Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of C3⋊((C3⋊S4)⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C32 |
| direct product | C2×C3⋊((C3⋊S4)⋊C3) |
| unambiguous semidirect product | (C2×C3)⋊((C3⋊S4)⋊C3) |
| Fitting | C23×C33 ⋊ D3 |
| derived | C32×A4 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊((C3⋊S4)⋊C3) |
| commutator, cosocle, hypocenter | C32×A4 | C22×C3 |
| Frattini | C3 | C2×C3×C3⋊S4 |
| Fitting | C23×C33 | D3 |
| socle | C23×C32 | C3×D3 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 ◁ C2 | |
508 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
78 conjugacy classes — view the full character table.
| maximal subgroup of | #72220517, C2×#13049222, #72220631, … |
| maximal normal subgroup of | #72220517, C2×#13049222, #72220631, … |
| maximal quotient of | #72220517 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3003);