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×#101 ⋊ C2 |
| Sylow | C3 ⋊ #1908 |
312 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 312 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) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(2,3)(8,11)(9,12)(10,16)(13,15)(14,18)(17,22)(19,20)(21,23);(1,2,3)(4,6,7,5)(8,10,13,21,12,18,19,17)(9,14,20,22,11,16,15,23);(4,5,7,6)(8,9,13,15,12,11,19,20)(10,16,17,23,18,14,21,22)Polycyclic presentation:
2^7,3|a^2=f,b^2=dfg,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=eg,f^c=fg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1908, C2×#70123, #10797123, … |
| maximal normal subgroup of | D3×#1908, C2×#70123, #10797123, … |
| maximal quotient of | #10690658, #10690672, #10691021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 2699);