Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#1304 ⋊ C2 |
| Sylow | C3 ⋊ #10336 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #4671 |
| commutator, cosocle | C25×C3 | C23 |
| Frattini | C25 | C22×D3 |
| Fitting | C3×#1304 | C2 |
| socle | C22×C3 | #392 |
| hypercenter | #1304 | D3 |
| hypocenter | C3 | #10336 |
| LCS | G ▷ C25×C3 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ #1304 | |
1566 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 1566 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|b^a=bd,c^a=ce,f^a=fgh,i^a=i^2,c^b=cf,e^b=eh,i^b=i^2,d^c=dg,i^c=i^260 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#10336, #72231807, #72248728, … |
| maximal normal subgroup of | D3×#10336, #72231807, #72248728, … |
| maximal quotient of | #72230367, #72231787, #72231790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 72521);