Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of GO+(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | D32⋊C16 |
| Sylow | C32 ⋊ C22⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | GO+(4,𝔽2) |
| commutator | C3⋊D3 | C2×C16 |
| Fitting | C32×C8 | D4 |
| socle | C2×C32 | C22⋊C8 |
| prime radical | C2×D32 | C8 |
| cosocle | C8×C3⋊D3 | C22 |
| hypocenter | C32 | C22⋊C16 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C8 | |
118 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 118 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3^2|b^2=d,d^2=e,e^2=f,b^a=bc,g^a=g^2,g^b=h,h^b=g,g^c=g^2,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D32⋊C16, #11795991, #11796019, … |
| maximal normal subgroup of | C2×D32⋊C16, #11795991, #11796019, … |
| maximal quotient of | #11771346, #11771446, #11771541, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 1894);