Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C4×SL(2,ℤ/4) |
| unambiguous semidirect product | (C2×A4)⋊C42 |
| derived | A4 ⋊ C2×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | S4 |
| commutator, hypocenter | A4 | C2×C42 |
| Frattini | C22 | C22×S4 |
| Fitting | C24×C4 | D3 |
| socle | C25 | C2×D3 |
| prime radical | C23×A4 | C22 |
| cosocle | C22×A4 | C23 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C22×C4 | |
579 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 579 conjugacy classes of subgroups — too many to draw. Only its 108 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 — 3 generators on 14 points:
(3,4,6,5)(11,12,13,14);(4,5,6)(7,8,9,10);(1,2)80 conjugacy classes — view the full character table.
| maximal subgroup of | #11711286, #11711298, #11711299, … |
| maximal normal subgroup of | #11711286, #11711298, #11711299, … |
| maximal quotient of | C42×SL(2,ℤ/4), C2×A4⋊#95, #11711683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18054);