Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ((C3⋊S4)⋊C3)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊H(3,𝔽3) | C22 |
| Frattini, socle | C2×C3 | D3×S4 |
| Fitting | C32×Q8 | C2×D3 |
| LCS | G ▷ Q8⋊H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
161 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 161 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^2,3^3,2^3|b^2=h,f^2=h,g^2=h,b^a=bh,d^a=d^2,e^a=e^2,c^b=c^2,e^b=e^2,f^b=gh,g^b=fh,d^c=de,f^c=g,g^c=fgh,g^f=gh28 conjugacy classes — view the full character table.
| maximal subgroup of | #13164197, C2×#11722256, #13164207, … |
| maximal normal subgroup of | #13164197, C2×#11722256, #13164207, … |
| maximal quotient of | #13146330, #13146333, #13146334, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2223);