Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#2821 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| prime radical | C24×C3 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 31 points:
(5,6,8,10)(7,9,11,12)(13,14)(15,19)(16,20)(17,18)(21,27)(22,25)(23,26)(24,28)(29,30,31);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(14,18)(16,23)(19,25)(21,24);(2,4)(6,10)(9,12)(13,15,17,22)(14,19,18,25)(16,24,23,21)(20,28,26,27);(7,11)(9,12)(13,16,17,23)(14,20,18,26)(15,21,22,24)(19,27,25,28)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17261);