Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #60398⋊C3 |
| Fitting, Sylow | #60398 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#120⋊C3 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C23×C3 |
| Frattini | C23 | C23×A4 |
| Fitting | #60398 | C3 |
| socle | C2 | C22×#120⋊C3 |
| hypercenter | C2×C4 | #120⋊C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
921 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 921 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|f^2=i,b^a=bhi,c^a=cdeghi,d^a=cfh,g^a=ghi,h^a=gi,c^b=cg,d^b=dh,e^b=ei,h^c=hi,g^d=gi46 conjugacy classes — view the full character table.
| maximal subgroup of | #72218407, #72233471, #72233783, … |
| maximal normal subgroup of | #72233471, #72233783, #72236824, … |
| maximal quotient of | C4×#3026⋊C3, C2×#60398⋊C3, C4×#3027⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084726);