Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3)×H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32×C4 · Schur multiplier (H2) = C25 |
| direct product | 3−1+2×#518 |
| unambiguous semidirect product | (C9×#518)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C22×C3 | C24×C32 |
| commutator | C2×C3 | C23×C32×C4 |
| prime radical | C22×C32×D4 | C2×C3 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
1352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,3,6,8,7,5,9)(10,11,12,13)(14,15)(16,20)(17,18)(19,21);(2,5,6)(4,8,9)(15,18)(19,20);(14,15)(16,19)(17,18)(20,21);(14,16)(15,19)(17,21)(18,20)374 conjugacy classes — view the full character table.
| Fitting subgroup of | #72268898, AGL(1,ℤ/9)×#518 |
| maximal subgroup of | 3−1+2×#3484, #72268898, AGL(1,ℤ/9)×#518 |
| maximal normal subgroup of | 3−1+2×#3484, #72268898, AGL(1,ℤ/9)×#518 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20601);