Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×C3×#479 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(3,4,5);(6,7,9,11)(8,12,13,10)(16,22)(17,24)(19,26)(20,28)(23,25)(27,29);(1,2)(6,8,9,13)(7,10,11,12)(14,15,18,21)(16,19,23,27)(17,20,24,28)(22,26,25,29);(6,7)(8,10)(9,11)(12,13)(14,16,18,23)(15,19,21,27)(17,25,24,22)(20,29,28,26)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17515);