Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C3×#2693 ⋊ C2 |
| Sylow | C3 ⋊ #47830 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator, socle, cosocle | C23×C3 | C25 |
| Fitting | C3×#2693 | C2 |
| hypercenter | #2693 | D3 |
| hypocenter | C3 | #47830 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #2693 | |
3240 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|d^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,i^e=i^284 conjugacy classes — view the full character table.
| maximal subgroup of | #72247956, #72248006, #72248094, … |
| maximal normal subgroup of | #72247956, #72248006, #72248094, … |
| maximal quotient of | #72255258, #72255509, #72255565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 676122);