Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | D3×#453 |
| Fitting | C3×#453 ⋊ C2 |
| Sylow | C3 ⋊ C2×#453 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×#120 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#453 | C2 |
| socle | C2×C3 | C2×#120 |
| hypercenter | #453 | D3 |
| hypocenter | C3 | C2×#453 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #453 | |
844 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 844 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 4 generators on 11 points:
(3,7)(5,8)(9,10,11);(1,2)(3,8)(4,6)(5,7);(2,5)(3,7)(6,8);(10,11)48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1402, #10715408, D3×#2196, … |
| maximal normal subgroup of | D3×#1402, #10715408, D3×#2196, … |
| maximal quotient of | D3×#1595, D3×#1596, D3×#1613, … |
| holomorph of | C3×D4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12852);