Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C42 |
| direct product | C4×C3⋊#95 |
| Sylow | C3 ⋊ C4×#95 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D3 |
| commutator | C2×C3 | C43 |
| Frattini | C24 | C22×D3 |
| Fitting | C22×C3×C42 | C2 |
| socle, prime radical, cosocle | C24×C3 | C23 |
| hypercenter | C22×C42 | D3 |
| hypocenter | C3 | C4×#95 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C42 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(12,13,14,15)(17,19);(2,3)(8,9,10,11)(16,17)(18,19);(4,5,6,7)144 conjugacy classes — view the full character table.
| maximal subgroup of | #10681718, #10681745, #10681747, … |
| maximal normal subgroup of | #10681718, #10681745, #10681747, … |
| maximal quotient of | #10683106, #10683107, C4×#4550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4109);