Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | #737 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #4207⋊C2 |
| commutator | C2×#737 | C2×C4 |
| Frattini, socle | C22×C3 | #1113 |
| Fitting, cosocle | C22×#737 | C22 |
| prime radical | C22×H(3,𝔽3) | C2×D3 |
| hypocenter | #737 | C4⋊C4 |
| LCS | G ▷ C2×#737 ▷ #737 | |
| UCS | C1 ◁ C22 | |
209 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 209 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^4,3^4|a^2=d,b^2=c,g^3=h^2,b^a=bd,e^a=e^2,g^a=g^2h,h^a=h^2,e^b=e^2,f^b=f^2,f^e=fh78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1493);