Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#459 ⋊ C2 |
| Sylow | C3 ⋊ #3214 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#459 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #459 | D3 |
| hypocenter | C3 | #3214 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #459 | |
434 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 434 conjugacy classes of subgroups — too many to draw. Only its 113 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 — 4 generators on 19 points:
(1,2)(5,9)(8,10)(12,13,15,16)(14,17,18,19);(6,11)(8,10)(12,14)(13,17)(15,18)(16,19);(2,3)(4,6,7,11)(5,8,9,10)(12,13,15,16)(14,17,18,19);(4,5)(6,10)(7,9)(8,11)(13,16)(14,17)(18,19)Polycyclic presentation:
2^7,3|b^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ceg,h^b=h^2,f^c=fg,h^d=h^245 conjugacy classes — view the full character table.
| maximal subgroup of | #10772040, #10772043, #10772106, … |
| maximal normal subgroup of | #10772040, #10772043, #10772106, … |
| maximal quotient of | #10804531, #10804939, #10805758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12914);