Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1645 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #953 |
| commutator, cosocle | C3×C42 | C23 |
| Frattini | C42 | C22×D3 |
| Fitting | C3×Q82 | C2 |
| socle, prime radical | C22×C3 | #120 |
| hypercenter | Q82 | D3 |
| hypocenter | C3 | #1645 |
| LCS | G ▷ C3×C42 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ Q82 | |
200 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 65 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:
(1,2,3)(12,14,16,21)(13,17,19,26)(15,25,24,22)(18,27,23,20);(4,5,7,9)(6,11,10,8)(15,24)(18,23)(20,27)(22,25);(2,3)(5,8)(6,10)(9,11)(12,13,16,19)(14,20,21,27)(15,23,24,18)(17,25,26,22)Polycyclic presentation:
2^7,3|a^2=g,b^2=f,c^2=g,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2,d^b=df,e^b=eg,d^c=dg,e^c=eg51 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1645, C2×#71941, #10771125, … |
| maximal normal subgroup of | D3×#1645, C2×#71941, #10771125, … |
| maximal quotient of | #10684025, #10684036, #10684071, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4517);