Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C3×#1355 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #329 |
| commutator | C42 | C2×C3×C4 |
| Frattini, cosocle | #97 | C22×C3 |
| prime radical | C3×#127 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#97 ◁ G | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3,8,4,6,12,14,5,9,11,16,13,7,10,15)(17,18,19);(2,6)(3,10)(4,13)(7,9)(8,16)(11,12)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 491);