Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #1763 |
324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(8,10,11,9)(12,15,17,16)(13,14,19,18);(2,3)(4,5,6,7)(12,14,16,19,17,18,15,13);(2,3)(8,9,11,10)(12,13)(14,15)(16,18)(17,19)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=eg,d^2=g,f^2=g,b^a=bd,d^a=dg,f^a=fg,h^a=h^2,c^b=cg,h^b=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10687077, #10687079, #10687091, … |
| maximal normal subgroup of | #10687077, #10687079, #10687091, … |
| maximal quotient of | #10685797, #10685807, #10685832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2051);