Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#1833 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #117 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C2×C3 | #382 |
| prime radical | C23×C3 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C3×#117 ◁ G | |
172 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 172 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)(3,11,14,7)(4,15,17,30)(5,10)(6,9)(8,16,21,31)(12,25)(13,24,28,20)(18,22)(19,26)(23,32)(27,29)(33,34,35);(1,3,5,13,6,14,18,28)(2,7,10,20,9,11,22,24)(4,12,16,26,17,27,31,32)(8,19,15,29,21,23,30,25);(1,4)(2,8,9,21)(3,12)(5,16)(6,17)(7,19,11,23)(10,15,22,30)(13,26)(14,27)(18,31)(20,29,24,25)(28,32)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5124);