Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| isoclinism | isoclinic class of #272 |
| automorphism | Aut(G) = C3×AGL(2,𝔽3) (order 1296) · Inn(G) = C3⋊D3 · Out(G) = C3×S4 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, Sylow | #737 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | C3⋊D3 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C2×C3 |
| Frattini, socle | C3 | C3×C3⋊D3 |
| Fitting | #737 | C2 |
| prime radical | #272 | C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C9 | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,7,23,4,17,5,11,2,13)(3,24,21,18,15,20,9,14,22,8,16,27,10,25,12,19,6,26);(3,9,10)(6,15,16)(8,18,19)(12,21,22)(14,24,25)(20,26,27);(1,3,8)(2,6,14)(4,10,18)(5,12,20)(7,16,24)(9,19,11)(13,22,26)(15,25,17)(21,27,23)Polycyclic presentation:
2,3^4|b^3=e,c^a=c^2,d^a=d^2,d^c=de30 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 44);