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×C4×#226 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Frattini | C22 | C22×S4 |
| Fitting | C2×C4×Q8 | D3 |
| socle | C23 | C2×S4 |
| prime radical | C22×SL(2,𝔽3) | C22 |
| cosocle | C2×SL(2,𝔽3) | C23 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22×C4 | |
287 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 287 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(7,12,16,19,11,21,18,13)(8,17,20,10,14,9,22,15);(3,4,5,6)(9,15,18)(10,16,17)(12,19,22)(13,20,21);(1,2)64 conjugacy classes — view the full character table.
| maximal subgroup of | #11711106, C2×C8×#226, C2×#226⋊C8, … |
| maximal normal subgroup of | #11711106, C2×C8×#226, C2×#226⋊C8, … |
| maximal quotient of | C2×C8×#226, C2×#73010, #11711857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17891);