Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = D3×D4×GL(2,𝔽3) (order 2304) · Inn(G) = C2×D3 · Out(G) = C22×GL(2,𝔽3) |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×D12 |
| unambiguous semidirect product | C4⋊(C32×D3) |
| Fitting | C33×C4 ⋊ C2 |
| Sylow | C33 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | C2×D3 |
| commutator, cosocle | C2×C3 | C22×C32 |
| Frattini | C2 | C2×C32×D3 |
| Fitting | C33×C4 | C2 |
| socle | C2×C33 | C22 |
| hypercenter | C32×C4 | D3 |
| hypocenter | C3 | C32×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C32 ◁ C32×C4 | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 13 points:
(4,5,6)(7,8,9)(10,13,12,11);(1,2,3)(8,9)(11,13)81 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#3713, C3×#3714, C3×#3715, … |
| maximal normal subgroup of | C3×#3713, C3×#3714, C3×#3715, … |
| maximal quotient of | C32×#204, C32×D24, C32×Dic12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 137);