Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| Fitting | C22×C3×C42 ⋊ C2 |
| Sylow | C3 ⋊ #1437 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C22×D3 |
| commutator | C22×C3 | C2×C42 |
| Fitting | C22×C3×C42 | C2 |
| socle, cosocle | C24×C3 | C23 |
| prime radical | C24×D3 | C22 |
| hypercenter | C22×C42 | D3 |
| hypocenter | C3 | #1437 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C22×C42 | |
828 conjugacy classes of subgroups; only the 241 normal subgroups are shown.
This group has 828 conjugacy classes of subgroups — too many to draw. Only its 241 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(8,10,11,9)(17,19);(4,5,6,7)(13,15);(2,3)(8,9,11,10)(12,13)(14,15)(16,17,18,19)Polycyclic presentation:
2^7,3|a^2=f,b^2=g,c^2=ef,b^a=bd,c^a=ce,h^a=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #10686699, D3×#1437, C2×#71537, … |
| maximal normal subgroup of | #10686699, D3×#1437, C2×#71537, … |
| maximal quotient of | #10683107, #10683108, #10683110, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4113);