Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1719 |
186 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 57 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)(5,10)(8,11)(12,14,16,22,15,21,23,27)(13,18,19,20,17,25,24,26);(2,3)(4,5,9,11,7,10,6,8)(13,17)(18,25)(19,24)(20,26);(5,8)(6,9)(10,11)(12,13)(14,20)(15,17)(16,24)(18,22)(19,23)(21,26)(25,27)Polycyclic presentation:
2^7,3|b^2=dfg,c^2=e,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,c^b=cf,h^b=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1719, C2×#71261, #10782955, … |
| maximal normal subgroup of | D3×#1719, C2×#71261, #10782955, … |
| maximal quotient of | #10688980, #10694957, #10694982, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3837);