Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C4×#985 |
| Fitting | C4×#41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Frattini, socle | C22 | C22×S4 |
| Fitting | C4×#41 | D3 |
| prime radical | C2×#985 | C2 |
| cosocle | C2×SL(2,𝔽3) | C23 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C42 | |
287 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 287 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 3 generators on 20 points:
(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);(1,2,3,4)64 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#985, #11711829, #11711838, … |
| maximal normal subgroup of | C8×#985, #11711829, #11711838, … |
| maximal quotient of | C8×#985, #11711842, #11711858, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17916);