Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #2017 ⋊ D3 |
| derived | #338⋊C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #72997 |
| commutator, cosocle, hypocenter | #338⋊C3 | C22 |
| Frattini | C22×C4 | C2×S4 |
| Fitting | #2017 | D3 |
| hypercenter | C4 | #857⋊C2 |
| LCS | G ▷ #338⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 | |
184 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
2^2,3,2^6|b^2=fg,d^2=f,e^2=fgh,f^2=i,b^a=bfgi,c^a=c^2,d^a=deh,e^a=efghi,f^a=fh,g^a=ghi,d^b=dgi,e^b=eh,d^c=eg,e^c=defg,f^c=fghi,g^c=hi,h^c=gh,e^d=efg,h^d=hi,f^e=fi,g^e=gi27 conjugacy classes — view the full character table.
| maximal subgroup of | #72213991 |
| maximal normal subgroup of | #72213991 |
| maximal quotient of | #72213212, #72213217 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085768);