Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×Q8⋊C8 ⋊ C2 |
| Sylow | C3 ⋊ #1479 |
248 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 248 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) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(2,3)(12,15,17,16)(13,19,20,25)(14,22,24,23)(18,21,26,27);(1,2,3)(12,14,17,24)(13,18,20,26)(15,23,16,22)(19,27,25,21);(4,5,6,8,7,9,10,11)(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=f,c^2=f,d^2=f,e^2=g,b^a=bd,d^a=df,c^b=cf,d^b=df,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1479, C2×#68231, #10761558, … |
| maximal normal subgroup of | D3×#1479, C2×#68231, #10761558, … |
| maximal quotient of | #10683174, #10683183, #10683211, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 807);