Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#347 ⋊ C2 |
| Sylow | C3 ⋊ #2180 |
142 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,7,9,8,10,21,26,23)(5,12,14,20,15,30,24,13)(6,17,25,22,18,34,19,35)(11,27,32,33,28,16,29,31);(2,3)(4,6)(5,11)(7,17)(8,22)(9,25)(10,18)(12,27)(13,31)(14,32)(15,28)(16,30)(19,26)(20,33)(21,34)(23,35)(24,29);(2,3)(4,5)(6,16,18,27)(7,20)(8,12)(9,24)(10,15)(11,17,28,34)(13,21)(14,26)(19,31,25,33)(22,32,35,29)(23,30)Polycyclic presentation:
2^7,3|a^2=e,c^2=ef,d^2=f,f^2=g,b^a=bd,c^a=cf,d^a=df,f^a=fg,h^a=h^2,d^b=dfg,e^b=eg,f^b=fg,h^b=h^2,d^c=dg45 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2180, C2×#69090, #10948634, … |
| maximal normal subgroup of | D3×#2180, C2×#69090, #10948634, … |
| maximal quotient of | #10707062, #10707068, #10707140, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1666);