Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×#2153 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | C2×D4 |
| commutator | C4 | C22×C3×C8 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle | C2×C3 | C2×C4⋊C8 |
| prime radical | C23×C3 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C22×C3×C8 ◁ G | |
152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,5,10,6,9,19,24,7,11,20,25,4,17,18,30)(3,12,15,27,16,22,29,31,13,26,28,32,14,8,21,23)(33,34,35);(1,3,4,14,7,13,6,16)(2,8,9,12,11,22,17,26)(5,15,18,21,20,28,19,29)(10,23,24,27,25,31,30,32);(3,13)(8,22)(12,26)(14,16)(15,28)(21,29)(23,31)(27,32)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5444);