Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#3288 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ G | |
464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(2,7)(4,5)(8,15)(9,13)(10,16)(11,14)(17,18,19);(1,2)(3,10)(4,13)(5,7)(6,9)(8,11)(12,15)(14,16);(1,2)(3,8)(4,7)(5,13)(6,9)(10,11)(12,16)(14,15);(1,3)(2,8)(4,14)(5,11)(6,12)(7,15)(9,16)(10,13)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17728);