Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#3069 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C2×#121 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C3×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,7,12)(3,13,15,24)(4,18,20,26)(5,11,6,10)(8,17,23,16)(9,21,25,19)(14,28,29,22)(27,31,32,30)(33,34,35);(1,3,5,16,7,15,6,17)(2,8,10,13,12,23,11,24)(4,14,19,30,20,29,21,31)(9,22,18,27,25,28,26,32);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,18)(11,26)(12,25)(13,27)(15,29)(16,30)(17,31)(23,28)(24,32);(3,15)(8,23)(13,24)(14,29)(16,17)(22,28)(27,32)(30,31)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17509);