Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #2275 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#857 |
| commutator, cosocle, hypocenter | #95⋊C3 | C23 |
| Frattini, socle | C23 | C22×S4 |
| Fitting | #2275 | D3 |
| hypercenter | D4 | #857 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 ◁ D4 | |
530 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 530 conjugacy classes of subgroups — too many to draw. Only its 33 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|e^2=g,f^2=ghi,d^a=d^2,e^a=efhi,f^a=fgh,g^a=gi,h^a=hi,c^b=cgh,e^d=fhi,f^d=efg,g^d=ghi,h^d=i,i^d=hi,f^e=fgh45 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086201);