Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2241 |
140 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 140 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) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(5,9)(8,11)(12,15,16,24,26,21,17,22,19,25,20,14,18,23,27,13);(2,3)(4,6,7,10)(5,8,9,11)(12,14,17,15,18,22,16,23,19,24,27,25,26,13,20,21);(4,5)(6,8)(7,9)(10,11)(12,13,19,22)(14,18,24,26)(15,17,25,27)(16,23,20,21)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,c^2=d,d^2=fg,e^2=f,f^2=g,b^a=bd,c^a=ce,d^a=df,e^a=efg,f^a=fg,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2241, C2×#69215, #10950485, … |
| maximal normal subgroup of | D3×#2241, C2×#69215, #10950485, … |
| maximal quotient of | #10711565, #10713107, #10713108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1791);