Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #11710660 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting, Sylow | #62694 ⋊ C3 |
| derived | C22×C42 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C42 | C22×C3 |
| Frattini, socle | C22 | C22×C22⋊A4 |
| Fitting | #62694 | C3 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 | |
1175 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 1175 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|f^2=i,g^2=h,b^a=bcdh,c^a=b,d^a=dh,f^a=g,g^a=fghi,h^a=hi,i^a=h,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei36 conjugacy classes — view the full character table.
| maximal subgroup of | #72235028, #72258041 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085041);