Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of C3⋊D3×C3⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C32 |
| direct product | C3⋊D3×C3⋊S4 |
| Fitting, socle | C22×C33 ⋊ C2×D3 |
| derived | C33×A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C33×A4 | C22 |
| Fitting, socle | C22×C33 | C2×D3 |
| LCS | G ▷ C33×A4 | |
| UCS | C1 | |
892 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 892 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×A4⋊S4, #72220655, C3⋊D3×C3⋊#1020, … |
| maximal normal subgroup of | D32×C3⋊S4, D3×C3⋊D3×S4, #72220997, … |
| maximal quotient of | C3⋊D3×A4⋊S4, C3⋊D3×C3⋊#1020, #72220728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3551);