Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C25×C4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×S4 |
| commutator, cosocle | C22×A4 | C24 |
| Frattini | C22 | C23×S4 |
| Fitting | C25×C4 | D3 |
| socle | C24 | C23×D3 |
| hypercenter | C23×C4 | S4 |
| hypocenter | A4 | #525 |
| LCS | G ▷ C22×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 | |
2419 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 2419 conjugacy classes of subgroups — too many to draw. Only its 153 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:
2^6,3,2^2|b^2=f,c^a=cf,d^a=de,g^a=g^2,h^a=hi,h^g=i,i^g=hi88 conjugacy classes — view the full character table.
| maximal subgroup of | #72228254, #72228287, #72258898, … |
| maximal normal subgroup of | #72228287, #72258898, #41×GL(2,ℤ/4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088843);