Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #2028 |
246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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)(12,14,16,22)(13,17,19,26)(15,25,24,21)(18,27,23,20);(2,3)(4,6,7,11)(5,8,9,10)(15,24)(17,26)(18,23)(21,25);(4,5,7,9)(6,10,11,8)(12,13)(14,20)(15,23)(16,19)(17,21)(18,24)(22,27)(25,26)Polycyclic presentation:
2^7,3|a^2=d,b^2=d,c^2=g,e^2=g,b^a=bd,c^a=ce,e^a=eg,f^a=fg,c^b=cf,e^b=eg,h^b=h^2,e^c=eg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2028, C2×#68769, #10822555, … |
| maximal normal subgroup of | D3×#2028, C2×#68769, #10822555, … |
| maximal quotient of | #10698161, #10698166, #10698181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1345);