Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#3018 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C2×C3 | C2×#120 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(2,6)(4,11)(7,13)(8,12)(9,15)(14,16)(17,18,19);(1,2)(3,7)(4,12)(5,8)(6,11)(9,16)(10,14)(13,15);(4,11)(6,12)(9,15)(13,16);(1,3,5,10)(2,7,8,14)(4,9,11,15)(6,13,12,16)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17458);