Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C4 |
| direct product | C3×#1410 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #353 |
| commutator | C42 | C2×C3×C4 |
| Frattini, cosocle | C4⋊Q8 | C22×C3 |
| prime radical | C3×#127 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊Q8 ◁ G | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3,8,5,9,12,16)(4,6,10,14,13,7,11,15)(17,18,19);(2,6)(3,10)(4,13)(7,9)(11,12)(14,15)51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 546);