Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | Q8×H(3,𝔽3) ⋊ D3 |
| derived | #10619580 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #10619757 |
| commutator, cosocle, hypocenter | #10619580 | C2 |
| Frattini | C2×C32 | C3⋊S4 |
| Fitting | Q8×H(3,𝔽3) | D3 |
| socle | C2×C3 | #6258 |
| LCS | G ▷ #10619580 | |
| UCS | C1 ◁ C2 | |
156 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3^4,2^3|f^2=h,g^2=h,b^a=b^2,c^a=c^2,d^a=de^2,e^a=e^2,f^a=fg,g^a=gh,c^b=cd,f^b=g,g^b=fgh,d^c=de,g^f=gh36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 720);