Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#3415 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C24×C3 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C3×#41 ◁ C3×#131 ◁ G | |
370 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 370 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)(3,8)(4,7)(5,18)(6,11)(9,15)(10,16)(12,22)(13,19)(14,28)(17,21)(20,29)(23,27)(24,26)(25,30)(31,32)(33,34,35);(2,7)(4,15)(5,16)(8,19)(9,10)(11,21)(13,27)(14,26)(17,29)(18,20)(22,30)(23,24)(25,32)(28,31);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,23)(10,24)(11,22)(15,27)(16,26)(17,25)(18,28)(20,31)(21,30)(29,32);(3,12)(8,22)(13,25)(14,26)(19,30)(23,31)(24,28)(27,32)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17855);