direct product, metacyclic, supersoluble, monomial, A-group
Aliases: Dic3×C27, C3⋊C108, C6.C54, C54.4S3, C32.2C36, (C3×C27)⋊1C4, C2.(S3×C27), C6.8(S3×C9), (C3×C6).5C18, (C3×C54).1C2, (C3×C9).5C12, (C3×Dic3).C9, (C9×Dic3).C3, C18.10(C3×S3), (C3×C18).21C6, C3.4(C9×Dic3), C9.4(C3×Dic3), SmallGroup(324,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C27 |
Generators and relations for Dic3×C27
G = < a,b,c | a27=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C27
►On 108 points
Generators in S108
(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)
(1 42 10 51 19 33)(2 43 11 52 20 34)(3 44 12 53 21 35)(4 45 13 54 22 36)(5 46 14 28 23 37)(6 47 15 29 24 38)(7 48 16 30 25 39)(8 49 17 31 26 40)(9 50 18 32 27 41)(55 108 73 99 64 90)(56 82 74 100 65 91)(57 83 75 101 66 92)(58 84 76 102 67 93)(59 85 77 103 68 94)(60 86 78 104 69 95)(61 87 79 105 70 96)(62 88 80 106 71 97)(63 89 81 107 72 98)
(1 68 51 85)(2 69 52 86)(3 70 53 87)(4 71 54 88)(5 72 28 89)(6 73 29 90)(7 74 30 91)(8 75 31 92)(9 76 32 93)(10 77 33 94)(11 78 34 95)(12 79 35 96)(13 80 36 97)(14 81 37 98)(15 55 38 99)(16 56 39 100)(17 57 40 101)(18 58 41 102)(19 59 42 103)(20 60 43 104)(21 61 44 105)(22 62 45 106)(23 63 46 107)(24 64 47 108)(25 65 48 82)(26 66 49 83)(27 67 50 84)
G:=sub<Sym(108)| (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), (1,42,10,51,19,33)(2,43,11,52,20,34)(3,44,12,53,21,35)(4,45,13,54,22,36)(5,46,14,28,23,37)(6,47,15,29,24,38)(7,48,16,30,25,39)(8,49,17,31,26,40)(9,50,18,32,27,41)(55,108,73,99,64,90)(56,82,74,100,65,91)(57,83,75,101,66,92)(58,84,76,102,67,93)(59,85,77,103,68,94)(60,86,78,104,69,95)(61,87,79,105,70,96)(62,88,80,106,71,97)(63,89,81,107,72,98), (1,68,51,85)(2,69,52,86)(3,70,53,87)(4,71,54,88)(5,72,28,89)(6,73,29,90)(7,74,30,91)(8,75,31,92)(9,76,32,93)(10,77,33,94)(11,78,34,95)(12,79,35,96)(13,80,36,97)(14,81,37,98)(15,55,38,99)(16,56,39,100)(17,57,40,101)(18,58,41,102)(19,59,42,103)(20,60,43,104)(21,61,44,105)(22,62,45,106)(23,63,46,107)(24,64,47,108)(25,65,48,82)(26,66,49,83)(27,67,50,84)>;
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), (1,42,10,51,19,33)(2,43,11,52,20,34)(3,44,12,53,21,35)(4,45,13,54,22,36)(5,46,14,28,23,37)(6,47,15,29,24,38)(7,48,16,30,25,39)(8,49,17,31,26,40)(9,50,18,32,27,41)(55,108,73,99,64,90)(56,82,74,100,65,91)(57,83,75,101,66,92)(58,84,76,102,67,93)(59,85,77,103,68,94)(60,86,78,104,69,95)(61,87,79,105,70,96)(62,88,80,106,71,97)(63,89,81,107,72,98), (1,68,51,85)(2,69,52,86)(3,70,53,87)(4,71,54,88)(5,72,28,89)(6,73,29,90)(7,74,30,91)(8,75,31,92)(9,76,32,93)(10,77,33,94)(11,78,34,95)(12,79,35,96)(13,80,36,97)(14,81,37,98)(15,55,38,99)(16,56,39,100)(17,57,40,101)(18,58,41,102)(19,59,42,103)(20,60,43,104)(21,61,44,105)(22,62,45,106)(23,63,46,107)(24,64,47,108)(25,65,48,82)(26,66,49,83)(27,67,50,84) );
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)], [(1,42,10,51,19,33),(2,43,11,52,20,34),(3,44,12,53,21,35),(4,45,13,54,22,36),(5,46,14,28,23,37),(6,47,15,29,24,38),(7,48,16,30,25,39),(8,49,17,31,26,40),(9,50,18,32,27,41),(55,108,73,99,64,90),(56,82,74,100,65,91),(57,83,75,101,66,92),(58,84,76,102,67,93),(59,85,77,103,68,94),(60,86,78,104,69,95),(61,87,79,105,70,96),(62,88,80,106,71,97),(63,89,81,107,72,98)], [(1,68,51,85),(2,69,52,86),(3,70,53,87),(4,71,54,88),(5,72,28,89),(6,73,29,90),(7,74,30,91),(8,75,31,92),(9,76,32,93),(10,77,33,94),(11,78,34,95),(12,79,35,96),(13,80,36,97),(14,81,37,98),(15,55,38,99),(16,56,39,100),(17,57,40,101),(18,58,41,102),(19,59,42,103),(20,60,43,104),(21,61,44,105),(22,62,45,106),(23,63,46,107),(24,64,47,108),(25,65,48,82),(26,66,49,83),(27,67,50,84)]])
162 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 9A | ··· | 9F | 9G | ··· | 9L | 12A | 12B | 12C | 12D | 18A | ··· | 18F | 18G | ··· | 18L | 27A | ··· | 27R | 27S | ··· | 27AJ | 36A | ··· | 36L | 54A | ··· | 54R | 54S | ··· | 54AJ | 108A | ··· | 108AJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 27 | ··· | 27 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C9 | C12 | C18 | C27 | C36 | C54 | C108 | S3 | Dic3 | C3×S3 | C3×Dic3 | S3×C9 | C9×Dic3 | S3×C27 | Dic3×C27 |
| kernel | Dic3×C27 | C3×C54 | C9×Dic3 | C3×C27 | C3×C18 | C3×Dic3 | C3×C9 | C3×C6 | Dic3 | C32 | C6 | C3 | C54 | C27 | C18 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 18 | 12 | 18 | 36 | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 |
Matrix representation of Dic3×C27 ►in GL3(𝔽109) generated by
| 78 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 108 | 0 | 0 |
| 0 | 45 | 0 |
| 0 | 105 | 63 |
| 33 | 0 | 0 |
| 0 | 75 | 44 |
| 0 | 1 | 34 |
G:=sub<GL(3,GF(109))| [78,0,0,0,7,0,0,0,7],[108,0,0,0,45,105,0,0,63],[33,0,0,0,75,1,0,44,34] >;
Dic3×C27 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{27} % in TeX
G:=Group("Dic3xC27"); // GroupNames label
G:=SmallGroup(324,11);
// by ID
G=gap.SmallGroup(324,11);
# by ID
G:=PCGroup([6,-2,-3,-2,-3,-3,-3,36,79,93,7781]);
// Polycyclic
G:=Group<a,b,c|a^27=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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