Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | #63093 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×C22⋊A4 |
| commutator, cosocle | #585 | C22×C3 |
| Frattini | C4 | C22×C22⋊A4 |
| Fitting | #63093 | C3 |
| hypercenter | Q16 | C22⋊A4 |
| hypocenter | H(5,𝔽2) | C3×D4 |
| LCS | G ▷ #585 ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
1883 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 1883 conjugacy classes of subgroups — too many to draw. Only its 49 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=i,c^2=i,h^2=i,b^a=bei,c^a=cei,d^a=dfghi,e^a=eg,f^a=de,g^a=ei,c^b=ch,h^b=hi,h^c=hi,g^d=gi,f^e=fi46 conjugacy classes — view the full character table.
| maximal subgroup of | #72227318, #72341995, #72342022, … |
| maximal normal subgroup of | #72227318, #72341995, #72342022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085283);