Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#2097 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
238 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 31 points:
(5,6,8,10)(7,11,12,9)(13,14,16,20,17,21,25,18)(15,22,23,27,24,28,26,19)(29,30,31);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,18)(16,25)(19,22)(20,21)(23,26)(27,28);(2,4)(13,15,17,24)(14,19,21,27)(16,26,25,23)(18,22,20,28)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5388);