Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ C8×C4⋊C4 |
296 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 165 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)(12,14,15,19,16,13,17,18);(2,3)(4,5,6,7)(9,11);(2,3)(8,9,10,11)(12,13,15,18,16,14,17,19)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=de,e^2=g,b^a=bd,h^a=h^2,h^b=h^2,h^c=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #10685948, #10685954, #10685955, … |
| maximal normal subgroup of | #10685948, #10685954, #10685955, … |
| maximal quotient of | C3⋊C8×C4⋊C8, #10686319, #10686336, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 3041);