Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #1446 ⋊ D3 |
| derived | C22⋊SL(2,𝔽3) ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #1020⋊C2 |
| commutator, hypocenter | C22⋊SL(2,𝔽3) | C2×C4 |
| Frattini, socle | C24 | C2×S4 |
| Fitting | #1446 | D3 |
| prime radical | C2×#5931 | C2 |
| cosocle | C2×C22⋊SL(2,𝔽3) | C22 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
372 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^3,3,2^5|b^2=c,e^2=i,f^2=i,d^a=d^2,e^a=ef,f^a=fi,g^a=gh,e^b=eg,f^b=fh,e^d=f,f^d=efi,g^d=h,h^d=gh,f^e=fi42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085521);