Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of C3⋊#157 |
| automorphism | Aut(G) = C2×D3×AΓL(1,𝔽9) (order 1728) · Inn(G) = C3⋊#157 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C3⋊#157 |
| unambiguous semidirect product | (C2×C3)⋊#157 |
| Fitting, socle | C2×C33 ⋊ C4 |
| derived, Sylow | C33 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊#157 |
| commutator, hypocenter | C33 | C2×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C33 | C4 |
| prime radical | C2×C3×C3⋊D3 | C2 |
| cosocle | C3×C3⋊D3 | C22 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 11 points:
(4,5)(6,7)(8,10,11,9);(1,2)(3,4,5)(6,8,11)(7,10,9)24 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×#157, (C2×D3)⋊#157, Dic3⋊#157, … |
| maximal normal subgroup of | Dic3×#157, (C2×D3)⋊#157, Dic3⋊#157, … |
| maximal quotient of | #88988, #88989, C4×C3⋊#157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 169);