Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C3×#356 ⋊ C2 |
| Sylow | C3 ⋊ #1405 |
106 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 23 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:
(1,3)(2,6)(4,7)(5,9)(8,15)(10,11)(12,14)(13,16)(18,19);(1,2,4,8,6,3,10,12)(5,9,15,7,16,13,14,11)(17,18,19)Polycyclic presentation:
2^7,3|a^2=d,c^2=g,d^2=g,e^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,c^b=cg,d^b=de,e^b=eg,f^b=fg,h^b=h^2,d^c=dfg,e^c=eg,e^d=eg30 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1405, C2×#67550, #10715452, … |
| maximal normal subgroup of | D3×#1405, C2×#67550, #10715452, … |
| maximal quotient of | #10682548, #10682615, #10682714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 126);