Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4451 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1272 |
166 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 27 points:
(2,3)(12,14)(13,16)(15,19)(17,18)(20,21,23,25)(22,26,27,24);(1,2,3)(4,5,6,8,7,9,10,11)(12,13)(14,18,19,16)(15,17)(21,24,25,26)Polycyclic presentation:
2^7,3|a^2=d,b^2=f,c^2=f,d^2=g,b^a=bc,c^a=ce,c^b=cf,d^b=def,h^b=h^278 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1272, C2×#67480, #10683132, … |
| maximal normal subgroup of | D3×#1272, C2×#67480, #10683132, … |
| maximal quotient of | #10681739, #10681998, #10682000, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 56);