Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C3×#102 ⋊ C2 |
| Sylow | C3 ⋊ #1907 |
332 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2)(8,11)(9,12)(10,16)(13,15)(14,19)(17,20)(18,22)(21,23);(2,3)(4,6,7,5)(8,10,17,18,12,19,13,21)(9,14,15,23,11,16,20,22);(2,3)(4,5,7,6)(8,9,13,15,12,11,17,20)(10,16,18,23,19,14,21,22)Polycyclic presentation:
2^7,3|a^2=f,b^2=df,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,h^a=h^2,d^b=dg,e^b=eg,f^b=fg,h^b=h^2,d^c=dg,e^c=eg,f^c=fg,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1907, C2×#70218, #10797122, … |
| maximal normal subgroup of | D3×#1907, C2×#70218, #10797122, … |
| maximal quotient of | #10690657, #10690671, #10691013, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2794);