Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#2179 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×D8 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C2×C3 | C4×D8 |
| prime radical | C3×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C8 ◁ C3×C4×C8 ◁ G | |
144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,5,11,7,12,6,22)(3,13,16,30,17,24,23,28)(4,18,19,29,21,9,20,10)(8,14,25,31,27,32,26,15)(33,34,35);(1,3)(2,8)(4,14)(5,16)(6,23)(7,17)(9,24)(10,28)(11,25)(12,27)(13,18)(15,20)(19,31)(21,32)(22,26)(29,30);(1,4,5,19,7,21,6,20)(2,9,11,10,12,18,22,29)(3,14,16,31,17,32,23,15)(8,24,25,28,27,13,26,30)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5470);