Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#1636 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
188 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 188 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:
(1,2,5,9)(3,10,13,16)(4,7,14,8)(6,11,15,12)(17,18,22,26)(19,27,20,29)(21,31,32,25)(23,30,24,28)(33,34,35);(1,3)(2,6)(4,11)(5,13)(7,10)(8,16)(9,15)(12,14)(17,19,22,20)(18,23,26,24)(21,30,32,28)(25,29,31,27);(1,4,5,14)(2,7,9,8)(3,11,13,12)(6,10,15,16)(17,20,22,19)(18,24,26,23)(21,28,32,30)(25,27,31,29)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4927);