Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #47815⋊C3 |
| Fitting, Sylow | #47815 ⋊ C3 |
| derived | #2845 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #3567⋊C3 |
| commutator, cosocle, hypocenter | #2845 | C2×C3 |
| Frattini, socle | C23 | C2×C22⋊A4 |
| Fitting | #47815 | C3 |
| LCS | G ▷ #2845 | |
| UCS | C1 ◁ C2 | |
339 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 339 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=gh,d^2=h,e^2=g,f^2=i,b^a=dgh,c^a=ci,d^a=bcdghi,e^a=fh,f^a=efg,g^a=i,h^a=hi,i^a=gi,c^b=cg,d^b=dh,f^b=fg,d^c=di,e^c=eg,f^c=fi,e^d=ei28 conjugacy classes — view the full character table.
| maximal subgroup of | #72231838, C2×#47815⋊C3 |
| maximal normal subgroup of | #72231838, C2×#47815⋊C3 |
| maximal quotient of | #72231844, #72240348, C2×#47815⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084477);