Finite Group DatabasebetaFinite Group Databasebeta| order | 1458 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C34 |
| unambiguous semidirect product | C3⋊#92300 |
| Fitting, Sylow | C3×C32⋊H(3,𝔽3) ⋊ C2 |
| derived | C32×H(3,𝔽3) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊(C3⋊((C3⋊D3)⋊C3)) |
| commutator, cosocle, hypocenter | C32×H(3,𝔽3) | C2×C3 |
| Frattini | C32 | C3×C3⋊(C3⋊D3) |
| Fitting | C3×C32⋊H(3,𝔽3) | C2 |
| socle | C33 | C3×C3⋊D3 |
| LCS | G ▷ C32×H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
1454 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 1454 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
Polycyclic presentation:
2,3^6|b^a=b^2,c^a=c^2,d^a=de,e^a=e^2,g^a=g^2,c^b=cf,d^b=dg90 conjugacy classes — view the full character table.
| maximal subgroup of | #72214232, #72224516, #72224518, … |
| maximal normal subgroup of | #72214232, #72224516, #72224518, … |
| maximal quotient of | C3×C3⋊#92300, #72259823, #72277776 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1458, 1470);