Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #2229 |
124 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3)(12,14,15,23)(13,19,18,33)(16,29,26,43)(17,32,27,25)(20,38,34,42)(21,39,31,36)(22,41,40,28)(24,37,35,30);(2,3)(4,6,7,11)(5,8,9,10)(13,18)(19,33)(20,34)(21,31)(22,40)(28,41)(36,39)(38,42);(4,5,7,9)(6,10,11,8)(12,13,15,18)(14,22,23,40)(16,28,26,41)(17,31,27,21)(19,35,33,24)(20,37,34,30)(25,42,32,38)(29,39,43,36)Polycyclic presentation:
2^7,3|a^2=d,b^2=dg,c^2=g,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,h^b=h^2,e^c=ef,f^c=fg51 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2229, C2×#69212, #10949828, … |
| maximal normal subgroup of | D3×#2229, C2×#69212, #10949828, … |
| maximal quotient of | #10711241, #10711784, #10711802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1788);