Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×A4×C3⋊D3 |
| commutator, cosocle, hypocenter | C32×Q8 | C22×C3 |
| Fitting | C32×#41 | C2×C3 |
| socle | C2×C32 | C22×A4 |
| hypercenter | C4 | A4×C3⋊D3 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C2 ◁ C4 | |
316 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^2,3,2^2,3^2,2|a^2=h,b^2=h,d^2=h,e^2=h,b^a=bh,f^a=f^2,g^a=g^2,d^c=de,e^c=d,e^d=eh75 conjugacy classes — view the full character table.
| maximal subgroup of | #13165983, #13165984, #13165985, … |
| maximal normal subgroup of | #13165983, #13165984, #13165985, … |
| maximal quotient of | SL(2,𝔽3)×#634, #13173143, C4⋊(C3⋊(C3⋊(#41⋊C3))), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4458);