p-group, metacyclic, nilpotent (class 3), monomial
Aliases: C27⋊C9, C9.53- 1+2, C32.73- 1+2, C27⋊C3.C3, C9⋊C9.1C3, C9.2(C3×C9), C3.3(C9⋊C9), (C3×C9).1C32, SmallGroup(243,22)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C27⋊C9
G = < a,b | a27=b9=1, bab-1=a7 >
Permutation representations of C27⋊C9
►On 27 points - transitive group 27T107
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 5 17 11 14 26 20 23 8)(3 9 6 21 27 24 12 18 15)(4 13 22)(7 25 16)
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), (2,5,17,11,14,26,20,23,8)(3,9,6,21,27,24,12,18,15)(4,13,22)(7,25,16)>;
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), (2,5,17,11,14,26,20,23,8)(3,9,6,21,27,24,12,18,15)(4,13,22)(7,25,16) );
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)], [(2,5,17,11,14,26,20,23,8),(3,9,6,21,27,24,12,18,15),(4,13,22),(7,25,16)]])
G:=TransitiveGroup(27,107);
C27⋊C9 is a maximal subgroup of
C27⋊C18
35 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 9A | ··· | 9F | 9G | ··· | 9L | 27A | ··· | 27R |
| order | 1 | 3 | 3 | 3 | 3 | 9 | ··· | 9 | 9 | ··· | 9 | 27 | ··· | 27 |
| size | 1 | 1 | 1 | 3 | 3 | 3 | ··· | 3 | 9 | ··· | 9 | 9 | ··· | 9 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 9 |
| type | + | ||||||
| image | C1 | C3 | C3 | C9 | 3- 1+2 | 3- 1+2 | C27⋊C9 |
| kernel | C27⋊C9 | C9⋊C9 | C27⋊C3 | C27 | C9 | C32 | C1 |
| # reps | 1 | 2 | 6 | 18 | 4 | 2 | 2 |
Matrix representation of C27⋊C9 ►in GL9(𝔽109)
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 63 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 63 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 63 |
| 0 | 45 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 45 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 45 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 63 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 63 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(9,GF(109))| [0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,45,0,0,0,0,0,0,0,0,0,45,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,0,63,0,0,0],[1,0,0,0,0,0,0,0,0,0,45,0,0,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,45,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,0,1,0] >;
C27⋊C9 in GAP, Magma, Sage, TeX
C_{27}\rtimes C_9 % in TeX
G:=Group("C27:C9"); // GroupNames label
G:=SmallGroup(243,22);
// by ID
G=gap.SmallGroup(243,22);
# by ID
G:=PCGroup([5,-3,3,-3,3,-3,135,121,36,1352,147,1268]);
// Polycyclic
G:=Group<a,b|a^27=b^9=1,b*a*b^-1=a^7>;
// generators/relations
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