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×#3565 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,5,7)(3,6,9,14)(4,11,12,8)(10,16,15,13)(17,18,22,25)(19,27,29,23)(20,26,21,24)(28,31,30,32)(33,34,35);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19)(18,23)(20,30)(21,28)(22,29)(24,32)(25,27)(26,31);(2,7)(3,9)(8,11)(10,15)(17,20,22,21)(18,24,25,26)(19,28,29,30)(23,31,27,32);(1,4,5,12)(2,8,7,11)(3,10,9,15)(6,13,14,16)(18,25)(19,29)(24,26)(28,30);(17,21,22,20)(18,26,25,24)(19,28,29,30)(23,31,27,32)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20025);