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×#1389 |
| 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 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
128 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 128 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:
(1,2,5,10,7,11,6,22)(3,12,14,29,17,25,16,31)(4,18,19,27,21,32,20,9)(8,23,24,15,26,30,28,13)(33,34,35);(1,3,4,15,5,14,19,30,7,17,21,13,6,16,20,23)(2,8,9,25,22,28,32,29,11,26,27,12,10,24,18,31)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 525);