Finite Group DatabasebetaFinite Group Databasebeta| order | 1458 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, Sylow | UT(4,𝔽3) ⋊ C2 |
| derived | C32⋊H(3,𝔽3) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | #92300 |
| commutator, cosocle, hypocenter | C32⋊H(3,𝔽3) | C2×C3 |
| Frattini | C33 | C3×C3⋊D3 |
| Fitting | UT(4,𝔽3) | C2 |
| hypercenter | C32 | C3⋊((C3⋊D3)⋊C3) |
| LCS | G ▷ C32⋊H(3,𝔽3) | |
| UCS | C1 ◁ C3 ◁ C32 | |
305 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 305 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.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
Polycyclic presentation:
2,3^6|b^a=b^2d^2f^2,c^a=c^2,f^a=f^2,c^b=ce,d^b=df,e^b=eg,f^c=fg,e^d=eg45 conjugacy classes — view the full character table.
| maximal subgroup of | #72212516, #72212523, #72212528, … |
| maximal normal subgroup of | #72212516, #72212523, #72212528, … |
| maximal quotient of | C2×#13091109, C3×#13091109, #72277654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1458, 1183);