Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#343 ⋊ C2 |
| Sylow | C3 ⋊ #1815 |
264 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 85 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)(8,11)(9,14)(10,18)(12,22)(13,16)(15,21)(17,20)(19,23);(2,3)(8,10,12,19,14,21,13,20)(9,15,16,17,11,18,22,23);(4,5,6,7)(8,9,13,16,14,11,12,22)(10,17,20,18,21,23,19,15)Polycyclic presentation:
2^7,3|a^2=f,b^2=d,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,h^b=h^2,e^c=eg,f^c=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1815, C2×#69000, #10794872, … |
| maximal normal subgroup of | D3×#1815, C2×#69000, #10794872, … |
| maximal quotient of | #10688060, #10688075, #10688158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1576);