Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #1404 |
112 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 25 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) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2)(3,10)(4,13)(5,9)(6,12)(7,11)(8,15)(14,16)(17,18,19);(1,2,3,7)(4,13,14,10)(5,9,12,16)(6,15,8,11)(18,19)Polycyclic presentation:
2^7,3|a^2=d,c^2=g,e^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,h^a=h^2,c^b=cg,d^b=de,e^b=eg,f^b=fg,d^c=dfg,e^c=eg,e^d=eg33 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1404, C2×#67771, #10715436, … |
| maximal normal subgroup of | D3×#1404, C2×#67771, #10715436, … |
| maximal quotient of | #10682549, #10682616, #10682712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 347);