Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#519 ⋊ C2 |
| Sylow | C3 ⋊ #3126 |
382 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 125 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) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(4,6,7,10)(5,9,8,11)(13,17)(18,24)(19,25)(26,27);(12,14,15,21)(13,18,17,24)(16,23,22,20)(19,26,25,27);(5,8)(9,11)(12,13)(14,18)(15,17)(16,19)(20,27)(21,24)(22,25)(23,26);(2,3)(4,5)(6,9)(7,8)(10,11)(13,17)(14,20)(16,22)(18,26)(21,23)(24,27)Polycyclic presentation:
2^7,3|c^2=g,d^2=eg,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,d^b=dg,f^c=fg66 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#3126, C2×#82553, #11663559, … |
| maximal normal subgroup of | D3×#3126, C2×#82553, #11663559, … |
| maximal quotient of | #10795287, #10798308, #10798345, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15129);