Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D3×C3⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| derived | C32×A4 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×C3⋊S4 |
| commutator, hypocenter | C32×A4 | C2×C4 |
| Fitting, socle | C23×C32 | C2×D3 |
| prime radical | C2×C3⋊(C3⋊S4) | C2 |
| cosocle | C2×C32×A4 | C22 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 ◁ C2 | |
381 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 381 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^3,3^3,2^2|a^2=c,b^2=c,d^a=d^2,f^a=f^2,g^a=h,h^a=g,e^b=e^2,g^d=h,h^d=gh54 conjugacy classes — view the full character table.
| maximal subgroup of | D3×(C3⋊S4)⋊C4, S4×#624, #13180282, … |
| maximal normal subgroup of | D3×(C3⋊S4)⋊C4, S4×#624, #13180282, … |
| maximal quotient of | #13165998, #13166000, #13166017, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4168);