Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | D4×SL(2,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×S4 |
| commutator | C2×A4 | C22×C4 |
| Fitting | C23×D4 | D3 |
| socle | C24 | C22×D3 |
| prime radical | C2×D4×A4 | C2 |
| cosocle | C22×A4 | C23 |
| hypercenter | C2×D4 | S4 |
| hypocenter | A4 | C4×D4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
511 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 511 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 48 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,3,4)(9,10,12,11);(6,8)(10,11,12);(5,6)(7,8)50 conjugacy classes — view the full character table.
| maximal subgroup of | #11712923, #11712925, D8×SL(2,ℤ/4), … |
| maximal normal subgroup of | #11712923, #11712925, D8×SL(2,ℤ/4), … |
| maximal quotient of | C22⋊C4×SL(2,ℤ/4), C4⋊C4×SL(2,ℤ/4), #11711695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18066);