Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived | Q8⋊H(3,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22×C3 | C3⋊S4 |
| commutator, hypocenter | Q8⋊H(3,𝔽3) | C4 |
| Fitting | C2×C32×Q8 | D3 |
| prime radical, cosocle | C2×Q8⋊H(3,𝔽3) | C2 |
| LCS | G ▷ Q8⋊H(3,𝔽3) | |
| UCS | C1 ◁ C22×C3 | |
149 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^2,3^3,2^3|a^2=b,f^2=h,g^2=h,c^a=c^2,d^a=d^2,f^a=gh,g^a=fh,d^c=de,f^c=g,g^c=fgh,g^f=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #13146327, #13146329, #13146330, … |
| maximal normal subgroup of | #13146327, #13146329, #13146330, … |
| maximal quotient of | #13136786, C2×#11720738 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 705);