Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C4×A4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×S4 |
| commutator | C2×A4 | C22×C4 |
| Frattini | C22 | C22×S4 |
| Fitting | C22×C42 | D3 |
| socle | C24 | C22×D3 |
| prime radical, cosocle | C22×A4 | C23 |
| hypercenter | C42 | S4 |
| hypocenter | A4 | C4×Q8 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
325 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 58 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 — 3 generators on 16 points:
(5,6,8)(9,11,12,16)(10,13,14,15);(6,7)(9,10,12,14)(11,15,16,13);(1,2,3,4)56 conjugacy classes — view the full character table.
| maximal subgroup of | #11711104, #11711144, C4×#5414, … |
| maximal normal subgroup of | #11711104, #11711144, C4×#5414, … |
| maximal quotient of | C4×SL(2,ℤ/4)⋊C4, C4×C4⋊SL(2,ℤ/4), #11711714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17858);