Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#335 ⋊ C2 |
| Sylow | C3 ⋊ #1568 |
168 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 168 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) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(5,8)(9,11)(12,15,17,16)(13,19,20,25)(14,22,24,23)(18,21,26,27);(2,3)(12,14,16,23,17,24,15,22)(13,18,19,21,20,26,25,27);(4,5,6,9,7,8,10,11)(12,13)(14,21)(15,19)(16,25)(17,20)(18,23)(22,26)(24,27)Polycyclic presentation:
2^7,3|a^2=e,b^2=d,c^2=f,d^2=f,e^2=g,b^a=bd,c^a=cg,d^a=df,h^b=h^2,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10705220, #10705222, #10705230, … |
| maximal normal subgroup of | #10705220, #10705222, #10705230, … |
| maximal quotient of | #10684654, #10686375, #10686382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1146);