Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | Aut(G) = D4×((C3⋊D3)⋊C3)⋊C2 (order 864) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×(C3⋊Dic3)⋊C3 |
| unambiguous semidirect product | (C2×C3⋊Dic3)⋊C3 |
| derived | C32 ⋊ C2×C3×C4 |
| Sylow | H(3,𝔽3) ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C2×C3×C4 |
| Frattini | C2×C3 | C2×C3×D3 |
| Fitting, prime radical | C22×H(3,𝔽3) | C2 |
| socle | C22×C3 | C3×D3 |
| cosocle | C2×C32 | C22×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C22 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 15 points:
(3,4,5,6)(8,13,15,10,11,14)(9,12);(1,2)(7,8,10)(9,11,14)(12,15,13)40 conjugacy classes — view the full character table.
| maximal subgroup of | #88447, #88448, #88449, … |
| maximal normal subgroup of | #88447, #88448, #88449, … |
| maximal quotient of | C2×(C3⋊(C3⋊C8))⋊C3, #3748⋊C3, C4×(C3⋊Dic3)⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 59);