Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived | #338⋊C3 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #4626 |
| commutator, hypocenter | #338⋊C3 | C4 |
| Frattini | C23×C4 | S4 |
| Fitting | #2091 | D3 |
| socle | C2 | #68003 |
| prime radical, cosocle | #2091⋊C3 | C2 |
| LCS | G ▷ #338⋊C3 | |
| UCS | C1 ◁ C4 | |
107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
Polycyclic presentation:
2^2,3,2^6|a^2=b,d^2=f,e^2=fgh,f^2=i,c^a=c^2,d^a=deh,e^a=efhi,f^a=fh,g^a=gh,d^b=dg,e^b=ehi,d^c=eghi,e^c=def,f^c=fghi,g^c=hi,h^c=ghi,e^d=efg,h^d=hi,f^e=fi,g^e=gi33 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085348);