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 ⋊ #1836 |
224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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)(9,13)(12,15)(16,18,19,21,20,22,23,17);(2,3)(8,10,11,14)(9,12,13,15)(16,17,19,18,20,21,23,22);(4,5,6,7)(8,9)(10,12)(11,13)(14,15)(16,17,20,21)(18,19,22,23)Polycyclic presentation:
2^7,3|a^2=f,b^2=e,c^2=d,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cg,h^b=h^2,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10689548, #10689551, #10689555, … |
| maximal normal subgroup of | #10689548, #10689551, #10689555, … |
| maximal quotient of | #10688853, #10688863, #10688884, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2124);