Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #4626 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| derived | C42⋊C3 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #4626 |
| commutator, hypocenter | C42⋊C3 | C2×C8 |
| Frattini | C22×C4 | C2×S4 |
| Fitting | C4×#127 | D3 |
| socle | C23 | C2×SL(2,ℤ/4) |
| prime radical | C2×#127⋊C3 | C4 |
| cosocle | #384⋊C3 | C22 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C4 | |
195 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 30 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:
2^4,3,2^4|a^2=c,c^2=d,f^2=h,g^2=i,e^a=e^2,f^a=fg,g^a=ghi,h^a=hi,f^b=fh,g^b=gi,f^c=fh,g^c=gi,f^e=gh,g^e=fg,h^e=i,i^e=hi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085722);