Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| Fitting | C3×#2403 ⋊ C2 |
| Sylow | C3 ⋊ #29634 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#937 |
| commutator, cosocle | C24×C3 | C24 |
| Frattini | C24 | C23×D3 |
| Fitting | C3×#2403 | C2 |
| socle | C22×C3 | C2×#120 |
| hypercenter | #2403 | D3 |
| hypocenter | C3 | #29634 |
| LCS | G ▷ C24×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ #2403 | |
1908 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|b^a=be,c^a=cf,g^a=gh,c^b=cg,f^b=fh,i^b=i^2,d^c=de75 conjugacy classes — view the full character table.
| maximal subgroup of | #72248828, #72248847, D3×#29634, … |
| maximal normal subgroup of | #72248828, #72248847, D3×#29634, … |
| maximal quotient of | #72253701, #72253763, #72270466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 275273);