Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1609 |
216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(4,6,8,7)(5,9,10,11)(12,15)(13,18)(14,20)(16,22)(17,23)(19,25)(21,27)(24,26);(4,5)(6,9)(7,11)(8,10)(12,14)(13,17)(15,20)(16,21)(18,23)(19,24)(22,27)(25,26);(2,3)(5,9)(6,7)(10,11)(12,13,16,19)(14,17,21,24)(15,18,22,25)(20,26,27,23)Polycyclic presentation:
2^7,3|a^2=g,c^2=f,d^2=f,b^a=bd,c^a=ce,d^a=df,h^a=h^2,d^b=df,e^b=eg,d^c=dg,h^c=h^242 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1609, C2×#68296, #10767345, … |
| maximal normal subgroup of | D3×#1609, C2×#68296, #10767345, … |
| maximal quotient of | #10684224, #10684233, #10684252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 872);