Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #4625 |
| commutator, hypocenter | C22⋊SL(2,𝔽3) | C8 |
| Frattini | C22×C8 | S4 |
| Fitting | #1780 | D3 |
| socle | C23 | A4⋊C8 |
| prime radical | #5948 | C4 |
| cosocle | #1780⋊C3 | C2 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C4 | |
96 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^3,3,2^5|a^2=b,b^2=c,c^2=i,e^2=i,f^2=i,d^a=d^2,e^a=efg,f^a=fghi,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, 1085326);