Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| direct product | C2×#5935 |
| Fitting | C2×H(5,𝔽2) ⋊ D3 |
| derived | #997 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | #1020 |
| commutator, cosocle, hypocenter | #997 | C22 |
| Frattini | C2 | C2×#1020 |
| Fitting | C2×H(5,𝔽2) | D3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22 | |
333 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 10 points:
(3,8,10,5,6,4,7,9);(1,2)(4,7,9)(5,8,10)26 conjugacy classes — view the full character table.
| maximal subgroup of | #11711591, #11711594, #11711630, … |
| maximal normal subgroup of | #11711591, #11711594, #11711630, … |
| maximal quotient of | C2×#85551, C2×#85550, #11715581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20090);