Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#2158 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | D8 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C22×C3 | C2×D8 |
| prime radical | C23×C3 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4)(5,6)(7,16)(8,12)(9,22)(10,15)(11,19)(13,17)(14,21)(18,29)(20,31)(23,27)(24,32)(25,36)(26,35)(28,33)(30,34)(37,38,39);(5,7,17,32,8,18,23,35,10,20,33,36,21,34,9,19)(6,11,22,30,14,25,28,31,15,26,27,29,12,24,13,16);(5,8,10,21)(6,12,15,14)(7,18,20,34)(9,17,23,33)(11,24,26,25)(13,27,28,22)(16,29,31,30)(19,32,35,36)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5449);