Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = D3×D4×GL(2,𝔽3) (order 2304) · Inn(G) = D3 · Out(G) = D4×GL(2,𝔽3) |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C32×Dic3 |
| unambiguous semidirect product | (C2×C3)⋊(C32×C4) |
| derived | C3 ⋊ C2×C32×C4 |
| Sylow | C33 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C32 | D3 |
| commutator, hypocenter | C3 | C2×C32×C4 |
| Frattini | C2 | C2×C32×D3 |
| Fitting, socle, prime radical | C22×C33 | C2 |
| cosocle | C2×C3 | C22×C32 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C22×C32 | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 15 points:
(4,5)(6,7,8)(12,13,14,15);(1,2)(3,4,5)(9,10,11)108 conjugacy classes — view the full character table.
| maximal subgroup of | C3×Dic32, C3×(C2×D3)⋊Dic3, C3×#3722, … |
| maximal normal subgroup of | C3×Dic32, C3×(C2×D3)⋊Dic3, C3×#3722, … |
| maximal quotient of | C2×C32×C3⋊C8, C32×#208, C32×C4×Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 138);