Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #157⋊(C3⋊D3) |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×#157⋊(C3⋊D3) |
| Fitting, socle | C2×C34 ⋊ D4 |
| Sylow | C34 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #157⋊(C3⋊D3) |
| commutator, cosocle | C32×C3⋊D3 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C2×C34 | D4 |
| hypocenter | C34 | C2×D4 |
| LCS | G ▷ C32×C3⋊D3 ▷ C34 | |
| UCS | C1 ◁ C2 | |
688 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
66 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13049799, C2×C3⋊D3×GO+(4,𝔽2), C2×AGL(1,𝔽9)⋊(C3⋊D3), … |
| maximal normal subgroup of | C2×#13049799, C2×C3⋊D3×GO+(4,𝔽2), C2×AGL(1,𝔽9)⋊(C3⋊D3), … |
| maximal quotient of | C22×#157⋊(C3⋊D3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3575);