Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×#1336 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #102 |
| commutator | C8 | C2×C3×C8 |
| Frattini, cosocle | C4×C8 | C22×C3 |
| socle | C2×C3 | D4⋊C8 |
| prime radical | C3×#135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C8 ◁ C3×C4×C8 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,5,3,7,11,9,14,8,12,16,10,6,13,4,15)(17,18,19);(1,3)(2,9)(4,12)(5,13)(6,15)(7,14)(8,10)(11,16)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 472);