Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #1873 |
360 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 109 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 — 3 generators on 27 points:
(1,2,3)(9,11)(12,15,17,16)(13,19,21,20)(14,23,25,24)(18,26,27,22);(4,6,7,5)(12,14)(13,18)(15,23)(16,24)(17,25)(19,26)(20,22)(21,27);(2,3)(4,5,7,6)(8,9)(10,11)(12,13,16,20,17,21,15,19)(14,22,23,18,25,26,24,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=df,c^2=g,d^2=g,f^2=g,b^a=bd,c^a=ce,h^a=h^2,d^b=dg,f^b=fg78 conjugacy classes — view the full character table.
| maximal subgroup of | #10687522, #10687523, D3×#1873, … |
| maximal normal subgroup of | #10687522, #10687523, D3×#1873, … |
| maximal quotient of | #10689713, #10689728, #10689927, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4161);