Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#3067 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| socle | C2×C3 | C2×#121 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
452 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)(3,13)(4,18)(5,10)(6,11)(7,12)(8,16)(9,20)(14,28)(15,24)(17,23)(19,25)(21,26)(22,31)(27,30)(29,32)(33,34,35);(1,3,6,16,7,17,5,15)(2,8,10,13,12,24,11,23)(4,14,20,30,21,31,19,29)(9,22,18,27,25,28,26,32);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,18)(11,26)(12,25)(13,27)(15,29)(16,30)(17,31)(23,32)(24,28);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,24,23)(9,18,25,26)(14,29,31,30)(22,27,28,32)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17507);