Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
| direct product | C4×#859 |
| derived | SL(2,𝔽3) ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C42 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | C2×C4×Q8 | D3 |
| prime radical, cosocle | C22×SL(2,𝔽3) | C22 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22×C4 | |
197 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 197 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) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 16 points:
(5,6,7,8)(9,10,16,11,12,15,14,13);(1,2,3,4)(10,13,16)(11,14,15)64 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#859, #11711000, #11711001, … |
| maximal normal subgroup of | C8×#859, #11711000, #11711001, … |
| maximal quotient of | C4×SL(2,𝔽3)⋊C8, #11710989, C8×#859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5634);