Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×D4⋊C8 ⋊ C2 |
| Sylow | C3 ⋊ #1494 |
230 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 67 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)(4,7,9,8)(5,10,6,11)(13,18)(14,20)(19,25)(21,26);(1,2,3)(4,6)(5,9)(7,10)(8,11)(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(4,5,8,10,9,6,7,11)(12,13,15,19,16,18,22,25)(14,17,21,23,20,24,26,27)Polycyclic presentation:
2^7,3|a^2=e,c^2=f,d^2=f,e^2=g,b^a=bd,c^a=cg,d^a=df,c^b=cg,d^b=df,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1494, C2×#68177, #10761796, … |
| maximal normal subgroup of | D3×#1494, C2×#68177, #10761796, … |
| maximal quotient of | #10683516, #10683521, #10683526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 753);