Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #1886 |
324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 324 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) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(8,11,13,15)(9,10,18,22)(12,19,14,23)(16,21,17,20);(4,6,7,5)(8,10)(9,15)(11,18)(12,21)(13,22)(14,20)(16,23)(17,19);(2,3)(4,5,7,6)(8,9,14,17,13,18,12,16)(10,19,21,15,22,23,20,11)Polycyclic presentation:
2^7,3|a^2=f,b^2=df,c^2=g,d^2=g,f^2=g,b^a=bd,c^a=ce,e^a=eg,h^a=h^2,d^b=dg,e^b=eg,f^b=fg,d^c=dg,f^c=fg,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1886, C2×#68765, #10796763, … |
| maximal normal subgroup of | D3×#1886, C2×#68765, #10796763, … |
| maximal quotient of | #10689836, #10689856, #10690108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1341);