Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C22×2−1+4⋊C3 |
| Fitting, Sylow | C22×2−1+4 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C24×C3 |
| Frattini | C2 | C24×A4 |
| Fitting | C22×2−1+4 | C3 |
| hypercenter | C22×D4 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
1287 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 1287 conjugacy classes of subgroups — too many to draw. Only its 212 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,12,13,11,8,19)(9,10)(14,18,15,17,16,20);(5,13,9,15)(6,20,10,19)(7,16,14,8)(11,12,17,18);(3,4);(1,2)76 conjugacy classes — view the full character table.
| maximal subgroup of | #34257⋊C3, #34259⋊C3, #35687⋊C3, … |
| maximal normal subgroup of | #34257⋊C3, #34259⋊C3, #35687⋊C3, … |
| maximal quotient of | C22×#519⋊C3, C22×#520⋊C3, C2×C4×2−1+4⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20110);