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 ⋊ C4×C4⋊C8 |
316 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 161 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:
(2,3)(4,6,7,5);(1,2,3)(9,11)(12,13,14,16,15,17,18,19);(4,5,7,6)(8,9,10,11)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=de,f^2=g,b^a=bd,h^c=h^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #10683123, #10683125, #10683131, … |
| maximal normal subgroup of | #10683123, #10683125, #10683131, … |
| maximal quotient of | #10686307, #10686311, #10686320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1178);