Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived | #10619584 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊((C22⋊C9)⋊C2) |
| commutator, cosocle, hypocenter | #10619584 | C2 |
| Frattini, socle | C2×C32 | C3⋊S4 |
| Fitting | C33×Q8 | D3 |
| LCS | G ▷ #10619584 | |
| UCS | C1 ◁ C2×C3 | |
171 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 29 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:
2,3^4,2^3|b^3=e,f^2=h,g^2=h,b^a=b^2e^2,c^a=c^2,e^a=e^2,f^a=fg,g^a=gh,c^b=cd,f^b=g,g^b=fgh,g^f=gh60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 741);