Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #2207 |
174 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 55 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:
(2,3)(8,9,12,15)(10,17,19,23)(11,20,18,14)(13,21,22,16);(1,2,3)(4,5,6,7);(5,7)(9,13)(10,16)(11,18)(14,23)(15,22)(17,20)(19,21)Polycyclic presentation:
2^7,3|b^2=d,c^2=g,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,e^c=ef,f^c=fg,h^c=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | #10708615, #10708616, #10708622, … |
| maximal normal subgroup of | #10708615, #10708616, #10708622, … |
| maximal quotient of | #10709307, #10709361, #10709480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3870);