Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C22×#120 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #85493 |
| commutator, cosocle | C23×A4 | C23 |
| Frattini | C23 | C22×S4 |
| Fitting | C22×#120 | D3 |
| socle | C23 | #937 |
| hypercenter | #120 | S4 |
| hypocenter | A4 | B(4,𝔽2) |
| LCS | G ▷ C23×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #120 | |
1244 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 51 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:
2^6,3,2^2|c^a=ce,d^a=df,g^a=g^2,h^a=hi,c^b=cd,e^b=ef,h^g=i,i^g=hi46 conjugacy classes — view the full character table.
| maximal subgroup of | #72219749, #72219755, #72222909, … |
| maximal normal subgroup of | #72219755, S4×B(4,𝔽2), #72252544, … |
| maximal quotient of | #72219749, #72220400, #72222909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087583);