Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #4626 |
| commutator, prime radical, hypocenter | #95⋊C3 | C8 |
| Frattini | C22×C8 | S4 |
| Fitting | #1843 | D3 |
| socle | C23 | A4⋊C8 |
| cosocle | #1843⋊C3 | C2 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C4 | |
63 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^3,3,2^5|a^2=b,b^2=c,c^2=gh,e^2=g,f^2=ghi,d^a=d^2,e^a=efi,f^a=fgi,g^a=gi,h^a=hi,e^b=eh,f^b=fi,e^d=fhi,f^d=efg,g^d=ghi,h^d=i,i^d=hi,f^e=fgh42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085328);