Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C4 |
| direct product | C3×#1412 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #354 |
| commutator | C42 | C2×C3×C4 |
| Frattini, cosocle | #127 | C22×C3 |
| prime radical | C3×#127 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#127 ◁ G | |
94 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)(3,10,12,16)(4,8,14,7)(5,9)(6,11,15,13)(17,18,19);(1,3,5,11)(2,6)(4,13,14,12)(7,10)(8,15)(9,16)42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 548);