Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| isoclinism | isoclinic class of #4174⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#4174⋊C2 |
| Fitting, Sylow | C3×AΣL(1,𝔽27) ⋊ C22 |
| derived | AΣL(1,𝔽27) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #4174⋊C2 |
| commutator, cosocle, hypocenter | AΣL(1,𝔽27) | C22×C3 |
| Frattini | C32 | C3×D32 |
| Fitting | C3×AΣL(1,𝔽27) | C22 |
| socle | C32 | ((C3⋊D3)⋊C3)⋊C2 |
| LCS | G ▷ AΣL(1,𝔽27) | |
| UCS | C1 ◁ C3 | |
236 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 18 |
51 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#4174⋊C2, C2×C3×#4174⋊C2, #72214618, … |
| maximal normal subgroup of | D3×#4174⋊C2, C2×C3×#4174⋊C2, C3×(#4174⋊C2)⋊C3, … |
| maximal quotient of | C3×#4183⋊C4, C3×#4174⋊C4, C3×#4175⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 436);