Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2061 |
230 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(4,7,8,6)(5,9,10,11)(13,18)(16,24)(17,26)(19,20)(21,27)(23,25);(2,3)(4,6,8,7)(5,9,10,11)(12,14,15,22)(13,17,19,21)(16,25,24,23)(18,27,20,26);(4,5,8,10)(6,11,7,9)(12,13,15,19)(14,21,22,17)(16,18,24,20)(23,27,25,26)Polycyclic presentation:
2^7,3|a^2=d,b^2=d,c^2=dg,e^2=g,b^a=bd,c^a=ce,e^a=eg,f^a=fg,c^b=cf,e^b=eg,h^b=h^2,e^c=eg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2061, C2×#68736, #10823296, … |
| maximal normal subgroup of | D3×#2061, C2×#68736, #10823296, … |
| maximal quotient of | #10699594, #10699599, #10699609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1312);