Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#1830 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #381 |
| commutator | C23 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C23×C3 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×C4 ◁ G | |
236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,5,10,6,9,19,24)(3,11,13,25,14,7,18,21)(4,15,17,29,16,8,22,23)(12,20,28,31,27,26,30,32)(33,34,35);(1,3,6,14)(2,7,9,11)(4,12,16,27)(5,18,19,13)(8,20,15,26)(10,25,24,21)(17,30,22,28)(23,32,29,31);(1,4)(2,8,9,15)(3,12)(5,17)(6,16)(7,20,11,26)(10,23,24,29)(13,28)(14,27)(18,30)(19,22)(21,31,25,32)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5121);