Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C4×#737 ⋊ C22 |
| Sylow | #737 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#4207⋊C2 |
| commutator, cosocle | C2×#737 | C23 |
| Frattini, socle | C2×C3 | C2×#1113 |
| Fitting | C4×#737 | C22 |
| hypercenter | C4 | #4207⋊C2 |
| hypocenter | #737 | C2×D4 |
| LCS | G ▷ C2×#737 ▷ #737 | |
| UCS | C1 ◁ C2 ◁ C4 | |
359 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^4,3^4|c^2=d,f^3=h,e^a=e^2,f^a=f^2h^2,h^a=h^2,c^b=cd,e^b=e^2,g^b=g^2,g^e=gh^263 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#4207⋊C2, #72259341 |
| maximal normal subgroup of | D4×#4207⋊C2, #72259341 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2480);