Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1722 |
198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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)(4,5,9,11,7,10,6,8)(14,21)(17,26)(20,27)(22,25);(2,3)(12,14,15,22,16,21,24,25)(13,17,18,20,19,26,23,27);(2,3)(5,8)(6,9)(10,11)(12,13,16,19)(14,20,21,27)(15,23,24,18)(17,25,26,22)Polycyclic presentation:
2^7,3|a^2=f,b^2=d,c^2=eg,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2,c^b=cf,h^b=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1722, C2×#71195, #10782976, … |
| maximal normal subgroup of | D3×#1722, C2×#71195, #10782976, … |
| maximal quotient of | #10688978, #10694986, #10694989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3771);