Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#2239 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C23×C3 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×M16 ◁ G | |
124 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(1,2,6,10)(3,11,12,29)(4,15,16,32)(5,20,18,9)(7,22,23,13)(8,25,26,19)(14,27,21,30)(17,24,31,28)(33,34,35);(4,16)(8,26)(11,29)(13,22)(15,32)(17,31)(19,25)(27,30);(1,3,5,21,6,12,18,14)(2,7,9,28,10,23,20,24)(4,17,19,22,16,31,25,13)(8,27,15,29,26,30,32,11)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5530);