Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #1530 |
256 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 81 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) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,8)(9,11)(12,14,16,15)(13,17,18,19);(5,8)(6,10)(12,13)(14,17)(15,19)(16,18);(2,3)(4,5,6,9,7,8,10,11)(13,17,18,19)Polycyclic presentation:
2^7,3|a^2=e,c^2=g,d^2=g,e^2=f,b^a=bd,c^a=cf,d^a=df,h^a=h^2,d^b=dg,e^b=efg60 conjugacy classes — view the full character table.
| maximal subgroup of | #10687069, #10687072, #10687090, … |
| maximal normal subgroup of | #10687069, #10687072, #10687090, … |
| maximal quotient of | #10683199, #10683243, #10683256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4374);