Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #2044 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #953 |
| commutator, cosocle | C24×C3 | C23 |
| Frattini | C24 | C22×D3 |
| Fitting, prime radical | C3×#534 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | #534 | D3 |
| hypocenter | C3 | #2044 |
| LCS | G ▷ C24×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #534 | |
382 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 382 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) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(2,3)(4,6,7,5)(12,15);(1,2,3)(4,5)(6,7)(8,10)(9,12)(11,14)(13,15);(5,6)(8,9)(10,12)(11,13)(14,15)Polycyclic presentation:
2^7,3|c^2=d,b^a=bd,c^a=ce,f^a=fg,c^b=cf,e^b=eg,h^c=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10707452, #10707454, #10707459, … |
| maximal normal subgroup of | #10707452, #10707454, #10707459, … |
| maximal quotient of | #10698452, #10698459, #10698466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4426);