Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #37475⋊C3 |
| Fitting, Sylow | #37475 ⋊ C3 |
| derived | #2840 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #3532⋊C3 |
| commutator, cosocle, hypocenter | #2840 | C2×C3 |
| Frattini, socle | C23 | C2×C22⋊A4 |
| Fitting | #37475 | C3 |
| LCS | G ▷ #2840 | |
| UCS | C1 ◁ C2 | |
622 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 622 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) = 24 |
Polycyclic presentation:
3,2^8|b^2=h,d^2=h,b^a=bcd,c^a=cefg,d^a=bcefg,e^a=eg,f^a=cehi,g^a=i,h^a=ghi,i^a=gi,c^b=cg,d^b=dh,e^b=eg,f^b=fgi,d^c=di,e^d=egi28 conjugacy classes — view the full character table.
| maximal subgroup of | #72209879, #72209918, #72226619, … |
| maximal normal subgroup of | #72209879, #72209918, #72247529, … |
| maximal quotient of | #72209874, #72226614, #72227882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084434);