Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#3414 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C3 |
| socle | C2×C3 | C22×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
364 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 180 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 — 4 generators on 35 points:
(1,2)(3,12)(4,9)(5,19)(6,8)(7,14)(10,17)(11,18)(13,26)(15,31)(16,28)(20,27)(21,25)(22,30)(23,29)(24,32)(33,34,35);(1,3)(2,7)(4,13)(5,20)(6,16)(8,22)(9,23)(10,26)(11,27)(12,19)(14,17)(15,18)(21,29)(24,31)(25,28)(30,32);(2,8)(7,22)(9,11)(10,24)(12,28)(19,25)(23,27)(26,31);(1,4,6,18)(2,9,8,11)(3,13,16,15)(5,17,21,32)(7,23,22,27)(10,25,24,19)(12,26,28,31)(14,29,30,20)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17854);