Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#342 ⋊ C2 |
| Sylow | C3 ⋊ #2021 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #882 |
| commutator, cosocle | C24×C3 | C23 |
| Frattini | C24 | C22×D3 |
| Fitting | C3×#342 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | #342 | D3 |
| hypocenter | C3 | #2021 |
| LCS | G ▷ C24×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #342 | |
638 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 638 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 48 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2)(5,7)(8,10)(9,12)(11,14)(13,15);(2,3)(12,15);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)Polycyclic presentation:
2^7,3|b^a=bd,c^a=ce,f^a=fg,h^a=h^2,c^b=cf,h^b=h^2,d^c=dg,h^c=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10707217, #10707222, #10707235, … |
| maximal normal subgroup of | #10707217, #10707222, #10707235, … |
| maximal quotient of | #10695386, #10695393, #10695400, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1541);