Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | C3⋊#5933 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#1020 |
| commutator, prime radical, cosocle, hypocenter | C3×#997 | C2 |
| Fitting | C3×H(5,𝔽2) | D3 |
| socle | C2×C3 | #1020 |
| LCS | G ▷ C3×#997 | |
| UCS | C1 ◁ C2 | |
160 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 17 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:
2,3^2,2^5|a^2=h,f^2=h,g^2=h,b^a=b^2,c^a=c^2,d^a=e,e^a=d,f^a=gh,g^a=fh,d^b=e,e^b=de,f^b=fg,g^b=f,f^d=fh,g^e=gh,g^f=gh24 conjugacy classes — view the full character table.
| maximal subgroup of | #11920726, #11920728, #11920734, … |
| maximal normal subgroup of | #11920726, #11920728, #11920734, … |
| maximal quotient of | C3⋊#85548, C3⋊#85549, C3⋊#85552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8485);