Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#1424 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | D16 |
| commutator | C16 | C2×C3×C4 |
| Frattini, cosocle | C2×C16 | C22×C3 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C2×C3×C16 ◁ G | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,3,4)(6,11)(7,16)(8,17)(9,20)(12,19)(13,28)(14,31)(15,25)(18,27)(21,34)(22,23)(24,29)(26,35)(30,36)(32,33)(37,38,39);(5,6,12,26,8,13,18,24,20,29,33,28,34,35,7,11,10,15,16,23,21,30,32,31,9,14,27,36,17,22,19,25)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 560);