Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | #732 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#4178⋊C2 |
| commutator, cosocle | C2×#732 | C23 |
| Frattini | C2×C32 | C2×D32 |
| Fitting | C4×#732 | C22 |
| socle | C2×C3 | C2×((C3⋊D3)⋊C3)⋊C2 |
| hypercenter | C4 | #4178⋊C2 |
| hypocenter | #732 | #41 |
| LCS | G ▷ C2×#732 ▷ #732 | |
| UCS | C1 ◁ C2 ◁ C4 | |
224 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 38 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:
2^4,3^4|b^2=d,c^2=d,f^3=h^2,c^a=cd,e^a=e^2,g^a=g^2h,c^b=cd,f^b=f^2h,g^b=g^2,h^b=h^2,f^e=fg,g^e=gh47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 857);