Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Fitting | C22×C34 ⋊ C22 |
| Sylow | C34 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊#1113 |
| commutator, socle, cosocle | C2×C34 | C23 |
| Fitting | C22×C34 | C22 |
| hypercenter | C22 | C3⋊#1113 |
| hypocenter | C34 | C2×D4 |
| LCS | G ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C2 ◁ C22 | |
1140 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 1140 conjugacy classes of subgroups — too many to draw. Only its 119 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|c^a=cd,e^a=e^2,f^a=f^2,g^a=g^2,c^b=cd,e^b=e^2,f^b=f^2,h^b=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#3811, C2×#13049721, #72220338, … |
| maximal normal subgroup of | C3⋊D3×#3811, C2×#13049721, #72220338, … |
| maximal quotient of | C2×#13049721 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3439);