Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #1672 |
150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 150 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)(12,14,16,22)(13,17,19,26)(15,24,25,21)(18,27,23,20);(4,5,6,8,7,9,10,11);(2,3)(5,8)(6,10)(9,11)(12,13,16,19)(14,20,22,27)(15,23,25,18)(17,24,26,21)Polycyclic presentation:
2^7,3|a^2=g,b^2=d,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,e^c=eg,h^c=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1672, C2×#68547, #10777551, … |
| maximal normal subgroup of | D3×#1672, C2×#68547, #10777551, … |
| maximal quotient of | #10688237, #10691332, #10691346, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1123);