Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#2231 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C3×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×C4×C8 ◁ G | |
120 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,14)(4,18,21,32)(5,12)(6,22)(7,11)(8,13)(9,16)(10,20,28,30)(15,27,23,29)(17,26)(19,25,24,31)(33,34,35);(1,3,7,17,8,6,5,16)(2,9,12,22,13,26,11,14)(4,15,24,20,21,23,19,30)(10,25,29,32,28,31,27,18);(1,4)(2,10)(3,15)(5,19)(6,23)(7,24)(8,21)(9,25)(11,27)(12,29)(13,28)(14,18)(16,30)(17,20)(22,32)(26,31)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5522);