Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #85645 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting, Sylow | C22×#379 ⋊ C3 |
| derived | C26 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #85645 |
| commutator, hypocenter | C26 | C3×C4 |
| Frattini | C23 | C2×C22⋊A4 |
| Fitting | C22×#379 | C3 |
| socle | C25 | C2×A4 |
| prime radical | C2×C22⋊(C22⋊A4) | C2 |
| cosocle | C27 | C2×C3 |
| LCS | G ▷ C26 | |
| UCS | C1 ◁ C2 | |
5649 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 5649 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
3,2^8|b^2=i,c^a=d,d^a=cdi,e^a=fhi,f^a=efg,g^a=h,h^a=gh,c^b=cg,d^b=dh48 conjugacy classes — view the full character table.
| maximal subgroup of | #72233785, #72234979, #72234980, … |
| maximal normal subgroup of | #72233785, #72234979, #72234980, … |
| maximal quotient of | #72233412, #72238259, C2×#11709759, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084140);