Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Sylow | C3 ⋊ #15949 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator, socle, cosocle | C24×C3 | C24 |
| Fitting | C3×#1438 | C2 |
| prime radical | C23×C3×D4 | C22 |
| hypercenter | #1438 | D3 |
| hypocenter | C3 | #15949 |
| LCS | G ▷ C24×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ #1438 | |
2354 conjugacy classes of subgroups; only the 315 normal subgroups are shown.
This group has 2354 conjugacy classes of subgroups — too many to draw. Only its 315 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^8,3|a^2=eh,c^2=e,b^a=be,c^a=cf,d^a=dh,i^a=i^2,c^b=cg,d^b=de,i^c=i^2102 conjugacy classes — view the full character table.
| maximal subgroup of | #72270947, #72271007, #72271022, … |
| maximal normal subgroup of | #72270947, #72271007, #72271022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 117094);