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 ⋊ #2163 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C8 | C22×D3 |
| commutator, socle | C22×C3 | C22×C8 |
| Fitting | C3×#346 | C2 |
| prime radical | C22×D3 | C2×C8 |
| cosocle | C2×C3×C8 | C23 |
| hypercenter | #346 | D3 |
| hypocenter | C3 | #2163 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C8 ◁ #346 | |
152 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 73 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) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(13,17)(18,23)(19,24)(25,27);(2,3)(5,8)(9,11);(4,5)(6,9)(7,8)(10,11)(12,13,14,18,15,19,20,25,16,17,21,23,22,24,26,27)Polycyclic presentation:
2^7,3|a^2=e,c^2=d,e^2=f,f^2=g,b^a=bd,c^a=cg,h^b=h^296 conjugacy classes — view the full character table.
| maximal subgroup of | #10704228, #10704230, #10704233, … |
| maximal normal subgroup of | #10704228, #10704230, #10704233, … |
| maximal quotient of | #10710188, #10710219, #10710497, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1649);