Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#1395 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #99 |
| commutator | C2×C4 | C3×C42 |
| Frattini, cosocle | C2×#41 | C22×C3 |
| socle | C2×C3 | #342 |
| prime radical | C2×C3×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ G | |
162 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,5,10)(3,13,15,29)(4,11,18,27)(6,9,20,26)(7,12,21,28)(8,22,23,14)(16,25,30,32)(17,24,19,31)(33,34,35);(1,3,6,16,7,17,4,14)(2,8,11,24,12,25,9,13)(5,19,20,22,21,15,18,30)(10,23,27,31,28,32,26,29)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 531);