Finite Group DatabasebetaFinite Group Databasebeta| order | 324 |
| isoclinism | isoclinic class of ((C3⋊D3)⋊C3)⋊D3 |
| automorphism | Aut(G) = ((((C3⋊D3)⋊C3)⋊D3)⋊C3)⋊C2 (order 1944) · Inn(G) = G · Out(G) = D3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | ((C3⋊D3)⋊C3)⋊D3 |
| Fitting, derived, Sylow | C3×H(3,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×H(3,𝔽3) | C22 |
| Frattini | C3 | D3×C3⋊D3 |
| socle | C32 | D32 |
| LCS | G ▷ C3×H(3,𝔽3) | |
| UCS | C1 | |
133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,3)(5,11,9,7,8,12)(6,10);(1,2,3)(6,10)(8,9)(11,12);(4,5,7)(6,9,11)(8,12,10)Polycyclic presentation:
2^2,3^4|e^a=e^2,f^a=f^2,c^b=c^2,d^b=d^2,f^b=f^2,e^c=ef21 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(324, 121);