Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C22×#120 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×S4 |
| commutator, cosocle | C22×A4 | C24 |
| Fitting | C22×#120 | D3 |
| socle | C24 | C23×D3 |
| hypercenter | #120 | S4 |
| hypocenter | A4 | D42 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ #120 | |
2652 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 2652 conjugacy classes of subgroups — too many to draw. Only its 139 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:
2^6,3,2^2|b^a=bf,g^a=g^2,h^a=hi,d^b=df,d^c=de,h^g=i,i^g=hi64 conjugacy classes — view the full character table.
| maximal subgroup of | #72215293, #72228056, C2×#11714540, … |
| maximal normal subgroup of | #72228056, C2×#11714540, D42×S4, … |
| maximal quotient of | #72226209, #72226217, #72226229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088921);