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 ⋊ #1920 |
230 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 97 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) = 384 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(8,10,11,14)(9,13,12,15)(17,20)(18,23)(21,22);(4,6,7,5)(9,12)(13,15);(2,3)(4,5,7,6)(8,9)(10,13)(11,12)(14,15)(16,17,19,22)(18,20,23,21)Polycyclic presentation:
2^7,3|a^2=f,b^2=fg,c^2=d,e^2=g,b^a=bd,c^a=ce,e^a=eg,h^a=h^2,e^c=eg78 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1920, C2×#69460, #10797343, … |
| maximal normal subgroup of | D3×#1920, C2×#69460, #10797343, … |
| maximal quotient of | #10693318, #10693346, #10700843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2036);