Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Fitting | C22×C3×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C3⋊S4 |
| commutator, cosocle | C2×C3×A4 | C23 |
| Fitting | C22×C3×Q8 | D3 |
| socle | C23×C3 | C22×D3 |
| hypercenter | Q8 | C3⋊S4 |
| hypocenter | C3×A4 | #41 |
| LCS | G ▷ C2×C3×A4 ▷ C3×A4 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
406 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^4,3^2,2^2|b^2=d,c^2=d,c^a=cd,e^a=e^2,f^a=f^2,g^a=h,h^a=g,c^b=cd,g^e=h,h^e=gh45 conjugacy classes — view the full character table.
| maximal subgroup of | #11922200, #11922203, #11922208, … |
| maximal normal subgroup of | #11922200, #11922203, #11922208, … |
| maximal quotient of | #11922855, #11922857, #11922858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8464);