Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #15172⋊C3 |
| Fitting, Sylow | #15172 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#127⋊C3 |
| commutator, cosocle | C2×#95 | C22×C3 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | #15172 | C3 |
| hypercenter | C2×C4 | #127⋊C3 |
| hypocenter | #95 | C3×D4 |
| LCS | G ▷ C2×#95 ▷ #95 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
541 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 541 conjugacy classes of subgroups — too many to draw. Only its 29 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:
3,2^8|c^2=f,d^2=g,e^2=hi,b^a=bcdeh,c^a=dehi,d^a=cdef,e^a=efghi,f^a=ghi,g^a=fgi,c^b=cf,d^b=dg,e^b=ei,d^c=dh48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#15172⋊C3, #72222667, #72222668, … |
| maximal normal subgroup of | C2×#15172⋊C3, #72222667, #72222668, … |
| maximal quotient of | C2×#15172⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083680);