Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3189 |
416 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 137 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 — 4 generators on 27 points:
(1,2,3)(5,9)(8,10);(6,11)(8,10)(12,14,16,22)(13,17,19,26)(15,24,25,21)(18,27,23,20);(4,6)(5,8)(7,11)(9,10);(2,3)(4,5)(6,10)(7,9)(8,11)(12,13,16,19)(14,20,22,27)(15,23,25,18)(17,24,26,21)Polycyclic presentation:
2^7,3|a^2=g,c^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^2,c^b=ce,f^c=fg60 conjugacy classes — view the full character table.
| maximal subgroup of | #10691750, #10691751, #10691753, … |
| maximal normal subgroup of | #10691750, #10691751, #10691753, … |
| maximal quotient of | #10804195, #10804199, #10805455, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15317);