Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | #60402 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C23×C3 |
| Frattini | C23 | C23×A4 |
| Fitting | #60402 | C3 |
| hypercenter | Q8 | #120⋊C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C2 ◁ Q8 | |
908 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 908 conjugacy classes of subgroups — too many to draw. Only its 54 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|b^2=i,e^2=i,f^2=i,b^a=bh,c^a=dfgh,d^a=cdfhi,e^a=eg,g^a=h,h^a=gh,c^b=cg,d^b=dh,h^c=hi,g^d=gi,f^e=fi46 conjugacy classes — view the full character table.
| maximal subgroup of | #72236799, #72236804, #72236805, … |
| maximal normal subgroup of | #72236799, #72236804, #72236805, … |
| maximal quotient of | #72341361, #72341793, C2×#11710352 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084733);