Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of D3×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C3⋊S4)⋊C8 |
| derived | C3×A4 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D3×S4 |
| commutator, hypocenter | C3×A4 | C2×C8 |
| Fitting | C22×C3×C4 | C2×D3 |
| socle | C23×C3 | C4×D3 |
| prime radical | C2×C3⋊S4 | C4 |
| cosocle | C3×C4×A4 | C22 |
| LCS | G ▷ C3×A4 | |
| UCS | C1 ◁ C4 | |
174 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 29 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|a^2=c,b^2=c,c^2=d,f^a=f^2,e^b=e^2,g^b=h,h^b=g,g^e=h,h^e=gh60 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C8×S4, #11921787, #11921793, … |
| maximal normal subgroup of | D3×C8×S4, #11921787, #11921793, … |
| maximal quotient of | #11920518, #11920519, #11920535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5056);