Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | C3×#3496 ⋊ C2 |
| Sylow | C3 ⋊ D4×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C25×D3 |
| commutator, socle, cosocle | C22×C3 | C26 |
| Fitting | C3×#3496 | C2 |
| hypercenter | #3496 | D3 |
| hypocenter | C3 | D4×H(5,𝔽2) |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #3496 | |
12414 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|b^a=bg,c^a=ch,i^b=i^2,f^c=fh,e^d=eh,i^f=i^2129 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D4×H(5,𝔽2), #72270965, #72270983, … |
| maximal normal subgroup of | D3×D4×H(5,𝔽2), #72270965, #72270983, … |
| maximal quotient of | #72248006, #72248018, #72248050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1081671);