Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C3×#1782 ⋊ C2 |
| Sylow | C3 ⋊ #23896 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #73852 |
| commutator, cosocle | C24×C3 | C24 |
| Frattini | C24 | C23×D3 |
| Fitting | C3×#1782 | C2 |
| socle | C2×C3 | #2403 |
| hypercenter | #1782 | D3 |
| hypocenter | C3 | #23896 |
| LCS | G ▷ C24×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ #1782 | |
1764 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 1764 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|b^2=h,b^a=be,c^a=cf,d^a=de,g^a=gh,i^a=i^2,c^b=cg,e^c=eh,i^c=i^2,f^d=fh72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#23896 |
| maximal normal subgroup of | D3×#23896 |
| maximal quotient of | #72253393, #72253488 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 201125);