Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2188 |
216 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(1,2,3)(5,7);(2,3)(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)(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=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^b=h^269 conjugacy classes — view the full character table.
| maximal subgroup of | #10707307, #10707308, #10707313, … |
| maximal normal subgroup of | #10707307, #10707308, #10707313, … |
| maximal quotient of | #10707308, #10707652, #10708412, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1689);