Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#2221 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#131 ◁ G | |
148 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 148 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 — 3 generators on 35 points:
(2,8)(3,14)(4,9)(5,6)(10,25)(11,22)(12,21)(13,23)(15,29)(16,19)(17,18)(20,26)(27,32)(28,31)(33,34,35);(1,2,9,22,6,12,18,23,7,13,17,21,5,11,4,8)(3,10,24,25,14,26,29,31,16,27,30,32,19,28,15,20);(1,3)(2,10)(4,15)(5,19)(6,14)(7,16)(8,20)(9,24)(11,28)(12,26)(13,27)(17,30)(18,29)(21,32)(22,25)(23,31)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5512);