Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C3×#131 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×C3⋊S4 |
| commutator, hypocenter | C3×SL(2,𝔽3) | C2×C4 |
| Fitting | C3×#131 | D3 |
| socle | C2×C3 | C4×S4 |
| prime radical | #64266 | C2 |
| cosocle | C3×#41⋊C3 | C22 |
| hypercenter | C8 | C3⋊S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C4 ◁ C8 | |
160 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 31 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:
2^3,3^2,2^3|b^2=c,c^2=h,f^2=h,g^2=h,b^a=bh,d^a=d^2,e^a=e^2,f^a=gh,g^a=fh,f^d=g,g^d=fgh,g^f=gh54 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#5405, #11921824, #11921831, … |
| maximal normal subgroup of | D3×#5405, #11921824, #11921831, … |
| maximal quotient of | #11920964, #11920966, #11920974, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5521);