Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2208 |
166 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(9,13)(10,17)(11,18)(14,15)(16,22)(19,23)(20,21);(2,3)(4,5,6,7);(5,7)(8,9,12,16)(10,13,19,22)(11,20,18,15)(14,23,21,17)Polycyclic presentation:
2^7,3|a^2=g,b^2=d,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,h^b=h^2,e^c=ef,f^c=fg51 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2208, C2×#71476, #10949450, … |
| maximal normal subgroup of | D3×#2208, C2×#71476, #10949450, … |
| maximal quotient of | #10709329, #10709333, #10709367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4052);