Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C1 |
| direct product | C3×#1398 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | C42 |
| commutator | C4 | C3×C4×C8 |
| Frattini, cosocle | C2×C16 | C22×C3 |
| socle | C2×C3 | #346 |
| prime radical | C22×C3 | M32 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3,8,4,10,13,23,5,11,14,21,15,25,27,31,6,9,12,22,16,24,26,32,17,7,18,19,28,20,29,30)(33,34,35);(2,7,9,11)(3,12)(8,21,22,19)(10,20,24,25)(13,26)(14,18)(23,31,32,30)(27,29)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 534);