Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting, derived, Sylow | #62849 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | C22⋊(C22⋊A4) |
| commutator, Fitting, cosocle, hypocenter | #62849 | C3 |
| LCS | G ▷ #62849 | |
| UCS | C1 ◁ C22 | |
1218 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 1218 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=h,d^2=i,e^2=i,b^a=bchi,c^a=bhi,d^a=defghi,e^a=dg,f^a=fghi,g^a=fi,c^b=ch,d^b=di,g^b=gi,f^c=fi,e^d=ei38 conjugacy classes — view the full character table.
| maximal subgroup of | #72210072, C2×#11710743 |
| maximal normal subgroup of | #72210072, C2×#11710743 |
| maximal quotient of | C2×#11710743, #72267532, #72267569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085124);