Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1759 |
320 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 117 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)(4,6,7,5)(10,17)(13,21)(16,22)(18,23);(8,10)(9,13,15,21)(11,18)(12,17)(14,16,19,22)(20,23);(2,3)(4,5,7,6)(8,9)(10,16)(11,19)(12,15)(13,18)(14,20)(17,22)(21,23)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=e,d^2=g,b^a=bd,d^a=dg,f^a=fg,h^a=h^2,c^b=cg,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1759, C2×#70354, C4×#4821, … |
| maximal normal subgroup of | D3×#1759, C2×#70354, C4×#4821, … |
| maximal quotient of | #10685791, #10685796, #10685811, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2930);