Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C4 |
| direct product | C3×#1407 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #353 |
| 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 | |
98 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)(4,15,16,31)(5,18,20,9)(6,10)(7,12,21,28)(8,17)(14,24,19,27)(22,29,23,30)(25,32)(33,34,35);(1,3,6,14)(2,7,10,23)(4,12,16,30)(5,19,20,13)(8,24,25,11)(9,21,18,22)(15,27,31,26)(17,28,32,29)42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 543);