Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#357 ⋊ C2 |
| Sylow | C3 ⋊ #2189 |
220 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2)(5,7)(8,11,14,13)(9,16,18,28)(10,20,22,29)(12,23,27,26)(15,31,33,32)(17,24,35,38)(19,36,37,34)(21,25,30,39);(8,10)(9,15)(11,20)(12,25)(13,29)(14,22)(16,31)(17,34)(18,33)(19,24)(21,26)(23,30)(27,39)(28,32)(35,36)(37,38);(2,3)(4,5)(6,7)(8,9,14,18)(10,19,22,37)(11,16,13,28)(12,24,27,38)(15,30,33,21)(17,23,35,26)(20,36,29,34)(25,31,39,32)Polycyclic presentation:
2^7,3|a^2=g,c^2=g,d^2=fg,f^2=g,b^a=bd,c^a=ce,d^a=df,f^a=fg,h^a=h^2,d^b=df,f^b=fg,h^c=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2189, C2×#69194, #10948960, … |
| maximal normal subgroup of | D3×#2189, C2×#69194, #10948960, … |
| maximal quotient of | #10707313, #10707657, #10708417, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1770);