Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C32 ⋊ #469 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×GO+(4,𝔽2) |
| commutator, cosocle | C2×C3⋊Dic3 | C23 |
| Fitting | C23×C32 | D4 |
| socle | C22×C32 | C2×D4 |
| hypercenter | C23 | GO+(4,𝔽2) |
| hypocenter | C32 | #469 |
| LCS | G ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C23 | |
224 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 33 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:
2^6,3^2|d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=g^2,c^b=cef,d^b=df,g^b=h,h^b=g,g^d=g^2,h^d=h^236 conjugacy classes — view the full character table.
| maximal subgroup of | #11771761, #11800002, C2×#10593522, … |
| maximal normal subgroup of | #11771761, #11800002, C2×#10593522, … |
| maximal quotient of | #11781723, #11781755, #11781946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5404);