Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#2143 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×D8 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C2×C3 | C2×#102 |
| prime radical | C2×C3×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
220 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 100 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:
(1,2,5,10,7,11,6,21)(3,12,15,29,13,28,22,26)(4,16,18,27,20,31,19,9)(8,17,24,30,23,32,25,14)(33,34,35);(1,3)(2,8)(4,17)(5,15)(6,22)(7,13)(9,26)(10,24)(11,23)(12,16)(14,19)(18,30)(20,32)(21,25)(27,29)(28,31);(3,13)(8,23)(12,28)(14,30)(15,22)(17,32)(24,25)(26,29)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5434);