Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1511 |
274 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 73 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) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(13,18)(17,25)(19,26)(20,27);(2,3)(4,6,7,11)(5,8,9,10)(12,14)(13,17)(15,23)(16,22)(18,25)(19,27)(20,26)(21,24);(2,3)(4,5)(6,10,11,8)(7,9)(12,13)(14,20,22,27)(15,19)(16,18)(17,21,25,23)(24,26)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,d^2=g,b^a=bd,c^a=cg,d^a=df,h^a=h^2,d^b=dg,e^b=efg,h^b=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10687519, #10687522, #10687684, … |
| maximal normal subgroup of | #10687519, #10687522, #10687684, … |
| maximal quotient of | #10683268, #10683275, #10683298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2557);