Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C23×SL(2,ℤ/4) |
| derived | A4 ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | S4 |
| commutator, hypocenter | A4 | C23×C4 |
| Frattini | C2 | C23×S4 |
| Fitting, socle | C26 | D3 |
| prime radical | C24×A4 | C2 |
| cosocle | C2×A4 | C24 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C24 | |
1655 conjugacy classes of subgroups; only the 252 normal subgroups are shown.
This group has 1655 conjugacy classes of subgroups — too many to draw. Only its 252 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) = 96 |
A minimal faithful permutation representation — 4 generators on 14 points:
(7,8,10,9)(11,12,13,14);(5,6)(8,9,10);(3,4);(1,2)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A4⋊#95, C22⋊C4×SL(2,ℤ/4), #11711695, … |
| maximal normal subgroup of | C2×A4⋊#95, C22⋊C4×SL(2,ℤ/4), #11711695, … |
| maximal quotient of | C22×C4×SL(2,ℤ/4), C22×#5432, C22×C4⋊SL(2,ℤ/4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20076);