Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1813 |
252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,7)(5,12)(6,16)(8,21)(9,24)(10,22)(11,25)(13,27)(14,20)(15,29)(17,32)(18,34)(19,33)(23,28)(26,35)(30,31);(2,3)(4,6,8,17,10,19,9,18)(5,11,13,15,14,26,23,30)(7,16,21,32,22,33,24,34)(12,25,27,29,20,35,28,31);(4,5,9,13,10,14,8,23)(6,15,18,26,19,30,17,11)(7,20,21,27,22,12,24,28)(16,31,32,35,33,29,34,25)Polycyclic presentation:
2^7,3|a^2=f,b^2=d,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,h^b=h^2,f^c=fg54 conjugacy classes — view the full character table.
| maximal subgroup of | #10711200, #10711201, #10711205, … |
| maximal normal subgroup of | #10711200, #10711201, #10711205, … |
| maximal quotient of | #10688046, #10688050, #10688054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2710);