Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Fitting | C3×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C3⋊S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C23 |
| Fitting | C3×H(5,𝔽2) | D3 |
| socle | C2×C3 | C22×S4 |
| hypercenter | Q8 | C3⋊S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ Q8 | |
350 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 53 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=h,c^2=h,f^2=h,g^2=h,d^a=d^2,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,f^d=g,g^d=fgh,g^f=gh42 conjugacy classes — view the full character table.
| maximal subgroup of | #11922461, #11922466, #11922469, … |
| maximal normal subgroup of | #11922461, #11922466, #11922469, … |
| maximal quotient of | #11922909, #11922910, #11922911, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8469);