Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of C3⋊((C3⋊D3)⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C3⋊((C3⋊Dic3)⋊C3) |
| derived | C33 ⋊ C3×C4 |
| Sylow | C3×H(3,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊((C3⋊D3)⋊C3) |
| commutator, hypocenter | C33 | C3×C4 |
| Frattini | C2×C3 | C3×C3⋊D3 |
| Fitting, prime radical | C2×C3×H(3,𝔽3) | C2 |
| socle | C2×C32 | C3×D3 |
| cosocle | C2×C33 | C2×C3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,4)(6,7)(9,14,16,11,12,15)(10,13);(5,6,7);(8,9,11)(10,12,15)(13,16,14)Polycyclic presentation:
2,3,2,3^3|a^2=c,d^a=d^2,e^a=e^2,f^a=f^2,d^b=df42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 98);