Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C4 |
| direct product | C3×#1718 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
164 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 164 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)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32)(33,34,35);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(18,22)(23,30)(24,29)(25,31);(3,10)(6,15)(9,16)(11,14)(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5009);