Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C2×#5966 |
| Fitting | C2×C3×H(5,𝔽2) ⋊ C2 |
| Sylow | C3 ⋊ C2×#585 |
| 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 | C2×#585 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×H(5,𝔽2) | |
5956 conjugacy classes of subgroups; only the 3205 normal subgroups are shown.
This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3205 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)(7,12)(8,15)(10,16)(11,19)(13,17)(14,20)(18,21);(6,8)(7,11)(9,15)(10,14)(12,19)(13,18)(16,20)(17,21);(6,7,10,13)(8,11,14,18)(9,17,16,12)(15,21,20,19);(8,14)(11,18)(15,20)(19,21);(4,5)(9,16)(12,17)(15,20)(19,21);(1,2)102 conjugacy classes — view the full character table.
| maximal subgroup of | #10794448, #10794456, #10794480, … |
| maximal normal subgroup of | #10794448, #10794456, #10794480, … |
| maximal quotient of | C2×#85856, C2×#85885, C2×#85857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20142);