Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#3584 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C25×C3 |
| socle | C2×C3 | C23×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×#41 ◁ G | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2)(3,12)(4,8)(5,9)(6,11)(7,15)(10,19)(13,26)(14,21)(16,25)(17,22)(18,23)(20,29)(24,30)(27,32)(28,31)(33,34,35);(1,3)(2,7)(4,13)(5,14)(6,16)(8,20)(9,12)(10,25)(11,21)(15,19)(17,27)(18,28)(22,31)(23,26)(24,32)(29,30);(1,4)(2,8)(3,13)(5,18)(6,17)(7,20)(9,23)(10,24)(11,22)(12,26)(14,28)(15,29)(16,27)(19,30)(21,31)(25,32);(4,17)(8,22)(13,27)(18,30)(20,31)(23,24)(26,32)(28,29);(1,5,6,19)(2,9,11,10)(3,14,16,15)(4,18,17,30)(7,12,21,25)(8,23,22,24)(13,28,27,29)(20,26,31,32)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20044);