Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #33564⋊C3 |
| Fitting, Sylow | #33564 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | B(4,𝔽2)⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽4) | C22×C3 |
| Frattini | C2×Q8 | C22×A4 |
| Fitting | #33564 | C3 |
| socle | C2 | C2×B(4,𝔽2)⋊C3 |
| hypercenter | C2×C4 | #120⋊C3 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
272 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
3,2^8|e^2=i,f^2=i,g^2=i,c^a=dh,d^a=cdhi,f^a=gi,g^a=fghi,c^b=cf,d^b=dg,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=ghi,f^d=fh,g^d=gi,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72209926, #72209927, #72268022 |
| maximal normal subgroup of | #72209927 |
| maximal quotient of | #72227888, #72287968 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084088);