Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#3287 |
| 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 ◁ C3×C4×D4 ◁ G | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(3,4)(6,10)(7,14)(8,16)(11,13)(12,17)(19,20)(21,22,23);(5,6,9,13)(7,11,14,10)(8,17,16,19)(12,15,20,18);(2,4)(5,7,9,14)(6,11,13,10)(8,15,16,18)(12,17,20,19);(5,8,9,16)(6,12,13,20)(7,15,14,18)(10,19,11,17)105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17727);