Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C4 |
| direct product | C3×#1287 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #95 |
| commutator | C2×C4 | C3×C42 |
| Frattini, cosocle | C2×C42 | C22×C3 |
| prime radical | C23×C3 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
126 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 192 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,4,9,6,8,5,10)(3,11,12,16,13,7,14,15)(17,18)(19,25,27,31)(20,23)(21,24)(22,26,29,32)(28,30)(33,34,35);(1,3,5,12,6,13,4,14)(2,7,10,15,8,11,9,16)(17,19)(18,22,24,29)(20,26)(21,27)(23,25,30,31)(28,32)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 423);