Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C4 |
| direct product | C3×#1406 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #353 |
| commutator | C42 | C2×C3×C4 |
| Frattini, cosocle | #124 | C22×C3 |
| prime radical | C3×#496 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#124 ◁ G | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3,8)(4,7,10,15)(5,9,11,16)(6,12,14,13)(17,18,19);(2,6)(3,10)(4,13)(7,9)(11,12)(14,15)42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 542);