Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #4625 |
| commutator, hypocenter | C22⋊SL(2,𝔽3) | C2×C4 |
| Frattini | C22×C4 | C2×S4 |
| Fitting | #2388 | D3 |
| socle | C23 | C2×SL(2,ℤ/4) |
| prime radical | #2388⋊C3 | C2 |
| cosocle | #395⋊C3 | C22 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C4 | |
199 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 199 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) = 48 |
Polycyclic presentation:
2^3,3,2^5|a^2=c,c^2=i,e^2=i,f^2=i,d^a=d^2,e^a=efh,f^a=fgi,g^a=gh,e^b=eg,f^b=fh,e^c=eg,f^c=fh,e^d=f,f^d=efi,g^d=h,h^d=gh,f^e=fi42 conjugacy classes — view the full character table.
| maximal subgroup of | #72268900 |
| maximal normal subgroup of | #72268900 |
| maximal quotient of | #72268905 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085747);