Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C23×C16 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #5402 |
| commutator, cosocle | C8×A4 | C23 |
| Frattini | C8 | C22×S4 |
| Fitting | C23×C16 | D3 |
| socle | C23 | C2×D24 |
| hypercenter | C2×C16 | S4 |
| hypocenter | A4 | #508 |
| LCS | G ▷ C8×A4 ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 | |
375 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 375 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3,2^2|b^2=f,c^2=de,d^2=ef,e^2=f,c^a=cd,d^a=de,e^a=ef,g^a=g^2,h^a=hi,h^g=i,i^g=hi76 conjugacy classes — view the full character table.
| maximal subgroup of | #72342674, S4×#508, #72342735, … |
| maximal normal subgroup of | #72342674, S4×#508, #72342735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088572);