Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2176 |
148 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 148 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) = 384 |
A minimal faithful permutation representation — 3 generators on 39 points:
(2,3)(4,6,7,5)(9,15)(14,29)(16,31)(17,27)(18,35)(22,34)(30,33)(32,38);(1,2,3)(8,10,13,21)(9,14,15,29)(11,23,26,39)(12,28,24,20)(16,32,31,38)(17,33,27,30)(18,34,35,22)(19,36,37,25);(4,5,7,6)(8,9)(10,18)(11,22)(12,27)(13,15)(14,19)(16,25)(17,24)(20,38)(21,35)(23,33)(26,34)(28,32)(29,37)(30,39)(31,36)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,c^2=e,d^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fg,d^b=df,f^b=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2176, C2×#69209, #10948492, … |
| maximal normal subgroup of | D3×#2176, C2×#69209, #10948492, … |
| maximal quotient of | #10706930, #10707128, #10707134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1785);