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×#3556 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C25 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| prime radical | C23×C3 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
890 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)(3,6)(4,11)(5,8)(7,13)(9,16)(10,15)(12,14)(17,18,22,25)(19,26,28,23)(20,24,21,30)(27,32,29,31)(33,34,35);(1,3,5,10)(2,6,8,15)(4,12,13,9)(7,16,11,14)(17,19,22,28)(18,23,25,26)(20,29,21,27)(24,32,30,31);(1,4,5,13)(2,7,8,11)(3,9,10,12)(6,14,15,16);(2,8)(6,15)(7,11)(14,16)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32);(17,21,22,20)(18,24,25,30)(19,27,28,29)(23,32,26,31)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20016);