Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#346 ⋊ C2 |
| Sylow | C3 ⋊ #2168 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C22×D3 |
| commutator, socle | C22×C3 | C22×C8 |
| Fitting | C3×#346 | C2 |
| prime radical | C2×D12 | C8 |
| cosocle | C2×C3×C8 | C23 |
| hypercenter | #346 | D3 |
| hypocenter | C3 | #2168 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C8 ◁ #346 | |
164 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(8,10,11,18,12,16,17,23,13,19,20,22,21,9,14,15);(2,3)(5,7);(2,3)(4,5)(6,7)(8,9,11,15,12,10,17,18,13,16,20,23,21,19,14,22)Polycyclic presentation:
2^7,3|a^2=e,c^2=e,e^2=f,f^2=g,b^a=bd,c^a=cg,h^a=h^2,h^b=h^296 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2168, C2×#69075, #10948355, … |
| maximal normal subgroup of | D3×#2168, C2×#69075, #10948355, … |
| maximal quotient of | #10706899, #10706962, #10706969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1651);