Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #634 |
| automorphism | Aut(G) = C23×AGL(2,𝔽3) (order 3456) · Inn(G) = C2×C3⋊D3 · Out(G) = C2×GL(2,𝔽3) |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×#638 |
| Fitting | C22×C33 ⋊ C2 |
| Sylow | C33 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×C3⋊D3 |
| commutator, cosocle | C2×C32 | C22×C3 |
| Frattini | C2 | C2×C3×C3⋊D3 |
| Fitting | C22×C33 | C2 |
| socle | C2×C33 | C22 |
| hypercenter | C22×C3 | C3⋊D3 |
| hypocenter | C32 | C3×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 13 points:
(2,4)(5,6,7)(11,12,13);(1,2)(3,4)(9,10)(12,13);(8,9,10)63 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#3804, C3×D3×#67, #89029, … |
| maximal normal subgroup of | C3×#3804, C3×D3×#67, #89029, … |
| maximal quotient of | C3×#3750, C3×#3752, C3×#3753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 144);