Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of C3⋊(C3⋊(#41⋊C3)) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C3⋊(C3⋊(#41⋊C3)) |
| unambiguous semidirect product | (C2×C3)⋊(C3⋊(#41⋊C3)) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | A4×C3⋊D3 |
| commutator, cosocle, hypocenter | C32×Q8 | C22×C3 |
| Frattini | C2 | C2×A4×C3⋊D3 |
| Fitting | C2×C32×Q8 | C2×C3 |
| socle | C22×C32 | C2×A4 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C22 | |
350 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | #13165941, #13165944, #13165947, … |
| maximal normal subgroup of | #13165941, #13165944, #13165947, … |
| maximal quotient of | C4×C3⋊(C3⋊(#41⋊C3)), C4⋊(C3⋊(C3⋊(#41⋊C3))), #13173146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4455);