Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#2133 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C2×#99 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C2×Q8 | C23×C3 |
| prime radical | C2×C3×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C23×C3 ◁ C22×C3×Q8 ◁ G | |
230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,5,9,6,10,21,28)(3,11,14,29,12,25,19,31)(4,15,17,27,18,8,20,26)(7,22,24,16,23,30,32,13)(33,34,35);(1,3)(2,7)(4,16)(5,19)(6,12)(8,25)(9,24)(10,23)(11,15)(13,18)(14,21)(17,22)(20,30)(26,31)(27,29)(28,32);(3,12)(7,23)(11,25)(13,16)(14,19)(22,30)(24,32)(29,31)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5424);