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) = C3 |
| direct product | C32×Dic6 |
| unambiguous semidirect product | C3⋊(C32×Q8) |
| Sylow | C33 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | C2×D3 |
| commutator, cosocle | C2×C3 | C22×C32 |
| Frattini | C2 | C2×C32×D3 |
| Fitting | C33×C4 | C2 |
| socle, prime radical | C2×C33 | C22 |
| hypercenter | C32×C4 | D3 |
| hypocenter | C3 | C32×Q8 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C32 ◁ C32×C4 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(4,5,6)(7,8,9)(10,17,13,12)(11,16,15,14);(1,2,3)(8,9)(10,15,13,11)(12,14,17,16)81 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#3715, C3×#3717, C3×#3718, … |
| maximal normal subgroup of | C3×#3715, C3×#3717, C3×#3718, … |
| maximal quotient of | C32×Dic3⋊C4, C32×C4⋊Dic3, C2×C32×Dic6, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 135);