Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C4×#731 ⋊ C2 |
| Sylow | #731 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#4184 |
| commutator, cosocle | C2×#731 | C23 |
| Frattini | C2×C32 | C22×C3⋊D3 |
| Fitting | C2×C4×#731 | C2 |
| socle | C2×C3 | C22×#272 |
| hypercenter | C2×C4 | #4184 |
| hypocenter | #731 | #41 |
| LCS | G ▷ C2×#731 ▷ #731 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
280 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 59 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,f^3=h^2,c^a=cd,e^a=e^2,f^a=f^2h,h^a=h^2,f^e=fg,g^e=gh,g^f=gh^286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1150);