Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#3203 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×#41 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,20)(13,17)(14,18)(16,22)(19,23)(21,24)(25,26,27);(1,3)(2,5)(4,8)(6,7)(9,11,13,18)(10,14,17,15)(12,21,22,19)(16,24,20,23);(3,8)(5,7)(10,15)(11,18)(14,17)(16,24)(19,21)(20,23);(2,6)(5,7)(9,12,13,22)(10,16,17,20)(11,19,18,21)(14,23,15,24)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17643);