Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1653 |
194 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 194 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) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(12,14,17,25)(13,19,21,27)(15,24,16,23)(18,26,20,22);(2,3)(4,5,7,9)(6,11,10,8)(13,18)(15,16)(19,26)(20,21)(22,27)(23,24);(2,3)(5,8)(6,10)(9,11)(12,13,17,21)(14,22,25,26)(15,18,16,20)(19,23,27,24)Polycyclic presentation:
2^7,3|a^2=g,b^2=fg,c^2=g,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,h^a=h^2,d^b=df,e^b=eg,h^b=h^2,d^c=dg,e^c=eg54 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1653, C2×#68275, #10771027, … |
| maximal normal subgroup of | D3×#1653, C2×#68275, #10771027, … |
| maximal quotient of | #10684031, #10684037, #10684079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 851);