Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #127⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C4 |
| direct product | C22×#127⋊C3 |
| Fitting, Sylow | C22×#127 ⋊ C3 |
| derived | C42 ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #127⋊C3 |
| commutator, cosocle, hypocenter | C42 | C23×C3 |
| Frattini | C22 | C23×A4 |
| Fitting | C22×#127 | C3 |
| socle | C24 | C2×A4 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C22 | |
580 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 580 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2)(3,5,9)(4,7,11)(6,12,13,8,10,14);(3,4)(5,12,7,10)(6,8)(9,13,11,14);(1,2)40 conjugacy classes — view the full character table.
| maximal subgroup of | #14661⋊C3, #14713⋊C3, C2×C4×#127⋊C3, … |
| maximal normal subgroup of | #14661⋊C3, #14713⋊C3, C2×C4×#127⋊C3, … |
| maximal quotient of | C22×#384⋊C3, C2×C4×#127⋊C3, C22×#392⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18167);