Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#1676 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
144 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 384 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,5,8)(3,9,12,16)(4,13,10,7)(6,14,15,11)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26)(33,34,35);(1,3,10,14,5,12,4,11)(2,6,13,16,8,15,7,9);(17,19,21,27)(18,22,24,31)(20,29,30,26)(23,32,28,25)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4967);