Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C2×D3×H(5,𝔽2) |
| Fitting | C2×C3×H(5,𝔽2) ⋊ C2 |
| Sylow | C3 ⋊ C22×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C24×D3 |
| commutator, cosocle | C2×C3 | C26 |
| Frattini | C2 | C25×D3 |
| Fitting | C2×C3×H(5,𝔽2) | C2 |
| socle | C22×C3 | C25 |
| hypercenter | C2×H(5,𝔽2) | D3 |
| hypocenter | C3 | C22×H(5,𝔽2) |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×H(5,𝔽2) | |
6560 conjugacy classes of subgroups; only the 3213 normal subgroups are shown.
This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3213 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 6 generators on 13 points:
(1,2)(3,4,5)(6,8)(7,11)(9,12)(10,13);(7,10)(11,13);(6,7)(8,11)(9,10)(12,13);(7,10)(8,12);(4,5);(1,2)102 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1790, #10794452, D3×#1792, … |
| maximal normal subgroup of | D3×#1790, #10794452, D3×#1792, … |
| maximal quotient of | C2×D3×#518, C2×D3×#520, C4×D3×H(5,𝔽2), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20143);