Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#547 ⋊ C2 |
| Sylow | C3 ⋊ #3330 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×#67 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#547 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #547 | D3 |
| hypocenter | C3 | #3330 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ #547 | |
420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(1,2,3)(6,11)(8,10)(12,14,16,15)(13,17,18,19);(2,3)(12,13)(14,17)(15,19)(16,18);(4,6)(5,8)(7,11)(9,10)(13,18)(17,19);(4,5,7,9)(6,10,11,8)(13,17)(14,15)(18,19)Polycyclic presentation:
2^7,3|a^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=dg,f^a=fg,c^b=cg,d^b=de,f^c=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10784790, #10784792, #10784850, … |
| maximal normal subgroup of | #10784790, #10784792, #10784850, … |
| maximal quotient of | #10810833, #10811610, #10812477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16103);