p-group, metacyclic, nilpotent (class 2), monomial
Aliases: C27⋊C3, C9.C9, C32.C9, C9.2C32, C3.3(C3×C9), (C3×C9).3C3, SmallGroup(81,6)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C27⋊C3
G = < a,b | a27=b3=1, bab-1=a10 >
Permutation representations of C27⋊C3
►On 27 points - transitive group 27T22
Generators in S27
(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)
(2 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 27)
G:=sub<Sym(27)| (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), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)>;
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), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27) );
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)], [(2,20,11),(3,12,21),(5,23,14),(6,15,24),(8,26,17),(9,18,27)]])
G:=TransitiveGroup(27,22);
C27⋊C3 is a maximal subgroup of
C27⋊C6 C9.4He3 C9.5He3 C9.6He3 C27⋊C9 C27○He3 C27⋊A4 C62.C9
C27⋊C3 is a maximal quotient of
C27⋊2C9 C32⋊C27 C9⋊C27 C27⋊A4 C62.C9
33 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 9A | ··· | 9F | 9G | 9H | 9I | 9J | 27A | ··· | 27R |
| order | 1 | 3 | 3 | 3 | 3 | 9 | ··· | 9 | 9 | 9 | 9 | 9 | 27 | ··· | 27 |
| size | 1 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 3 | ··· | 3 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 3 |
| type | + | |||||
| image | C1 | C3 | C3 | C9 | C9 | C27⋊C3 |
| kernel | C27⋊C3 | C27 | C3×C9 | C9 | C32 | C1 |
| # reps | 1 | 6 | 2 | 12 | 6 | 6 |
Matrix representation of C27⋊C3 ►in GL3(𝔽109) generated by
| 0 | 1 | 0 |
| 0 | 0 | 63 |
| 16 | 0 | 0 |
| 1 | 0 | 0 |
| 0 | 63 | 0 |
| 0 | 0 | 45 |
G:=sub<GL(3,GF(109))| [0,0,16,1,0,0,0,63,0],[1,0,0,0,63,0,0,0,45] >;
C27⋊C3 in GAP, Magma, Sage, TeX
C_{27}\rtimes C_3 % in TeX
G:=Group("C27:C3"); // GroupNames label
G:=SmallGroup(81,6);
// by ID
G=gap.SmallGroup(81,6);
# by ID
G:=PCGroup([4,-3,3,-3,-3,36,241,46]);
// Polycyclic
G:=Group<a,b|a^27=b^3=1,b*a*b^-1=a^10>;
// generators/relations
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