Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3×H(3,𝔽3) ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×((C3⋊D3)⋊C3)⋊C2 |
| commutator, Fitting, cosocle | C2×C3×H(3,𝔽3) | C23 |
| Frattini | C2×C3 | D33 |
| socle | C2×C32 | C2×D32 |
| hypocenter | C3×H(3,𝔽3) | C2×D4 |
| LCS | G ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C2 | |
496 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 496 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) = 72 |
Polycyclic presentation:
2^4,3^4|g^a=g^2,h^a=h^2,c^b=cd,e^b=e^2,h^b=h^2,f^c=f^2,g^e=gh54 conjugacy classes — view the full character table.
| maximal subgroup of | #72259648, C3×#13048339 |
| maximal normal subgroup of | C3×#13048339 |
| maximal quotient of | C3×#13048339 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2057);