Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1998 |
228 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 83 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:
(2,3)(4,6,7,11)(5,8,9,10)(13,18)(17,24)(19,26)(25,27);(1,2,3)(12,14,20,22,16,21,15,23)(13,17,19,27,18,24,26,25);(4,5,7,9)(6,10,11,8)(12,13,15,19,16,18,20,26)(14,17,22,25,21,24,23,27)Polycyclic presentation:
2^7,3|a^2=f,b^2=deg,c^2=efg,e^2=g,f^2=g,b^a=bd,c^a=cef,d^a=dg,d^b=dg,e^b=eg,f^b=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1998, C2×#68975, #10799854, … |
| maximal normal subgroup of | D3×#1998, C2×#68975, #10799854, … |
| maximal quotient of | #10695232, #10695234, #10695237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1551);