Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#3040 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C2×C3×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(2,4)(5,6)(7,13)(8,17)(9,10)(11,16)(12,18)(14,19)(15,20)(21,22,23);(1,2,3,4)(6,10)(11,19)(12,17)(13,20);(5,7)(6,11)(8,14)(9,15)(10,19)(12,20)(13,17)(16,18);(5,8,9,18)(6,12,10,17)(7,14,15,16)(11,20,19,13)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17480);