metabelian, supersoluble, monomial, A-group
Aliases: C27⋊S3, C3⋊D27, C9.2D9, C32.3D9, C9.(C3⋊S3), (C3×C27)⋊3C2, (C3×C9).7S3, C3.2(C9⋊S3), SmallGroup(162,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C27 — C27⋊S3 |
Generators and relations for C27⋊S3
G = < a,b,c | a27=b3=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C27⋊S3
►On 81 points
Generators in S81
(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)
(1 31 65)(2 32 66)(3 33 67)(4 34 68)(5 35 69)(6 36 70)(7 37 71)(8 38 72)(9 39 73)(10 40 74)(11 41 75)(12 42 76)(13 43 77)(14 44 78)(15 45 79)(16 46 80)(17 47 81)(18 48 55)(19 49 56)(20 50 57)(21 51 58)(22 52 59)(23 53 60)(24 54 61)(25 28 62)(26 29 63)(27 30 64)
(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(28 68)(29 67)(30 66)(31 65)(32 64)(33 63)(34 62)(35 61)(36 60)(37 59)(38 58)(39 57)(40 56)(41 55)(42 81)(43 80)(44 79)(45 78)(46 77)(47 76)(48 75)(49 74)(50 73)(51 72)(52 71)(53 70)(54 69)
G:=sub<Sym(81)| (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), (1,31,65)(2,32,66)(3,33,67)(4,34,68)(5,35,69)(6,36,70)(7,37,71)(8,38,72)(9,39,73)(10,40,74)(11,41,75)(12,42,76)(13,43,77)(14,44,78)(15,45,79)(16,46,80)(17,47,81)(18,48,55)(19,49,56)(20,50,57)(21,51,58)(22,52,59)(23,53,60)(24,54,61)(25,28,62)(26,29,63)(27,30,64), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)>;
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), (1,31,65)(2,32,66)(3,33,67)(4,34,68)(5,35,69)(6,36,70)(7,37,71)(8,38,72)(9,39,73)(10,40,74)(11,41,75)(12,42,76)(13,43,77)(14,44,78)(15,45,79)(16,46,80)(17,47,81)(18,48,55)(19,49,56)(20,50,57)(21,51,58)(22,52,59)(23,53,60)(24,54,61)(25,28,62)(26,29,63)(27,30,64), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,60)(37,59)(38,58)(39,57)(40,56)(41,55)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69) );
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)], [(1,31,65),(2,32,66),(3,33,67),(4,34,68),(5,35,69),(6,36,70),(7,37,71),(8,38,72),(9,39,73),(10,40,74),(11,41,75),(12,42,76),(13,43,77),(14,44,78),(15,45,79),(16,46,80),(17,47,81),(18,48,55),(19,49,56),(20,50,57),(21,51,58),(22,52,59),(23,53,60),(24,54,61),(25,28,62),(26,29,63),(27,30,64)], [(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15),(28,68),(29,67),(30,66),(31,65),(32,64),(33,63),(34,62),(35,61),(36,60),(37,59),(38,58),(39,57),(40,56),(41,55),(42,81),(43,80),(44,79),(45,78),(46,77),(47,76),(48,75),(49,74),(50,73),(51,72),(52,71),(53,70),(54,69)]])
C27⋊S3 is a maximal subgroup of
S3×D27 C32⋊D27 He3.D9 He3.2D9 C9⋊D27 C81⋊S3 C33.5D9 He3.5D9 C32⋊4D27
C27⋊S3 is a maximal quotient of
C27⋊Dic3 C9⋊D27 C32⋊2D27 C81⋊S3 C32⋊4D27
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 9A | ··· | 9I | 27A | ··· | 27AA |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 9 | ··· | 9 | 27 | ··· | 27 |
| size | 1 | 81 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | S3 | S3 | D9 | D9 | D27 |
| kernel | C27⋊S3 | C3×C27 | C27 | C3×C9 | C9 | C32 | C3 |
| # reps | 1 | 1 | 3 | 1 | 6 | 3 | 27 |
Matrix representation of C27⋊S3 ►in GL4(𝔽109) generated by
| 63 | 79 | 0 | 0 |
| 30 | 93 | 0 | 0 |
| 0 | 0 | 58 | 87 |
| 0 | 0 | 22 | 80 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 108 | 108 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 108 | 108 | 0 | 0 |
| 0 | 0 | 108 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(109))| [63,30,0,0,79,93,0,0,0,0,58,22,0,0,87,80],[1,0,0,0,0,1,0,0,0,0,108,1,0,0,108,0],[1,108,0,0,0,108,0,0,0,0,108,1,0,0,0,1] >;
C27⋊S3 in GAP, Magma, Sage, TeX
C_{27}\rtimes S_3 % in TeX
G:=Group("C27:S3"); // GroupNames label
G:=SmallGroup(162,18);
// by ID
G=gap.SmallGroup(162,18);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,221,456,182,1803,138,2704]);
// Polycyclic
G:=Group<a,b,c|a^27=b^3=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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