Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #2785 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4625⋊C2 |
| commutator, cosocle | #395⋊C3 | C22 |
| Frattini | C22×C4 | C2×S4 |
| Fitting | #2785 | D3 |
| socle | C23 | GL(2,ℤ/4) |
| hypocenter | C22⋊SL(2,𝔽3) | D4 |
| LCS | G ▷ #395⋊C3 ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
172 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^3,3,2^5|c^2=i,e^2=i,f^2=i,b^a=bc,c^a=ci,d^a=d^2,e^a=ef,f^a=fi,g^a=gh,c^b=ci,e^b=eh,f^b=fgh,e^c=eg,f^c=fh,e^d=f,f^d=efi,g^d=h,h^d=gh,f^e=fi24 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085777);