Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting | C3×#357 ⋊ C2 |
| Sylow | C3 ⋊ #2234 |
142 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 45 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)(4,6,7,10)(5,9,8,11)(12,14,15,21)(13,18,17,25)(16,24,22,26)(19,27,23,20);(2,3)(5,8)(9,11)(12,13,15,17)(14,18,21,25)(16,19,22,23)(20,24,27,26);(4,5)(6,9)(7,8)(10,11)(13,17)(14,20)(16,23)(18,26)(19,22)(21,27)(24,25)Polycyclic presentation:
2^7,3|b^2=g,c^2=d,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,h^b=h^2,e^c=ef,f^c=fg,h^c=h^251 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2234, C2×#69182, #10949983, … |
| maximal normal subgroup of | D3×#2234, C2×#69182, #10949983, … |
| maximal quotient of | #10711483, #10711698, #10712127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1758);