Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C3×#3566 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
954 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32)(33,34,35);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,23)(19,25)(24,30)(26,31);(3,11)(6,9)(10,16)(13,15)(18,23)(22,28)(24,30)(29,32);(2,8)(6,9)(7,14)(13,15)(17,20,21,27)(18,24,23,30)(19,26,25,31)(22,29,28,32)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20026);