Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3080 |
384 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 141 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)(12,15,16,17)(13,20,18,21)(14,23,24,25)(19,22,26,27);(12,14)(13,19)(15,23)(16,24)(17,25)(18,26)(20,22)(21,27);(2,3)(4,6,7,11)(5,8,9,10)(13,18)(19,26)(20,21)(22,27);(4,5,7,9)(6,10,11,8)(12,13,16,18)(14,22,24,27)(15,21,17,20)(19,25,26,23)Polycyclic presentation:
2^7,3|a^2=e,b^2=eg,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=dg,f^a=fg,h^b=h^2,f^c=fg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10836681, #10836683, #10836689, … |
| maximal normal subgroup of | #10836681, #10836683, #10836689, … |
| maximal quotient of | #10794687, #10796160, #10796166, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11447);