102,122,108,128),(80,124,98,142),(83,109,101,127),(86,130,104,112),(89,115,107,133)]])
41 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 18A | 18B | 18C | 24A | 24B | 24C | 24D | 36A | ··· | 36I |
| order | 1 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 3 | 3 | 2 | 4 | 36 | 2 | 3 | 3 | 18 | 18 | 6 | 6 | 6 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 36 | 36 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 12 | ··· | 12 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | + | + | - | - | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | Dic18.C6 | S3 | D4 | D6 | Q16 | C3×S3 | C3×D4 | C3⋊D4 | S3×C6 | C3×Q16 | C3×C3⋊D4 | C3⋊Q16 | C3×C3⋊Q16 | C9⋊C6 | C2×C9⋊C6 | Dic9⋊C6 |
| kernel | Dic18.C6 | C9⋊C24 | C36.C6 | Q8×3- 1+2 | C9⋊Q16 | C9⋊C8 | Dic18 | Q8×C9 | C1 | Q8×C32 | C2×3- 1+2 | C3×C12 | 3- 1+2 | C3×Q8 | C18 | C3×C6 | C12 | C9 | C6 | C32 | C3 | Q8 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 1 | 2 |
Matrix representation of Dic18.C6 ►in GL10(𝔽73)
| 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 72 | 72 |
| 0 | 0 | 0 | 0 | 72 | 72 | 1 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 1 | 1 | 0 | 0 |
| 41 | 45 | 22 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 32 | 23 | 51 | 0 | 0 | 0 | 0 | 0 | 0 |
| 22 | 1 | 32 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 23 | 51 | 60 | 41 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 71 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 | 0 | 0 | 62 | 60 |
| 0 | 0 | 0 | 0 | 62 | 0 | 0 | 0 | 71 | 11 |
| 0 | 0 | 0 | 0 | 0 | 11 | 62 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 0 | 71 | 11 | 0 | 0 |
| 65 | 16 | 58 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 57 | 8 | 43 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 58 | 30 | 8 | 57 | 0 | 0 | 0 | 0 | 0 | 0 |
| 43 | 15 | 16 | 65 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 72 | 72 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(10,GF(73))| [0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,0,0,0,72,72,0,72,0,72,0,0,0,0,72,72,0,72,72,0,0,0,0,0,2,1,1,1,1,1,0,0,0,0,1,2,1,1,1,1,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0,0,0],[41,13,22,23,0,0,0,0,0,0,45,32,1,51,0,0,0,0,0,0,22,23,32,60,0,0,0,0,0,0,1,51,28,41,0,0,0,0,0,0,0,0,0,0,62,60,0,62,0,62,0,0,0,0,71,11,11,0,11,0,0,0,0,0,0,0,0,0,62,71,0,0,0,0,0,0,0,0,60,11,0,0,0,0,0,0,62,71,0,0,0,0,0,0,0,0,60,11,0,0],[65,57,58,43,0,0,0,0,0,0,16,8,30,15,0,0,0,0,0,0,58,43,8,16,0,0,0,0,0,0,30,15,57,65,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,72,0] >;
Dic18.C6 in GAP, Magma, Sage, TeX
{\rm Dic}_{18}.C_6 % in TeX
G:=Group("Dic18.C6"); // GroupNames label
G:=SmallGroup(432,162);
// by ID
G=gap.SmallGroup(432,162);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,176,1011,514,80,10085,2035,292,14118]);
// Polycyclic
G:=Group<a,b,c|a^36=1,b^2=c^6=a^18,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b>;
// generators/relations