Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | C22×C3×H(5,𝔽2) ⋊ C2 |
| Sylow | C3 ⋊ #62842 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C25×D3 |
| commutator, socle, cosocle | C22×C3 | C26 |
| Fitting | C22×C3×H(5,𝔽2) | C2 |
| hypercenter | C22×H(5,𝔽2) | D3 |
| hypocenter | C3 | #62842 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×H(5,𝔽2) | |
11770 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|b^a=bg,c^a=ch,f^a=fh,e^b=eh,i^b=i^2,d^c=dh150 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#62842, #72253851, #72253896, … |
| maximal normal subgroup of | D3×#62842, #72253851, #72253896, … |
| maximal quotient of | #72253746, #72253846, #72253882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1078581);