Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#5424 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Frattini | C22 | C22×S4 |
| Fitting | C22×#41 | D3 |
| socle | C23 | C2×S4 |
| prime radical | C22×#41⋊C3 | C2 |
| cosocle | C2×SL(2,𝔽3) | C23 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22×C4 | |
343 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2)(7,12,18,14,9,20,11,21,16,8,17,22)(10,13,15,19);(3,4,5,6)(9,10)(12,13)(15,17)(16,18)(19,21)(20,22);(1,2)64 conjugacy classes — view the full character table.
| maximal subgroup of | #11711311, #11711314, #11711321, … |
| maximal normal subgroup of | #11711311, #11711314, #11711321, … |
| maximal quotient of | C2×#73068, C2×#73069, #11711792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18091);