Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#2405 ⋊ C2 |
| Sylow | C3 ⋊ #25563 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#937 |
| commutator, cosocle | C24×C3 | C24 |
| Frattini | C24 | C23×D3 |
| Fitting | C3×#2405 | C2 |
| socle | C22×C3 | C2×#120 |
| hypercenter | #2405 | D3 |
| hypocenter | C3 | #25563 |
| LCS | G ▷ C24×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ #2405 | |
1608 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|b^2=e,c^2=f,b^a=be,c^a=cf,d^a=de,g^a=gh,i^a=i^2,c^b=cg,f^b=fh,g^c=gh,i^c=i^281 conjugacy classes — view the full character table.
| maximal subgroup of | #72271807, #72271810, #72271833, … |
| maximal normal subgroup of | #72271807, #72271810, #72271833, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 222712);