Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | C8×GL(2,𝔽3) |
| Fitting | C8×Q8 ⋊ D3 |
| derived | SL(2,𝔽3) ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C8 |
| Frattini | C2×C4 | C2×S4 |
| Fitting | C8×Q8 | D3 |
| socle | C22 | C4×S4 |
| prime radical | C2×GL(2,𝔽3) | C4 |
| cosocle | C4×SL(2,𝔽3) | C22 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C8 | |
127 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 127 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 64 |
A minimal faithful permutation representation — 2 generators on 16 points:
(9,10,16,11,12,15,14,13);(1,2,3,5,4,6,7,8)(10,13,16)(11,14,15)64 conjugacy classes — view the full character table.
| maximal subgroup of | C16×GL(2,𝔽3), #11711379, GL(2,𝔽3)⋊C16, … |
| maximal normal subgroup of | C16×GL(2,𝔽3), #11711379, GL(2,𝔽3)⋊C16, … |
| maximal quotient of | C8×#859, #11711087, #11711107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5593);