Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #64209 |
| commutator, cosocle | C3×#41⋊C3 | C22 |
| Frattini, hypercenter | C4 | D3×S4 |
| Fitting | C3×#41 | C2×D3 |
| socle | C2×C3 | GL(2,ℤ/4) |
| hypocenter | C3×SL(2,𝔽3) | D4 |
| LCS | G ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
151 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 151 conjugacy classes of subgroups — too many to draw. Only its 21 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|c^2=h,f^2=h,g^2=h,b^a=bc,c^a=ch,e^a=e^2,c^b=ch,d^b=d^2,f^b=gh,g^b=fh,f^d=g,g^d=fgh,g^f=gh30 conjugacy classes — view the full character table.
| maximal subgroup of | #11921842, #11921843, #11921844, … |
| maximal normal subgroup of | #11921842, #11921843, #11921844, … |
| maximal quotient of | #11920502, #11920510, #11920544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5111);