Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#3290 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ G | |
454 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)(3,12)(4,9)(5,22)(6,11)(7,15)(8,20)(10,23)(13,25)(14,31)(16,27)(17,26)(18,24)(19,29)(21,28)(30,32)(33,34,35);(1,3)(2,7)(4,18)(5,16)(6,17)(8,25)(9,30)(10,26)(11,27)(12,22)(13,21)(14,19)(15,23)(20,32)(24,29)(28,31);(1,4)(2,8)(3,13)(5,19)(6,21)(7,24)(9,22)(10,28)(11,29)(12,31)(14,15)(16,32)(17,18)(20,23)(25,27)(26,30);(1,4)(2,9)(3,14)(5,20)(6,21)(7,25)(8,22)(10,29)(11,28)(12,31)(13,15)(16,18)(17,32)(19,23)(24,27)(26,30)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17730);