Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×C3×#107 ⋊ C2 |
| Sylow | C3 ⋊ #2139 |
232 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 232 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:
(1,2,3)(9,14)(13,21)(15,20)(17,22);(2,3)(8,9)(10,17)(11,20)(12,14)(13,16)(15,18)(19,22)(21,23);(4,5,6,7)(9,13)(10,16)(11,18)(14,21)(15,22)(17,20)(19,23)Polycyclic presentation:
2^7,3|a^2=e,d^2=fg,f^2=g,b^a=bd,d^a=df,f^a=fg,c^b=cg,d^b=df,f^b=fg,h^b=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10710479, #10710482, #10710861, … |
| maximal normal subgroup of | #10710479, #10710482, #10710861, … |
| maximal quotient of | #10704930, #10705174, #10705226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3174);