Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#857⋊C2 |
| commutator, cosocle, hypocenter | #95⋊C3 | C23 |
| Frattini, socle | C23 | C22×S4 |
| Fitting | #2397 | D3 |
| prime radical | #2397⋊C3 | C2 |
| hypercenter | C4 | #857⋊C2 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 | |
474 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 474 conjugacy classes of subgroups — too many to draw. Only its 32 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|a^2=gh,c^2=gh,e^2=g,f^2=ghi,c^a=cgh,d^a=d^2,e^a=ef,f^a=fghi,g^a=gi,h^a=hi,e^b=eh,f^b=fi,e^d=fhi,f^d=efg,g^d=ghi,h^d=i,i^d=hi,f^e=fgh36 conjugacy classes — view the full character table.
| maximal subgroup of | #72340817, #72340820, #72340832 |
| maximal normal subgroup of | #72340817, #72340832 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086325);