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×#3416 |
| 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 | |
360 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 360 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:
(1,2,6,11)(3,8,12,22)(4,7,17,21)(5,18,16,10)(9,25,20,15)(13,19,26,30)(14,29,27,24)(23,32,31,28)(33,34,35);(2,7)(4,15)(5,16)(8,19)(9,10)(11,21)(13,28)(14,27)(17,25)(18,20)(22,30)(23,24)(26,32)(29,31);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,23)(10,24)(11,22)(15,28)(16,27)(17,26)(18,29)(20,31)(21,30)(25,32);(3,12)(8,22)(13,26)(14,27)(19,30)(23,31)(24,29)(28,32)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17856);