Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C92 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D92 |
| commutator | C2×C92 | C2×C4 |
| Frattini, socle | C22×C32 | D32 |
| Fitting, cosocle | C22×C92 | C22 |
| prime radical | C22×C9⋊D9 | C2 |
| hypocenter | C92 | C22⋊C4 |
| LCS | G ▷ C2×C92 ▷ C92 | |
| UCS | C1 ◁ C22 | |
265 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 265 conjugacy classes of subgroups — too many to draw. Only its 55 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|a^2=d,b^2=cd,e^3=h,f^3=g,b^a=bc,f^a=f^2g^2,g^a=g^2,e^b=e^2h^2,h^b=h^2138 conjugacy classes — view the full character table.
| maximal subgroup of | #72219982, #72220023, #72220039, … |
| maximal normal subgroup of | #72219982, #72220023, #72220039, … |
| maximal quotient of | #72219980, #72220085 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 181);