Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#498 ⋊ C2 |
| Sylow | C3 ⋊ #3184 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#498 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #498 | D3 |
| hypocenter | C3 | #3184 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #498 | |
446 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 446 conjugacy classes of subgroups — too many to draw. Only its 125 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)(4,6,7,10)(5,9,8,11)(13,17)(16,19);(12,13,18,19,15,17,14,16);(2,3)(5,8)(9,11)(13,17)(16,19);(2,3)(4,5)(6,9)(7,8)(10,11)(13,16)(14,18)(17,19)Polycyclic presentation:
2^7,3|c^2=fg,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,c^b=cg,h^b=h^2,d^c=dg,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10771520, #10771522, #10771580, … |
| maximal normal subgroup of | #10771520, #10771522, #10771580, … |
| maximal quotient of | #10795783, #10799497, #10800437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 14464);