Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×#985 |
| Fitting | C22×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C24 |
| Frattini | C2 | C23×S4 |
| Fitting | C22×#41 | D3 |
| socle | C23 | C2×S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22×C4 | |
803 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 803 conjugacy classes of subgroups — too many to draw. Only its 110 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)(5,10,16,12,7,18,9,19,14,6,15,20)(8,11,13,17);(7,8)(10,11)(13,15)(14,16)(17,19)(18,20);(3,4);(1,2)64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#73127, #11712719, #11712722, … |
| maximal normal subgroup of | C2×#73127, #11712719, #11712722, … |
| maximal quotient of | C22×#5424, C22×C4×GL(2,𝔽3), C22×C4×#226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20061);