Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C8×SL(2,ℤ/4) |
| derived | A4 ⋊ C4×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | S4 |
| commutator, hypocenter | A4 | C4×C8 |
| Frattini | C2×C4 | C2×S4 |
| Fitting | C23×C8 | D3 |
| socle | C24 | C4×D3 |
| prime radical | C22×A4 | C2×C4 |
| cosocle | C2×C4×A4 | C22 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C2×C8 | |
187 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 187 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,4)(5,6,8,7);(6,7,8)(9,10,11,13,12,14,15,16)80 conjugacy classes — view the full character table.
| maximal subgroup of | C16×SL(2,ℤ/4), #11711487, SL(2,ℤ/4)⋊C16, … |
| maximal normal subgroup of | C16×SL(2,ℤ/4), #11711487, SL(2,ℤ/4)⋊C16, … |
| maximal quotient of | C8×A4⋊C8, #11710985, C8×GU(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5682);