Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#454 ⋊ C2 |
| Sylow | C3 ⋊ #3288 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #5590 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#454 | C2 |
| socle | C2×C3 | D42 |
| hypercenter | #454 | D3 |
| hypocenter | C3 | #3288 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #454 | |
464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(4,6)(5,11)(7,17)(8,14)(9,15)(10,18)(12,19)(13,16);(2,3)(4,5)(6,11)(7,10)(8,16)(9,12)(13,14)(15,19)(17,18);(4,5)(6,13)(7,16)(8,10)(9,12)(11,14)(15,18)(17,19);(2,3)(5,10)(7,8)(11,18)(12,16)(13,19)(14,17)Polycyclic presentation:
2^7,3|e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,h^a=h^2,d^b=de,e^b=eg,f^c=fg,h^c=h^2,e^d=eg,h^d=h^242 conjugacy classes — view the full character table.
| maximal subgroup of | #10715916, #10715920, #10715930, … |
| maximal normal subgroup of | #10715916, #10715920, #10715930, … |
| maximal quotient of | #10784537, #10784542, #10784598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12863);