Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C2×D3×2−1+4 |
| Fitting | C2×C3×2−1+4 ⋊ C2 |
| Sylow | C3 ⋊ C22×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C24×D3 |
| commutator, cosocle | C2×C3 | C26 |
| Frattini | C2 | C25×D3 |
| Fitting | C2×C3×2−1+4 | C2 |
| socle | C22×C3 | C25 |
| hypercenter | C2×2−1+4 | D3 |
| hypocenter | C3 | C22×2−1+4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×2−1+4 | |
5872 conjugacy classes of subgroups; only the 3213 normal subgroups are shown.
This group has 5872 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) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 6 generators on 21 points:
(1,2)(3,4,5)(6,9)(7,13)(8,15)(10,17)(11,18)(12,19)(14,20)(16,21);(6,8,10,16)(7,12,11,14)(9,15,17,21)(13,19,18,20);(6,7,10,11)(8,14,16,12)(9,13,17,18)(15,20,21,19);(7,11)(9,17)(12,14)(15,21);(4,5);(1,2)102 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1793, #10794493, D3×#1796, … |
| maximal normal subgroup of | D3×#1793, #10794493, D3×#1796, … |
| maximal quotient of | C2×D3×#519, C2×D3×#520, C4×D3×2−1+4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20146);