Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #5450 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×#5450 |
| Fitting, Sylow | C2×H(3,𝔽4) ⋊ C3 |
| derived | C24 ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #5450 |
| commutator, cosocle, hypocenter | C24 | C23×C3 |
| Frattini | C22 | C23×A4 |
| Fitting | C2×H(3,𝔽4) | C3 |
| socle | C23 | C22×A4 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C2 | |
318 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 318 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) = 18 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(4,13,10,9,5,15)(6,7,14)(8,18,17,11,12,16);(3,10,7,17)(4,18,11,12)(5,9,13,8)(6,16,14,15);(1,2)34 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#5450, #45439⋊C3, C4⋊#5450, … |
| maximal normal subgroup of | C4×#5450, #45439⋊C3, C4⋊#5450, … |
| maximal quotient of | C2×#2375⋊C3, C4×#5450, C2×#2812⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18199);