Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | #41×S4 |
| Fitting | C22×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×S4 |
| commutator, cosocle | C2×A4 | C24 |
| Frattini | C2 | C23×S4 |
| Fitting | C22×#41 | D3 |
| socle | C23 | C23×D3 |
| hypercenter | #41 | S4 |
| hypocenter | A4 | C2×#41 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C4 ◁ #41 | |
853 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 853 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2,4,3);(2,3,4)(5,7,8,11)(6,10,9,12);(6,9)(10,12);(5,6)(7,10)(8,9)(11,12)50 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#104, #11712920, S4×#131, … |
| maximal normal subgroup of | S4×#104, #11712920, S4×#131, … |
| maximal quotient of | #41×SL(2,ℤ/4), S4×#117, #11714306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20063);