Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#1414 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #354 |
| commutator | C42 | C2×C3×C4 |
| Frattini, cosocle | #124 | C22×C3 |
| prime radical | C2×C3×D4 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#124 ◁ G | |
84 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)(3,11,13,26)(4,15,17,31)(5,19,21,9)(6,10)(7,12,22,29)(8,18)(14,28,20,27)(16,23,30,24)(25,32)(33,34,35);(1,3)(2,7,10,24)(4,16)(5,20)(6,14)(8,11,25,28)(9,22,19,23)(12,17)(13,21)(15,26,31,27)(18,30)(29,32)51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 550);