direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×D27, C27⋊3C6, C32.2D9, (C3×C27)⋊2C2, C9.2(C3×S3), (C3×C9).5S3, C3.2(C3×D9), SmallGroup(162,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — C3×D27 |
Generators and relations for C3×D27
G = < a,b,c | a3=b27=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D27
►On 54 points
Generators in S54
(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)(28 37 46)(29 38 47)(30 39 48)(31 40 49)(32 41 50)(33 42 51)(34 43 52)(35 44 53)(36 45 54)
(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)
(1 51)(2 50)(3 49)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 39)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(25 54)(26 53)(27 52)
G:=sub<Sym(54)| (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,37,46)(29,38,47)(30,39,48)(31,40,49)(32,41,50)(33,42,51)(34,43,52)(35,44,53)(36,45,54), (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), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,39)(14,38)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(25,54)(26,53)(27,52)>;
G:=Group( (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,37,46)(29,38,47)(30,39,48)(31,40,49)(32,41,50)(33,42,51)(34,43,52)(35,44,53)(36,45,54), (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), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,39)(14,38)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28)(25,54)(26,53)(27,52) );
G=PermutationGroup([[(1,19,10),(2,20,11),(3,21,12),(4,22,13),(5,23,14),(6,24,15),(7,25,16),(8,26,17),(9,27,18),(28,37,46),(29,38,47),(30,39,48),(31,40,49),(32,41,50),(33,42,51),(34,43,52),(35,44,53),(36,45,54)], [(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)], [(1,51),(2,50),(3,49),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,42),(11,41),(12,40),(13,39),(14,38),(15,37),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31),(22,30),(23,29),(24,28),(25,54),(26,53),(27,52)]])
C3×D27 is a maximal subgroup of
C27⋊3C18 C81⋊C6 C32⋊2D27 He3.3D9 He3.4D9 He3.5D9
C3×D27 is a maximal quotient of C32⋊D27 C81⋊C6
45 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 9A | ··· | 9I | 27A | ··· | 27AA |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 27 | ··· | 27 |
| size | 1 | 27 | 1 | 1 | 2 | 2 | 2 | 27 | 27 | 2 | ··· | 2 | 2 | ··· | 2 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | C3×S3 | D9 | D27 | C3×D9 | C3×D27 |
| kernel | C3×D27 | C3×C27 | D27 | C27 | C3×C9 | C9 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 3 | 9 | 6 | 18 |
Matrix representation of C3×D27 ►in GL2(𝔽109) generated by
| 63 | 0 |
| 0 | 63 |
| 22 | 0 |
| 0 | 5 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(109))| [63,0,0,63],[22,0,0,5],[0,1,1,0] >;
C3×D27 in GAP, Magma, Sage, TeX
C_3\times D_{27} % in TeX
G:=Group("C3xD27"); // GroupNames label
G:=SmallGroup(162,7);
// by ID
G=gap.SmallGroup(162,7);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,452,237,1803,138,2704]);
// Polycyclic
G:=Group<a,b,c|a^3=b^27=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export