Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #1726 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#857 |
| commutator, hypocenter | #95⋊C3 | C2×C4 |
| Frattini | C22×C4 | C2×S4 |
| Fitting | #1726 | D3 |
| socle | C23 | C4×S4 |
| prime radical | #4622 | C4 |
| cosocle | #376⋊C3 | C22 |
| hypercenter | C8 | #857 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C4 ◁ C8 | |
195 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^3,3,2^5|b^2=c,c^2=gh,e^2=g,f^2=ghi,b^a=bgh,d^a=d^2,e^a=efi,f^a=fghi,g^a=gi,h^a=hi,e^d=fh,f^d=efgh,g^d=ghi,h^d=i,i^d=hi,f^e=fgh48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085389);