Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #5751⋊C3 |
| Fitting | C3×#131 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | A4×D12 |
| commutator, cosocle | C3×#41 | C22×C3 |
| Frattini | C4 | C2×D3×A4 |
| Fitting | C3×#131 | C2×C3 |
| socle | C2×C3 | D4×A4 |
| hypercenter | C8 | D3×A4 |
| hypocenter | C3×Q8 | C3×D4 |
| LCS | G ▷ C3×#41 ▷ C3×Q8 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
153 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^2,3,2^3,3,2|b^2=d,d^2=h,e^2=h,f^2=h,b^a=bd,d^a=dh,g^a=g^2,e^c=ef,f^c=e,f^e=fh54 conjugacy classes — view the full character table.
| maximal subgroup of | #11921837, #11921838, #11921841, … |
| maximal normal subgroup of | #11921837, #11921838, #11921841, … |
| maximal quotient of | #79911⋊C3, SL(2,𝔽3)×#204, #79924⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5575);