Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #548⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | #41×SL(2,𝔽3) |
| Fitting, Sylow | Q8×#41 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Frattini, socle | C22 | C23×A4 |
| Fitting | Q8×#41 | C3 |
| hypercenter | C2×#41 | A4 |
| hypocenter | Q8 | C3×#41 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
348 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 104 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 — 3 generators on 16 points:
(1,3,4,7)(2,6,5,8)(10,11,13)(14,16,15);(2,5)(6,8)(9,13,12,15)(10,16,14,11);(1,2)(3,6)(4,5)(7,8)70 conjugacy classes — view the full character table.
| maximal subgroup of | SL(2,𝔽3)×#104, SL(2,𝔽3)×#131, #60253⋊C3, … |
| maximal normal subgroup of | SL(2,𝔽3)×#104, SL(2,𝔽3)×#131, #60253⋊C3, … |
| maximal quotient of | SL(2,𝔽3)×#117, C4×D4×SL(2,𝔽3), C4×Q8×SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18176);