Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #32899⋊C3 |
| Fitting, Sylow | #32899 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120⋊C3 |
| commutator, cosocle | C2×H(5,𝔽2) | C22×C3 |
| Frattini | C24 | C22×A4 |
| Fitting | #32899 | C3 |
| hypercenter | C2×C4 | #120⋊C3 |
| hypocenter | H(5,𝔽2) | C3×D4 |
| LCS | G ▷ C2×H(5,𝔽2) ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
736 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
3,2^8|e^2=i,b^a=bg,c^a=cdh,d^a=ci,f^a=fg,g^a=fi,c^b=cf,d^b=dg,e^b=eh,g^c=gi,f^d=fi43 conjugacy classes — view the full character table.
| maximal subgroup of | #72241384, #72245678, #72245692, … |
| maximal normal subgroup of | #72241384, #72245678, #72245692, … |
| maximal quotient of | #72341925, C2×#32899⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083985);