Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#2110 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | C42 |
| commutator | C4 | C2×C3×C42 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C2×C3 | C2×#97 |
| prime radical | C23×C3 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ G | |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,4,8,5,10,16,24,6,9,17,23,18,25,31,32)(3,11,13,27,14,20,29,22,12,26,15,30,28,7,19,21)(33,34,35);(1,3)(2,7,9,20)(4,15)(5,14)(6,12)(8,22,23,21)(10,11,25,26)(13,17)(16,19)(18,28)(24,30,32,27)(29,31);(3,12)(7,20)(11,26)(13,15)(14,28)(19,29)(21,22)(27,30)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5401);