Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#5919 |
| Fitting | C23×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×S4 |
| commutator, cosocle | C2×A4 | C24 |
| Frattini | C2 | C23×S4 |
| Fitting | C23×Q8 | D3 |
| socle | C24 | C22×D3 |
| hypercenter | C2×Q8 | S4 |
| hypocenter | A4 | C2×#41 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
845 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 845 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) = 48 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2)(3,4,6)(7,9,10,13)(8,11,12,14);(7,8,10,12)(9,14,13,11);(4,5)(9,13)(11,14);(1,2)50 conjugacy classes — view the full character table.
| maximal subgroup of | #11712898, #11712903, #11712907, … |
| maximal normal subgroup of | #11712898, #11712903, #11712907, … |
| maximal quotient of | C2×Q8×SL(2,ℤ/4), C2×#85307, C2×#85308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20058);