Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2063 |
228 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,3)(5,7)(9,15)(10,18)(11,12)(14,22)(17,23)(19,20);(2,3)(4,5)(6,7)(8,10,11,18,13,20,12,19)(9,14,21,17,15,23,16,22);(8,9,11,16,13,15,12,21)(10,17,18,14,20,22,19,23)Polycyclic presentation:
2^7,3|a^2=d,b^2=de,d^2=g,f^2=g,b^a=bde,c^a=ce,e^a=eg,f^a=fg,c^b=cf,h^b=h^2,d^c=dg,f^c=fg48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2063, C2×#70075, #10823731, … |
| maximal normal subgroup of | D3×#2063, C2×#70075, #10823731, … |
| maximal quotient of | #10699700, #10699714, #10699728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2651);