Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #3230 |
360 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 141 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 — 4 generators on 27 points:
(1,2,3)(5,10)(9,11)(13,19)(17,20)(18,27)(25,26);(2,3)(5,9)(6,8)(10,11)(12,15,16,24)(13,18,19,27)(14,21,22,23)(17,25,20,26);(4,6,7,8)(5,9,10,11)(12,14,16,22)(13,17,19,20)(15,23,24,21)(18,26,27,25);(2,3)(4,5,7,10)(6,11,8,9)(12,13)(14,20)(15,18)(16,19)(17,22)(21,26)(23,25)(24,27)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,c^2=eg,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^2,c^b=ce,f^c=fg,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10691710, #10691712, #10691836, … |
| maximal normal subgroup of | #10691710, #10691712, #10691836, … |
| maximal quotient of | #10804676, #10804867, #10805709, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11848);