direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C6×D27, C54⋊3C6, C32.3D18, C27⋊3(C2×C6), (C3×C54)⋊2C2, C9.2(S3×C6), (C3×C9).7D6, (C3×C6).7D9, C6.5(C3×D9), C3.2(C6×D9), C18.5(C3×S3), (C3×C27)⋊3C22, (C3×C18).21S3, SmallGroup(324,65)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — C6×D27 |
Generators and relations for C6×D27
G = < a,b,c | a6=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C6×D27
►On 108 points
Generators in S108
(1 38 19 29 10 47)(2 39 20 30 11 48)(3 40 21 31 12 49)(4 41 22 32 13 50)(5 42 23 33 14 51)(6 43 24 34 15 52)(7 44 25 35 16 53)(8 45 26 36 17 54)(9 46 27 37 18 28)(55 89 64 98 73 107)(56 90 65 99 74 108)(57 91 66 100 75 82)(58 92 67 101 76 83)(59 93 68 102 77 84)(60 94 69 103 78 85)(61 95 70 104 79 86)(62 96 71 105 80 87)(63 97 72 106 81 88)
(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 97)(2 96)(3 95)(4 94)(5 93)(6 92)(7 91)(8 90)(9 89)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 108)(18 107)(19 106)(20 105)(21 104)(22 103)(23 102)(24 101)(25 100)(26 99)(27 98)(28 55)(29 81)(30 80)(31 79)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 70)(41 69)(42 68)(43 67)(44 66)(45 65)(46 64)(47 63)(48 62)(49 61)(50 60)(51 59)(52 58)(53 57)(54 56)
G:=sub<Sym(108)| (1,38,19,29,10,47)(2,39,20,30,11,48)(3,40,21,31,12,49)(4,41,22,32,13,50)(5,42,23,33,14,51)(6,43,24,34,15,52)(7,44,25,35,16,53)(8,45,26,36,17,54)(9,46,27,37,18,28)(55,89,64,98,73,107)(56,90,65,99,74,108)(57,91,66,100,75,82)(58,92,67,101,76,83)(59,93,68,102,77,84)(60,94,69,103,78,85)(61,95,70,104,79,86)(62,96,71,105,80,87)(63,97,72,106,81,88), (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), (1,97)(2,96)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,88)(11,87)(12,86)(13,85)(14,84)(15,83)(16,82)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,55)(29,81)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(41,69)(42,68)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56)>;
G:=Group( (1,38,19,29,10,47)(2,39,20,30,11,48)(3,40,21,31,12,49)(4,41,22,32,13,50)(5,42,23,33,14,51)(6,43,24,34,15,52)(7,44,25,35,16,53)(8,45,26,36,17,54)(9,46,27,37,18,28)(55,89,64,98,73,107)(56,90,65,99,74,108)(57,91,66,100,75,82)(58,92,67,101,76,83)(59,93,68,102,77,84)(60,94,69,103,78,85)(61,95,70,104,79,86)(62,96,71,105,80,87)(63,97,72,106,81,88), (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), (1,97)(2,96)(3,95)(4,94)(5,93)(6,92)(7,91)(8,90)(9,89)(10,88)(11,87)(12,86)(13,85)(14,84)(15,83)(16,82)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,55)(29,81)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(41,69)(42,68)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,56) );
G=PermutationGroup([[(1,38,19,29,10,47),(2,39,20,30,11,48),(3,40,21,31,12,49),(4,41,22,32,13,50),(5,42,23,33,14,51),(6,43,24,34,15,52),(7,44,25,35,16,53),(8,45,26,36,17,54),(9,46,27,37,18,28),(55,89,64,98,73,107),(56,90,65,99,74,108),(57,91,66,100,75,82),(58,92,67,101,76,83),(59,93,68,102,77,84),(60,94,69,103,78,85),(61,95,70,104,79,86),(62,96,71,105,80,87),(63,97,72,106,81,88)], [(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)], [(1,97),(2,96),(3,95),(4,94),(5,93),(6,92),(7,91),(8,90),(9,89),(10,88),(11,87),(12,86),(13,85),(14,84),(15,83),(16,82),(17,108),(18,107),(19,106),(20,105),(21,104),(22,103),(23,102),(24,101),(25,100),(26,99),(27,98),(28,55),(29,81),(30,80),(31,79),(32,78),(33,77),(34,76),(35,75),(36,74),(37,73),(38,72),(39,71),(40,70),(41,69),(42,68),(43,67),(44,66),(45,65),(46,64),(47,63),(48,62),(49,61),(50,60),(51,59),(52,58),(53,57),(54,56)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 9A | ··· | 9I | 18A | ··· | 18I | 27A | ··· | 27AA | 54A | ··· | 54AA |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 1 | 27 | 27 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 27 | 27 | 27 | 27 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | C3×S3 | D9 | S3×C6 | D18 | D27 | C3×D9 | D54 | C6×D9 | C3×D27 | C6×D27 |
| kernel | C6×D27 | C3×D27 | C3×C54 | D54 | D27 | C54 | C3×C18 | C3×C9 | C18 | C3×C6 | C9 | C32 | C6 | C6 | C3 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 3 | 2 | 3 | 9 | 6 | 9 | 6 | 18 | 18 |
Matrix representation of C6×D27 ►in GL3(𝔽109) generated by
| 46 | 0 | 0 |
| 0 | 45 | 0 |
| 0 | 0 | 45 |
| 1 | 0 | 0 |
| 0 | 49 | 0 |
| 0 | 0 | 89 |
| 108 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(109))| [46,0,0,0,45,0,0,0,45],[1,0,0,0,49,0,0,0,89],[108,0,0,0,0,1,0,1,0] >;
C6×D27 in GAP, Magma, Sage, TeX
C_6\times D_{27} % in TeX
G:=Group("C6xD27"); // GroupNames label
G:=SmallGroup(324,65);
// by ID
G=gap.SmallGroup(324,65);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,1443,381,5404,208,7781]);
// Polycyclic
G:=Group<a,b,c|a^6=b^27=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export