Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of (C3⋊S4)⋊C3 |
| automorphism | Aut(G) = ((C3⋊S4)⋊C3)⋊C2 (order 432) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | (C3⋊S4)⋊C3 |
| Fitting | C22×C32 ⋊ D3 |
| derived | C3×A4 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×A4 | C2×C3 |
| Frattini | C3 | C3×S4 |
| Fitting | C22×C32 | D3 |
| socle | C22×C3 | C3×D3 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 | |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,4,3)(6,11,13,8,9,12)(7,10);(2,3,4)(5,6,8)(7,9,12)(10,13,11)Polycyclic presentation:
2,3^3,2^2|c^a=c^2,d^a=d^2,e^a=f,f^a=e,c^b=cd,e^c=f,f^c=ef19 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 92);