Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Sylow | #62624 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | C22⋊A4 |
| commutator, hypocenter | H(5,𝔽2) | C3×C8 |
| Frattini | C8 | C2×C22⋊A4 |
| Fitting | #62624 | C3 |
| socle | C2 | C8×C22⋊A4 |
| prime radical | #5950 | C4 |
| cosocle | #3573 | C2×C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C16 | |
393 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 393 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.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|b^2=g,g^2=h,h^2=i,c^a=cei,d^a=cdfh,e^a=ci,f^a=cdeh,f^c=fi,e^d=ei88 conjugacy classes — view the full character table.
| maximal subgroup of | #72227314, #72254591, #72319766, … |
| maximal normal subgroup of | #72227314, #72254591, #72319766, … |
| maximal quotient of | C16×#997, C2×#11710557 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084938);