Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C16 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C4×C8 |
| Frattini | C2×C8 | C2×S4 |
| Fitting | C2×#504 | D3 |
| socle | C22 | C8×S4 |
| prime radical | C2×#41⋊C3 | C2×C4 |
| cosocle | C2×#131⋊C3 | C22 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C16 | |
211 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 211 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^5,3,2^3|a^2=d,b^2=c,c^2=e,e^2=i,g^2=i,h^2=i,f^a=f^2,g^a=gh,h^a=hi,g^f=h,h^f=ghi,h^g=hi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085869);