Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2209 |
154 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2)(8,10,13,21)(9,14,17,31)(11,23,26,39)(12,28,24,20)(15,33,34,38)(16,35,27,30)(18,36,37,22)(19,32,29,25);(2,3)(4,5,6,7);(5,7)(8,9,13,17)(10,18,21,37)(11,22,26,36)(12,27,24,16)(14,29,31,19)(15,32,34,25)(20,38,28,33)(23,35,39,30)Polycyclic presentation:
2^7,3|a^2=g,b^2=d,c^2=g,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,h^b=h^2,e^c=ef,f^c=fg,h^c=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2209, C2×#69236, #10949451, … |
| maximal normal subgroup of | D3×#2209, C2×#69236, #10949451, … |
| maximal quotient of | #10709337, #10709341, #10709373, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1812);