Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C22×C32 |
No aliases for this group.
| Sylow | C24 ⋊ #737 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×A42 |
| commutator, socle, cosocle | C24×C3 | C33 |
| Fitting | C24×C32 | C32 |
| prime radical | C32⋊(C22⋊A4) | C3 |
| hypercenter | C32 | A42 |
| hypocenter | C24 | #737 |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C3 ◁ C32 | |
217 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 217 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
Polycyclic presentation:
3^4,2^4|a^3=d,b^3=d,e^a=ef,f^a=e,c^b=cd,g^b=gh,h^b=g80 conjugacy classes — view the full character table.
| maximal subgroup of | #72259666, #72259686, #72276439, … |
| maximal normal subgroup of | #72259666, #72259686, #72276439, … |
| maximal quotient of | C22⋊C9×#1094, #72276309 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2957);