Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C3×#1356 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #329 |
| commutator | C42 | C2×C3×C4 |
| Frattini, cosocle | #97 | C22×C3 |
| prime radical | C22×C3 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#97 ◁ G | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,4,9,5,10,18,28,6,11,17,29,20,8,25,26)(3,12,13,30,14,7,21,22,15,23,31,27,19,24,16,32)(33,34,35);(1,3,6,15)(2,7,11,24)(4,16,17,21)(5,19,20,14)(8,23,10,12)(9,27,29,30)(13,18,31,25)(22,26,32,28)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 492);