Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#3022 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C3×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
704 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 212 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 — 4 generators on 35 points:
(1,2,6,10)(3,11,15,22)(4,8,18,26)(5,19,20,9)(7,13,23,14)(12,28,30,27)(16,25,17,24)(21,31,32,29)(33,34,35);(1,3)(2,7)(4,12)(5,13)(6,15)(8,21)(9,22)(10,23)(11,19)(14,20)(16,29)(17,31)(18,30)(24,28)(25,27)(26,32);(1,4)(2,8)(3,12)(5,16)(6,18)(7,21)(9,24)(10,26)(11,27)(13,29)(14,31)(15,30)(17,20)(19,25)(22,28)(23,32);(1,5,6,20)(2,9,10,19)(3,13,15,14)(4,16,18,17)(7,22,23,11)(8,24,26,25)(12,29,30,31)(21,28,32,27)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17462);