Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#1838 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D4 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C22×C3 | C2×C4⋊C4 |
| prime radical | C23×C3 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,18,14,19,23,17)(11,16,20,15,21,24,12,22)(25,26,27);(1,3,4,7)(2,5,6,8)(9,11,13,20,14,21,23,12)(10,15,17,16,19,22,18,24);(2,6)(5,8)(9,12,13,11,14,20,23,21)(10,16,17,22,19,24,18,15)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5129);