Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| Fitting | C22×C3×#120 ⋊ C2 |
| Sylow | C3 ⋊ #15258 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator, socle, cosocle | C24×C3 | C24 |
| Fitting | C22×C3×#120 | C2 |
| hypercenter | C22×#120 | D3 |
| hypocenter | C3 | #15258 |
| LCS | G ▷ C24×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C22×#120 | |
5134 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|b^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,i^b=i^2114 conjugacy classes — view the full character table.
| maximal subgroup of | #72215002, D3×#15258, #72248446, … |
| maximal normal subgroup of | D3×#15258, #72248446, #72248449, … |
| maximal quotient of | #72215002, #72248906, #72249228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 107264);