Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #2183 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C22×D3 |
| commutator, socle, prime radical | C22×C3 | C22×C8 |
| Fitting | C3×C4⋊C16 | C2 |
| cosocle | C2×C3×C8 | C23 |
| hypercenter | C4⋊C16 | D3 |
| hypocenter | C3 | #2183 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C8 ◁ C4⋊C16 | |
126 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(12,14,15,22,16,20,21,27,17,23,24,26,25,13,18,19);(2,3)(4,6,7,11)(5,8,9,10);(4,5,7,9)(6,10,11,8)(12,13,15,19,16,14,21,22,17,20,24,27,25,23,18,26)Polycyclic presentation:
2^7,3|a^2=e,b^2=d,c^2=de,e^2=f,f^2=g,b^a=bd,c^a=cg,h^b=h^2,h^c=h^296 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2183, C2×#69282, #10948676, … |
| maximal normal subgroup of | D3×#2183, C2×#69282, #10948676, … |
| maximal quotient of | #10710533, #10710540, #10710544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1858);