Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting, Sylow | C22×B(4,𝔽2) ⋊ C3 |
| derived | C22×H(5,𝔽2) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #85645 |
| commutator, cosocle, hypocenter | C22×H(5,𝔽2) | C2×C3 |
| Frattini | C23 | C2×C22⋊A4 |
| Fitting | C22×B(4,𝔽2) | C3 |
| socle | C23 | #120⋊C3 |
| LCS | G ▷ C22×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
1279 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 1279 conjugacy classes of subgroups — too many to draw. Only its 23 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|c^a=cdf,d^a=cef,e^a=fi,f^a=ef,g^a=ghi,h^a=g,c^b=cg,d^b=dh,h^c=hi,g^d=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72209660, #72209714, #72209719, … |
| maximal normal subgroup of | #72209660, #72209719, #72209770, … |
| maximal quotient of | #72209714, #72209717, #72209873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084718);