Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#2009 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #392 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C3×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
348 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,13)(4,17)(5,11)(6,10)(7,12)(8,15)(9,19)(14,22)(16,27)(18,26)(20,25)(21,23)(24,30)(28,29)(31,32)(33,34,35);(1,3)(2,8)(4,14)(5,21)(6,15)(7,16)(9,24)(10,27)(11,13)(12,23)(17,28)(18,31)(19,29)(20,30)(22,26)(25,32);(1,4,7,20)(2,9,12,26)(3,14,16,30)(5,19,6,18)(8,22,23,24)(10,17,11,25)(13,28,27,32)(15,31,21,29)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5300);