Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #126⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C4×#126⋊C3 |
| unambiguous semidirect product | (C4×#126)⋊C3 |
| Fitting, Sylow | C4×#126 ⋊ C3 |
| derived | C42 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #126⋊C3 |
| commutator, hypocenter | C42 | C2×C3×C4 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | C4×#126 | C3 |
| prime radical | C2×#126⋊C3 | C2 |
| cosocle | C2×C42 | C22×C3 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C4 | |
126 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 20 points:
(6,14,15,10,7,17)(8,9,11)(12,13,16,20,18,19);(1,2,3,4)(5,17,9,20)(6,14,13,7)(8,12,11,15)(10,19,18,16)40 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#126⋊C3, #21666⋊C3, (#126⋊C3)⋊C8, … |
| maximal normal subgroup of | C8×#126⋊C3, #21666⋊C3, (#126⋊C3)⋊C8, … |
| maximal quotient of | C4×#383⋊C3, C4×#396⋊C3, C4×#401⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5748);