Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#3233 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C3×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,25)(20,29)(21,22)(23,28)(24,30)(26,32)(27,31)(33,34,35);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(18,22)(19,26)(24,29)(27,28);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,21,26)(18,23,22,31)(20,27,30,28)(24,32,29,25);(2,8)(6,9)(7,16)(14,15)(17,20,21,30)(18,24,22,29)(19,27,26,28)(23,32,31,25)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17673);