Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Sylow | #63001 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×C22⋊A4 |
| commutator, cosocle, hypocenter | Q82 | C22×C3 |
| Fitting | #63001 | C3 |
| hypercenter | C4⋊C4 | C22⋊A4 |
| LCS | G ▷ Q82 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
1239 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 1239 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 256 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=h,e^2=hi,f^2=i,b^a=cdh,c^a=bcgh,d^a=dfgi,e^a=defhi,g^a=dh,c^b=ch,d^b=di,g^d=gi,f^e=fi55 conjugacy classes — view the full character table.
| maximal subgroup of | #72268197, #72268198, #72268254, … |
| maximal normal subgroup of | #72268197, #72268198, #72268254, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085194);