Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C22×GU(2,𝔽3) |
| derived | SL(2,𝔽3) ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Frattini | C4 | C22×S4 |
| Fitting | C22×#41 | D3 |
| socle | C23 | SL(2,ℤ/4) |
| prime radical | C22×#41⋊C3 | C2 |
| cosocle | #41⋊C3 | C23 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22×C4 | |
359 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 359 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) = 28 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,14,11,7)(8,25,18,26)(13,24,23,22)(15,27,21,28)(16,19,17,20);(3,4)(5,8,17)(6,13,16)(7,15,22)(9,20,28)(10,21,26)(11,19,25)(12,18,24)(14,23,27);(1,2)64 conjugacy classes — view the full character table.
| maximal subgroup of | #11711312, #11711313, #11711320, … |
| maximal normal subgroup of | #11711312, #11711313, #11711320, … |
| maximal quotient of | C22×SL(2,𝔽3)⋊C8, C2×#73068, C2×#73057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18086);