Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C22×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×GL(2,ℤ/4) |
| commutator, cosocle | C4×A4 | C24 |
| Frattini | C4 | C23×S4 |
| Fitting | C22×H(5,𝔽2) | D3 |
| socle | C23 | C22×#67 |
| hypercenter | H(5,𝔽2) | S4 |
| hypocenter | A4 | #576 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ H(5,𝔽2) | |
1793 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 1793 conjugacy classes of subgroups — too many to draw. Only its 135 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|e^2=f,d^a=de,e^a=ef,g^a=g^2,h^a=hi,c^b=cf,e^d=ef,h^g=i,i^g=hi61 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1089081);