Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #2228 |
136 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 49 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:
(1,2,3)(13,18)(15,23)(16,22)(17,25)(19,20)(21,27)(24,26);(2,3)(4,6,7,11)(5,8,9,10)(13,17)(18,25)(19,26)(20,24);(2,3)(4,5,7,9)(6,10,11,8)(12,13,14,17)(15,18,21,25)(16,24,22,20)(19,27,26,23)Polycyclic presentation:
2^7,3|a^2=d,b^2=dg,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,h^a=h^2,h^b=h^2,e^c=ef,f^c=fg51 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2228, C2×#69181, #10949829, … |
| maximal normal subgroup of | D3×#2228, C2×#69181, #10949829, … |
| maximal quotient of | #10711236, #10711777, #10711795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1757);