Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C23 |
| direct product | C3×#2115 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | C22⋊C4 |
| commutator | C22 | C22×C3×C8 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C2×C3 | C2×C22⋊C8 |
| prime radical | C2×C3×#41 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C22×C3×C8 ◁ G | |
220 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 116 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 — 3 generators on 35 points:
(1,2,4,9,5,10,16,24,6,8,17,21,18,22,29,31)(3,11,13,23,14,25,27,32,12,7,15,19,26,20,28,30)(33,34,35);(1,3)(2,7)(4,15)(5,14)(6,12)(8,11)(9,23)(10,20)(13,17)(16,28)(18,26)(19,21)(22,25)(24,32)(27,29)(30,31);(3,12)(7,11)(13,15)(14,26)(19,23)(20,25)(27,28)(30,32)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5406);