Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
| direct product | D3×M64 |
| Fitting | C3×M64 ⋊ C2 |
| Sylow | C3 ⋊ C2×M64 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C16 | C22×D3 |
| commutator, socle | C2×C3 | C22×C16 |
| Fitting | C3×M64 | C2 |
| prime radical | C22×D3 | C16 |
| cosocle | C3×C16 | C23 |
| hypercenter | M64 | D3 |
| hypocenter | C3 | C2×M64 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C16 ◁ M64 | |
112 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(2,7)(8,17)(9,18)(10,19)(20,27)(21,28)(22,29)(30,32)(33,34,35);(1,2,3,8,4,9,11,20,5,10,12,21,14,22,23,30,6,7,13,17,15,18,24,27,16,19,25,28,26,29,31,32);(34,35)120 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1398, #10715373, D3×#1400, … |
| maximal normal subgroup of | D3×#1398, #10715373, D3×#1400, … |
| maximal quotient of | D3×#1397, Dic3×M64, #10715363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1947);