Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | C22×#458 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #85645 |
| commutator, cosocle, hypocenter | C22×H(5,𝔽2) | C2×C3 |
| Frattini | C23 | C2×C22⋊A4 |
| Fitting | C22×#458 | C3 |
| socle | C23 | #120⋊C3 |
| prime radical | C22⋊#997 | C2 |
| LCS | G ▷ C22×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
969 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 969 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) = 48 |
Polycyclic presentation:
3,2^8|b^2=i,b^a=bghi,c^a=cdei,d^a=cf,e^a=fi,f^a=ef,g^a=ghi,h^a=gi,c^b=cg,d^b=dh,h^c=hi,g^d=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72258331, #72258338, #72258348, … |
| maximal normal subgroup of | #72258331, #72258338, #72258348, … |
| maximal quotient of | #72267337 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084721);