Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×C4×C4⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ #1509 |
242 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 242 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) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(5,9)(8,11)(12,14,16,15)(13,17,18,19);(2,3)(4,5)(6,11)(7,9)(8,10)(12,13)(14,17)(15,19)(16,18);(5,8)(6,10)(9,11)(13,17,18,19)Polycyclic presentation:
2^7,3|a^2=e,c^2=fg,d^2=g,b^a=bd,d^a=df,c^b=cf,d^b=dg,e^b=efg,h^b=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10687067, #10687072, #10687074, … |
| maximal normal subgroup of | #10687067, #10687072, #10687074, … |
| maximal quotient of | #10683179, #10683192, #10683227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2046);