Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #11709221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting, Sylow | #13827 ⋊ C3 |
| derived | C22×C42 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C42 | C22×C3 |
| Frattini, socle | C24 | C22×A4 |
| Fitting | #13827 | C3 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 | |
548 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|b^2=h,c^2=i,b^a=bcgi,c^a=befgh,d^a=dfgh,e^a=ehi,f^a=fg,g^a=f,h^a=fhi,i^a=fgh,c^b=cf,d^b=dg,e^b=efg,d^c=dfg,e^c=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #72235187 |
| maximal quotient of | #72235184, #72235416 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083602);