Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C24×D4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #87524 |
| commutator, cosocle | C4×C22⋊A4 | C22 |
| Frattini | C4 | C2×#1020 |
| Fitting | C24×D4 | D3 |
| socle | C25 | #67 |
| hypercenter | D4 | #1020 |
| hypocenter | C22⋊A4 | D8 |
| LCS | G ▷ C4×C22⋊A4 ▷ C2×C22⋊A4 ▷ C22⋊A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
1333 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 1333 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^4,3,2^4|c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=fg,h^a=hi,c^b=cd,f^e=g,g^e=fg,h^e=i,i^e=hi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72211581, #72226657, #72228089, … |
| maximal normal subgroup of | #72228285, #72234216, #72252247, … |
| maximal quotient of | #72238100, C2×#11715692, #72267839, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090073);