Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2064 |
208 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 65 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)(5,7)(8,10,12,20)(9,14,17,22)(11,21,18,19)(13,16,23,15);(2,3)(4,5)(6,7)(8,10,18,21,12,20,11,19)(9,13,15,14,17,23,16,22);(2,3)(8,9,11,15,12,17,18,16)(10,14,19,23,20,22,21,13)Polycyclic presentation:
2^7,3|a^2=d,b^2=de,c^2=g,d^2=g,f^2=g,b^a=bde,c^a=ce,e^a=eg,f^a=fg,h^a=h^2,c^b=cf,h^b=h^2,d^c=dg,f^c=fg,h^c=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2064, C2×#68999, #10823731, … |
| maximal normal subgroup of | D3×#2064, C2×#68999, #10823731, … |
| maximal quotient of | #10699712, #10699726, #10699740, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1575);