Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#1388 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #102 |
| commutator | C8 | C3×C42 |
| Frattini, cosocle | C22×C8 | C22×C3 |
| socle | C2×C3 | #340 |
| prime radical | C23×C3 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
124 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,5,11,8,13,7,24)(3,14,16,26,19,28,18,32)(4,20,22,29,23,12,6,10)(9,17,25,21,27,31,30,15)(33,34,35);(1,3,6,17,7,18,23,15,8,19,22,31,5,16,4,21)(2,9,12,26,11,25,10,28,13,27,20,32,24,30,29,14)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 524);